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Conferences

2010-11-10T13:52:33Z

Noe: new conference icsc 2011

The following is a chronological list of conferences, seminars, or meetings related to thermodynamics, statistical mechanics, soft condensed matter, many-body problems, complex fluids etc.
===2010===
====December====
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP10)] 14-16 December 2010, Warwick University (United Kingdom)
===2011===
====January====
*[https://sites.google.com/site/wintermeetingstatphys/ XL Winter Meeting on Statistical Physics] January 4-7, 2011, Taxco, Guerrero (Mexico)
====March====
*[http://nithep.ac.za/2fq.htm Equilibration and Equilibrium: 2nd Stellenbosch Workshop on Statistical Physics] 07-18 March 2011, Stellenbosch (South Africa)
====August====
*[http://19ectp.cheng.auth.gr 19th European Conference on Thermophysical Properties] August 28 - September 1 2011 Thessaloniki (Greece)
*[http://www.thermodynamics2011.org/ Thermodynamics 2011] August 31-September 2, 2011, Athens (Greece)
*[http://www.icsc2011.fr/ 32nd International Conference on Solution Chemistry -ICSC 2011- ] La Grande Motte, France, 27th August - 2nd September 2011
====September====
*[http://lmc2011.univie.ac.at 8th Liquid Matter Conference] September 6-11, 2011 Wien (Austria)
*[http://www.thphys.uni-heidelberg.de/~nowak/FINESS2011/index.html Finite-Temperature Non-Equilibrium Superfluid Systems] 18 to 21 September (2011) Heidelberg (Germany)

=Previous conferences=
Note: over time it is natural that more and more of these links will become broken.
=====2007=====
*[http://www.simbioma.cecam.org/ Simulation of Hard Bodies] April 16 to April 19 2007 in Lyon.
*[http://www.math.rutgers.edu/events/smm/ 97th Statistical Mechanics Conference] May 6-8, 2007 Rutgers University
*[http://www.ucm.es/info/molecsim/luis/workshop.htm Workshop on Theory and Computer Simulations of Inhomogenoeus Fluids] May 16-18, 2007, Universidad Complutense, Madrid.
*[http://www-thphys.physics.ox.ac.uk/users/JuliaYeomans/OxfordWorkshop/home.php Mesoscale Modelling for Complex Fluids and Flows] June 25--27, 2007 University of Oxford, UK.
*[http://www.ecns2007.org 4th European Conference on Neutron Scattering] Lund, Sweden 25-29 June 2007
*[http://www.realitygrid.org/CompSci07 Computational Science 2007] 25-26 June 2007 The Royal Society, London
*[https://www.cecam.fr/index.php?content=activities/workshop New directions in liquid state theory] CECAM workshop, 2 - 4 July, 2007 at ENS, Lyon, France.
*[http://www.cecam.fr/index.php?content=activities/workshop&action=details&wid=157 Fluid phase behaviour and critical phenomena from liquid state theories and simulations] 5-7 July 2007 CECAM workshop, Lyon, France.
*[http://www.statphys23.org/ STATPHYS 23] Genova, Italy, from July 9 to 13, 2007.
*[http://www.iupac2007.org/ 41st IUPAC World Chemistry Conference] Turin, Italy, August 5-11th
*[http://www.srcf.ucam.org/~jae1001/ccp5_2007 CCP5 Annual Conference] 29th-31st August 2007 New Hall, Cambridge, UK
*[http://ccp2007.ulb.ac.be CCP 2007] Brussels 5-8 September 2007
*[http://www.castep.org CASTEP Workshop] 17th - 21st September 2007 University of York, UK
*[http://thermo2007.ifp.fr Thermodynamics 2007] 26-28 September 2007, IFP - Rueil-Malmaison (France)
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2007] Corfu, Greece, 25-30 September 2007
*[http://www.chem.unisa.it/polnan/index.html Polymers in Nanotechnology] 27-28th September 2007, Salerno, Italy.
*[http://www.fz-juelich.de/iff/ismc2007/ International Soft Matter Conference 2007] 1 - 4 October 2007, Eurogress, Aachen (Germany)
*[http://www.escet.urjc.es/~fisica/encuentro_complejos/index.html II Meeting on Modelling of Complex Systems] Universidad Rey Juan Carlos, Mostoles (Madrid), October 25-26 2007
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/SCD/event_22663.html Structuring Colloidal Dispersions by External Fields] 21 November 2007, Institute of Physics, London, United Kingdom
*[http://www.math.rutgers.edu/events/smm/ 98th Statistical Mechanics Conference] December 16-18, 2007, Rutgers University, New Jersey, USA.

=====2008=====
*[http://www.iop.org/activity/groups/subject/lcf/Events/file_25833.pdf Recent Advances in the Understanding of Confined Fluids: from Superfluids to Oil Reservoirs] 9-11 January, Cosener's House, Abingdon UK
*[http://academic.sun.ac.za/summerschool/2008.html Workshop on Soft Condensed Matter and Physics of Biological Systems - Perspectives and topics for South Africa] 23 Jan 2008 - 1 Feb 2008 National Institute for Theoretical Physics at Stellenbosch Institute of Advanced Study, Stellenbosch, Western Cape, South Africa
*[http://bifi.unizar.es/events/bifi2008/main.htm Bifi 2008 Large Scale Simulations of Complex Systems, Condensed Matter and Fusion Plasma] 6–8 February 2008, Zaragoza, Spain
*[http://events.dechema.de/Tagungen/MolMod+Workshop.html International Workshop Molecular Modeling and Simulation in Applied Material Science] March 10-11, DECHEMA-Haus, Frankfurt am Main, Germany
*[http://www.aps.org/meetings/march/index.cfm APS March Meeting] March 10-14, 2008. New Orleans, Louisiana, USA
*[http://users.physik.tu-muenchen.de/metz/jerusalem.html Modelling anomalous diffusion and relaxation] 23–28 March 2008, Jerusalem, Israel
*[http://www.newton.cam.ac.uk/programmes/CSM/csmw02.html Markov-Chain Monte Carlo Methods] 25 March to 28 March 2008 Isaac Newton Institute for Mathematical Sciences, Cambridge, UK
*[http://www.usal.es/~fises/ XV Congreso de Física Estadística (Fises' 08)] 27-29 March 2008, Salamanca, Spain
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/CMMP08/event_26545.html Condensed Matter and Materials Physics (CMMP)] 26-28 March 2008 Royal Holloway, University of London, UK
*[http://www.icmab.es/softmatter2008/index.html SoftMatter 2008] "Workshop on Electrostatic Effects in Soft Matter: Bringing Experiments, Theory and Simulation Together" 10 – 11 April ICMAB-CSIC, Barcelona, Spain.
*[http://www.icmab.es/11emscf/index.html 11th European Meeting on Supercritical Fluids] "New Perspectives in Supercritical Fluids: Nanoscience, Materials and Processing" 4-7 May 2008, ICMAB-CSIC, Barcelona, Spain
*[http://www.math.rutgers.edu/events/smm/index.html 99th Statistical Mechanics Conference] May 11-13, 2008 Rutgers University, USA
*[http://www.fondation-pgg.org/events/degennesdays/ DeGennesDays] 15-17 May 2008 Collège de France, Paris.
*[http://www.ill.fr/Events/rktsymposium/ Surfaces and Interfaces in Soft Matter and Biology SISMB 2008] "The impact and future of neutron reflectivity - A Symposium in Honor of Robert K. Thomas" 21-23 May 2008, Institut Laue-Langevin (Grenoble, France).
*[http://www.plmmp.univ.kiev.ua/ Physics of Liquid Matter: Modern Problems] May 23-26, 2008, Kyiv National Taras Shevchenko University, Ukraine
*[http://www.ffn.ub.es/sitges/ XXI Sitges Conference] XXI Sitges Conference on Statistical Mechanics. Statistical Mechanics of Molecular Biophysics 2nd-6th June 2008, Sitges, Spain
*[http://www1.ci.uc.pt/gcpi/poly2008 Polyelectrolytes 2008] 16th-19th June 2008, Coimbra, Portugal
*[http://www-spht.cea.fr/Meetings/BegRohu2008/index.html The Beg Rohu Summer School: Manifolds in random media, random matrices and extreme value statistics] 16th - 28th June 2008, French National Sailing School, Quiberon peninsula, France.
*[http://www.chm.bris.ac.uk/cms/ Computational Molecular Science 2008] 22nd – 25th June 2008, Cirencester, UK.
*[http://www.liquids2008.se/ 7th Liquid Matter Conference] 27 June - 1 July 2008 Lund, Sweden
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School 2008] University of Sheffield, England July 6 - July 15 2008.
*Potentials Workshop: Recent developments in interatomic potentials 7-8 July 2008 Oxford, UK
*[http://www.lcc-toulouse.fr/molmat2008/ MOLMAT2008 International Symposium on Molecular Materials based on Chemistry, Solid State Physics, Theory and Nanotechnology] July 8-11th 2008 Toulouse, France
*[http://www2.polito.it/eventi/sigmaphi2008/ SigmaPhi2008] 14th -18th July 2008 Kolympari, Crete, Greece
*[http://www.cecam.org/workshop-188.html Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers] July, 16th-18th 2008 CECAM-EPFL, Lausanne, Switzerland
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/event_22243.html Molecular Dynamics for Non-Adiabatic Processes] 21 July 2008 to 22 July 2008 Institute of Physics, London
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcitaa_08/announcement.html Quantum Monte Carlo in the Apuan Alps IV] 26th July - Sat 2nd August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.cecam.org/workshop-215.html New directions in the theory and modelling of liquid crystals] July, 28th 2008 to July, 30th 2008, CECAM-EPFL, Lausanne, Switzerland
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcatcp_08/announcement.html Quantum Monte Carlo and the CASINO program III] 3rd August - Sun 10th August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.icct2008.org/ 20th International Conference on Chemical Thermodynamics] August 3-8, 2008, Warsaw, Poland.
*[http://perso.ens-lyon.fr/thierry.dauxois/LORIS/LesHouchesSummerSchool2008.html Long-Range Interacting Systems] Summer School in Les Houches (France), 4-29 August 2008
*[http://ctbp.ucsd.edu/summer_school08/apply2008.html Coarse-Grained Physical Modeling of Biological Systems: Advanced Theory and Methods] August 11-15, 2008 University of California San Diego
*[http://www.chem.ucl.ac.uk/astrosurf/SIPML.html Surface and Interface Processes at the Molecular Level] 17 - 23 August 2008, Lucca, Italy
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD141/index.asp Faraday Discussion 141: Water - From Interfaces to the Bulk] 27 - 29 August 2008 Heriot-Watt University, Edinburgh, United Kingdom
*[http://ectp18.conforganizer.net 18th European Conference on Thermophysical Properties] 31 Aug-4 Sep 2008 Pau, France
*[http://iber2008.df.fct.unl.pt/ 9th Iberian Joint Meeting on Atomic and Molecular Physics - IBER 2008] 7-9th September, Capuchos, Portugal
*[http://www.cmmp.ucl.ac.uk/~dmd/ccp5.htm CCP5 Annual Meeting: Surfaces and Interfaces] 8-10th September, London, UK
*[http://ergodic.ugr.es/cp/ 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials] September 15-19, 2008 Granada, Spain
*[http://www.cecam.org/workshop-222.html Standardisation and databasing of ab-initio and classical simulations] September, 18th 2008 to September, 19th 2008 CECAM-ETHZ, Zurich, Switzerland
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2008 (ICCMSE 2008)] 25-30 September, Crete, Greece
*[http://www.mmm2008.org/bin/view.pl/Main/WebHome Multscale Materials Modeling] 27–31 October 2008, Tallahassee, Florida USA
*[http://www.aiche.org/Conferences/AnnualMeeting/index.aspx 2008 AIChE Annual Meeting] November 16-21 2008 Philadelphia, Pennsylvania USA
*[http://www.ihp.jussieu.fr/ceb/Trimestres/T08-4/C3/index.html Statistical mechanics] Paris (France) 8-12 December 2008
*[http://complex.ffn.ub.es/bcnetworkshop BCNet Workshop] Trends and perspectives in complex networks. Barcelona (Spain) 10-12 December 2008
*[http://www.math.rutgers.edu/events/smm/ 100th Statistical Mechanics Conference] Rutgers University, (USA) 13-18 December 2008
=====2009=====
*[http://www.ccpb.ac.uk/events/conference/2009/ Biomolecular Simulation 2009] 6-8 January, Yorkshire Museum and Gardens, York, United Kingdom 2009
*[http://www.fisica.unam.mx/externos/wintermeeting/ XXXVIII edition of the Winter Meeting on Statistical Physics] Taxco, Guerrero (Mexico) 6th - 9th January, 2009.
*[http://www.ucl.ac.uk/msl/events/2009/workshop09.htm 2009 MSL Workshop: Accessing large length and time scales with accurate quantum methods] 12th - 13th January 2009 University College London (United Kingdom)
*[http://euler.us.es/%7Eopap/stochgame/index-en.html Stochastic Models in Physics, Biology, and Social Sciences] Carmona (Sevilla), Spain February 12-14 (2009)
*[http://hera.physik.uni-konstanz.de/igk/news/workshops/homepage/index.html Frontiers of Soft Condensed Matter 2009] Les Houches () 15-20 February 2009
*[http://www.mpipks-dresden.mpg.de/~mbsffe09/ Many-body systems far from equilibrium: Fluctuations, slow dynamics and long-range interactions] Max Planck Institute for the Physics of Complex Systems Dresden, Germany February 16 - 27 (2009)
*[http://www.formulation.org.uk/Conference_flyers_Sept2007_on/Flyer-sims.pdf Workshop on advances in modelling for formulations] 25th of March 2009 at GSK Waybridge (United Kingdom)
*[http://www.physik.uni-leipzig.de/~janke/meco34/ 34th Conference of the Middle European Cooperation in Statistical Physics] 30 March - 01 April 2009 Universität Leipzig (Germany)
*[http://www.ipam.ucla.edu/programs/ktws2/ Workshop II: The Boltzmann Equation: DiPerna-Lions Plus 20 Years] April 15-17 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-313.html Computational Studies of Defects in Nanoscale Carbon Materials] May 11, 2009 to May 13, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.cecam.org/workshop-320.html Modeling of Carbon and Inorganic Nanotubes and Nanostructures] May 13, 2009 to May 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ima.umn.edu/2008-2009/W5.18-22.09/ Molecular Simulations: Algorithms, Analysis, and Applications] Institute for Mathematics and its Applications, University of Minnesota (USA), May 18-22, 2009
*[http://www.ipam.ucla.edu/programs/ktws4/ Workshop IV: Asymptotic Methods for Dissipative Particle Systems] May 18-22 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-310.html Computer Simulation in Food Science: CFD meets Soft Matter] May 25, 2009 to May 27, 2009 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://complenet09.diit.unict.it CompleNet 2009] International Workshop on Complex Networks. Catania (Italy), May 26-28, 2009
*[http://go.warwick.ac.uk/maths/research/events/2008_2009/symposium/wks5/ EPSRC Symposium Workshop on Molecular Dynamics] Monday 1 – Friday 5 June (2009) Warwick (United Kingdom)
*[http://www.cecam.org/workshop-272.html Theoretical Modeling of Transport in Nanostructures] June 2, 2009 to June 5, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.mpip-mainz.mpg.de/mmsd/ Mainz Materials Simulation Days] 3-5 June 2009 Max Planck Institute for Polymer Research (Germany)
*[http://www.soms.ethz.ch/workshop2009 Coping with Crises in Complex Socio-Economic Systems] ETH Zurich (Switzerland), June 8-13, 2009
*[[MOSSNOHO Workshop 2009]] 16 of June 2009, Universidad Complutense de Madrid (Spain)
*[http://cftc.cii.fc.ul.pt/Workshops/flowNjam/index.html Flow(ers) and jam(mers): from liquid crystals to grains] Lisbon (Portugal) 17-19 June 2009
*[http://ipht.cea.fr/Meetings/BegRohu2009/ Beg Rohu Summer School: Quantum Physics Out of Equilibrium] Ecole Nationale de Voile (France) 15-27 June 2009
*[http://symp17.boulder.nist.gov Seventeenth Symposium on Thermophysical Properties] Boulder, Colorado (USA), June 21-26, 2009
*[http://www.fhi-berlin.mpg.de/th/Meetings/DFT-workshop-Berlin2009/ Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions] Berlin (Germany) June 22-July 1 2009
*[https://www.irphe.univ-mrs.fr/~fe09 Fluid and Elasticity] Carry-le-Rouet, near Marseilles (France) June 23-26, 2009
*[http://www.icmp.lviv.ua/statphys2009/ Statistical Physics: Modern Trends and Applications] June 23-25, 2009 Lviv, (Ukraine)
*[http://www.fuw.edu.pl/~wssph/ 3rd Warsaw School of Statistical Physics] Kazimierz Dolny (Poland), 27 June - 4 July, 2009
*[http://www.cecam.org/workshop-298.html Modeling and Simulation of Water at Interfaces from Ambient to Supercooled Conditions] June 29, 2009 to July 1, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School] University of Sheffield (England) July 5 - July 14 2009
*[http://math.arizona.edu/~goriely/leshouches/home.html New trends in the Physics and Mechanics of Biological Systems"] Les Houches (France), 6-31 July 2009
*[http://research.yale.edu/boulder/Boulder-2009/index.html Nonequilibrium Statistical Mechanics: Fundamental Problems and Applications] July 6-24 2009 Boulder (USA)
*[http://www.xrqtc.cat/index.php/ca/homew New trends in Computational Chemistry for Industry Applications] July 6-7 Barcelona (Spain)
*[http://www.frias.uni-freiburg.de/BFF Computational Methods for Soft Matter and Biological Systems] July 8-11, 2009, FRIAS, Freiburg (Germany)
*[http://www.cecam.org/workshop-286.html Structural Transitions in Solids: Theory, Simulations, Experiments and Visualization Techniques] July 8, 2009 to July 11, 2009 CECAM-USI, Lugano (Switzerland)
*[http://fomms.org FOMMS 2009, Fourth International Conference Foundations of Molecular Modeling and Simulation] Semiahmoo Resort, Blaine, WA (USA) 12-16 July 2009
*[http://www.cecam.org/workshop-308.html Computer Simulation Approaches to Study Self-Assembly: From Patchy Nano-Colloids to Virus Capsids] July 13, 2009 to July 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.md-net.org.uk/events/bath_jul_2009.htm EPSRC Network Mathematical Challenges of Molecular Dynamics] 13-15 July (2009) Bath, United Kingdom.
*[http://www.cecam.org/workshop-279.html New Trends in Simulating Colloids: from Models to Applications] July 15, 2009 to July 18, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD144/index.asp Faraday Discussion 144: Multiscale Modelling of Soft Matter] University of Groningen (The Netherlands) 20 - 22 July 2009
*[http://www2.yukawa.kyoto-u.ac.jp/~ykis2009/Welcome.html Frontiers in Nonequilibrium Physics: Fundamental Theory, Glassy & Granular Materials, and Computational Physics] July 21 - August 21, 2009 Kyoto (Japan)
*[http://www.cecam.org/workshop-293.html Fundamental Aspects of Deterministic Thermostats] July 27, 2009 to July 29, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program IV] 2nd-9th August 2009 Towler Institute, Tuscany, (Italy)
*[http://www.warwick.ac.uk/go/quantsim09 EPSRC Symposium Workshop on Quantum Simulations] 24th-28th August 2009 University of Warwick (United Kingdom)
*[http://itf.fys.kuleuven.be/~fpspXII/ Fundamental Problems in Statistical Physics XII] August 31 - September 11, 2009 Leuven (Belgium)
*[http://denali.phys.uniroma1.it/~idmrcs6/ 6th International Discussion Meeting on Relaxations in Complex Systems] August 31- September 5, 2009, Roma (Italy)
*[http://www.dft09.org/ International Conference on the Applications of Density Functional Theory in Chemistry and Physics] August 31st to September 4th 2009 Lyon (France)
*[http://www.dfrl.ucl.ac.uk/CCP5/ccp5.htm CCP5 Annual Meeting 2009 Structure Prediction] 7th to 9th, September 2009 London (United Kingdom)
*[http://fises.dfa.uhu.es/fises09/ XVI Congreso de Física Estadística] Huelva, 10-12 September 2009, Spain
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/ 22nd Marian Smoluchowski Symposium On Statistical Physics] 12-17 September 2009 Zakopane (Poland)
*[http://www.isis.rl.ac.uk/largescale/loq/SAS2009/SAS2009.htm SAS-2009] XIV International Conference on Small-Angle Scattering, Sunday 13 - Friday 18 September, 2009, Oxford (UK)
*[http://www.esc.sandia.gov/dsmc09/dsmc09.html Direct Simulation Monte Carlo workshop] September 13-16, 2009 Santa Fe, New Mexico, USA,
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/PHYSASPECTS_09/index.html Physical Aspects of Polymer Science] 14-16 September 2009 Bristol (United Kingdom)
*[http://www.thermodynamics2009.org/ Thermodynamics 2009] September 23-25 Imperial College London , U.K. (2009)
*[http://www.crm.cat/wkstatisticalphysics/ Techniques and Challenges from Statistical Physics] October 14 to 16, 2009 Barcelona, (Spain)
*[http://paginas.fe.up.pt/~equifase/ EQUIFASE 2009] VIII Iberoamerican Conference on Phase Equilibria and Fluid Properties for Process Design, Praia da Rocha, Portugal 17-21 October 2009
*[http://cint.lanl.gov/workshop2009/ Multiple Length Scales in Polymers and Complex Fluids] October 18-21, 2009 Bishop's Lodge, Santa Fe (USA)
*[http://www.cecam.org/workshop-290.html Classical Density Functional Theory Methods in Soft and Hard Matter] October 21, 2009 to October 23, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://seneca.fis.ucm.es/parr/ktlog2_09 kTlog2 '09: Computing Matters Workshop] 22-24 October, 2009, Toledo (Spain)
*[https://www.cecam.org/workshop-325.html Linking Systems Biology and Biomolecular Simulations] November 16-19 2009 CECAM Lausanne, (Switzerland)
*[http://www.lorentzcenter.nl/lc/web/2009/357/info.php3?wsid=357 Subdivide and Tile: Triangulating spaces for understanding the world] November 16 to 20 2009, Leiden (Netherlands)
*[http://www.dfi.uchile.cl/~granular09/Welcome.html Southern Workshop on Granular Materials 2009 - SWGM09] November 30 to December 4 2009, Viña del Mar (Chile)
*[http://www.math.rutgers.edu/events/smm/index.html 102nd Statistical Mechanics Conference] Rutgers University, December 13-15, 2009 (USA)
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP)] 15-17 December 2009 Warwick University (United Kingdom)
=====2010=====
*[http://ddays2010.northwestern.edu Dynamics Days] January 4-7, 2010 Evanston Illinois (USA)
*[http://www2.surrey.ac.uk/maths/news/events/2010/emerging_interfaces_workshop.htm Emerging Interfacial Dynamics] January 6-8 University of Surrey (UK)
*[http://www.comphys.ethz.ch/jstat/ Journées de Physique Statistique 2010] 28-29 January 2010 Paris (France)
*[http://www.theory.caltech.edu/~ooguri/CMP-HEP/CMP-HEP.htm Condensed Matter Physics Meets High Energy Physics] University of Tokyo, February 8-12, 2010 (Japan)
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/2010/CFFI10/index.html Complex Fluid-Fluid Interfaces] 25 February 2010 Institute of Physics, London (UK)
*[http://tfy.tkk.fi/soft/levi2010/ International Workshop on Coarse-Grained Biomolecular Modeling] 7-12 March, Levi, (Finland)
*[http://www.lpm.u-nancy.fr/webperso/chatelain.c/GrpPhysStat/Meco35/index.php 35th Conference of the Middle European Cooperation in Statistical Physics] March 15th - 19th, 2010 Pont-à-Mousson (France)
*[https://www.cecam.org/workshop-445.html Interatomic potentials for transition metals and their compounds] April 12-14 2010 CECAM-ETHZ, Zurich, (Switzerland)
*[https://www.cecam.org/workshop-442.html Materials Informatics: Tools for Design and Discovery] April 19-21 2010 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[https://www.cecam.org/workshop-413.html Molecular Simulation of Clathrate Hydrates] May 6-8 2010 ACAM, Dublin, (Ireland)
*[http://www.math.rutgers.edu/events/smm/index.html 103rd Statistical Mechanics Conference] 9-11 May, Rutgers, The State University of New Jersey (USA)
*[http://www.ppeppd2010.cn/ Properties and Phase Equilibria for Product and Process Design] May 16-21, 2010 Suzhou, Jiangsu (China)
*[https://www.cecam.org/workshop-458.html Dynamic coarse-graining: Towards quantitative mesoscale modeling of complex fluids] May 19-21 2010 ACAM, Dublin, (Ireland)
*[http://bifi.es/events/cel2010/ Complex Energy Landscapes: Computational and Statistical Methods for Soft Matter] June 2-4, 2010 Zaragoza (Spain)
*[https://www.cecam.org/workshop-382.html Advances in the Implementation of Polarizable Force Fields for Molecular Simulations] June 7-9, 2010 CECAM-HQ-EPFL, Lausanne,(Switzerland)
*[http://www.ffn.ub.es/sitges/index.php XXII Sitges Conference on Statistical Mechanics] 7-11 June 2010, Sitges (Spain)
*[http://liblice.icpf.cas.cz/2010/2010.php Eighth Liblice Conference on the Statistical Mechanics of Liquids] June 13-18, 2010 Brno (Czech Republic)
*[http://www.mace.manchester.ac.uk/5thspheric 5th International SPHERIC SPH Workshop] June 23-25, 2010 Manchester (UK)
*[http://ismc2010.ugr.es International Soft Matter Conference 2010] 5th-8th July 2010, Granada, Spain
*[http://www.statphys.org.au StatPhys 24: XXIV International Conference on Statistical Physics of the International Union for Pure and Applied Physics (IUPAP)] 19-23 July, (2010). Cairns (Australia)
*[https://www.cecam.org/workshop-450.html Mesoscale methods for colloidal hydrodynamics] July 19-21 2010 CECAM-EPFL, Lausanne (Switzerland)
*[https://www.cecam.org/workshop-472.html Crystallisation: from colloids to pharmaceuticals] July 22-24, 2010 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo in the Apuan Alps VI] 24th-31st July 2010, Vallico Sotto, Italy
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program V] 1st-8th August 2010, Vallico Sotto (Italy)
*[http://ipht.cea.fr/Meetings/BegRohu2010/ Concepts and Methods of Statistical Mechanics] 23 August - 4 September 2010 Saint Pierre Quiberon (France)
*[http://neptuno.unizar.es/events/constraints2010/ Constraints in molecular simulation] September 2-4, 2010 at ZCAM, Zaragoza, (Spain)
*[http://ergodic.ugr.es/cp 11th Granada Seminar: Foundations of Nonequilibrium Statistical Physics — From Basic Science to Future Challenges] 13-17 September, (2010). La Herradura, Granada (Spain)
*[http://www.ccp5.ac.uk/AGM/agm_2010.shtml CCP5 Annual Meeting 2010] 13-15 September 2010, Sheffield (UK)
*[http://www.matgas.com/saft2010/ 20 YEARS OF THE SAFT EQUATION: RECENT ADVANCES AND CHALLENGES] September 19-21, 2010, Barcelona (Spain)
*[http://tnt.phys.uniroma1.it/~transport/ Anomalous Transport: from Billiards to Nanosystems] 20-24 September 2010 Sperlonga (Italy)
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/index.html Random Matrices, Statistical Physics and Information Theory] 26-30 September, Kraków (Poland)
*[https://www.cecam.org/workshop-410.html Protein Folding Dynamics: Bridging the Gap between Theory and Experiment] October 4 to October 7, 2010 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.pks.mpg.de/~fifcm10/ Fluctuation-Induced Forces in Condensed Matter] 11-15 October 2010 Dresden (Germany)
*[https://www.cecam.org/workshop-339.html Multiscale Modeling and Simulation: Bridging Scales and Disciplines] October 20-22 2010 Kartause Ittingen, Warth (Switzerland)
*[https://www.cecam.org/workshop-437.html Multiscale modeling of lipid bilayers under equilibrium and non-equilibrium conditions] October 27 to October 29, 2010 CECAM-HQ-EPFL, Lausanne (Switzerland)

__NOTOC__
[[category: Miscellaneous]]

Conferences

2010-09-09T15:19:59Z

Noe: /* 2011 */

The following is a chronological list of conferences, seminars, or meetings related to thermodynamics, statistical mechanics, soft condensed matter, many-body problems, complex fluids etc.
===2010===
====September====
*[http://ergodic.ugr.es/cp 11th Granada Seminar: Foundations of Nonequilibrium Statistical Physics — From Basic Science to Future Challenges] 13-17 September, (2010). La Herradura, Granada (Spain)
*[http://www.ccp5.ac.uk/AGM/agm_2010.shtml CCP5 Annual Meeting 2010] 13-15 September 2010, Sheffield (UK)
*[http://www.matgas.com/saft2010/ 20 YEARS OF THE SAFT EQUATION: RECENT ADVANCES AND CHALLENGES] September 19-21, 2010, Barcelona (Spain)
*[http://tnt.phys.uniroma1.it/~transport/ Anomalous Transport: from Billiards to Nanosystems] 20-24 September 2010 Sperlonga (Italy)
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/index.html Random Matrices, Statistical Physics and Information Theory] 26-30 September, Kraków (Poland)

====October====
*[http://www.pks.mpg.de/~fifcm10/ Fluctuation-Induced Forces in Condensed Matter] 11-15 October 2010 Dresden (Germany)
*[https://www.cecam.org/workshop-339.html Multiscale Modeling and Simulation: Bridging Scales and Disciplines] October 20-22 2010 Kartause Ittingen, Warth (Switzerland)
====December====
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP10)] 14-16 December 2010, Warwick University (United Kingdom)
===2011===
====March====
*[http://19ectp.cheng.auth.gr 19th European Conference on Thermophysical Properties] August 28 - September 1 2011 Thessaloniki (Greece)
====September====
*[http://lmc2011.univie.ac.at 8th Liquid Matter Conference] September 6-11, 2011 Wien (Austria)

==Previous conferences==
Note: over time it is natural that more and more of these links will become broken.
=====2007=====
*[http://www.simbioma.cecam.org/ Simulation of Hard Bodies] April 16 to April 19 2007 in Lyon.
*[http://www.math.rutgers.edu/events/smm/ 97th Statistical Mechanics Conference] May 6-8, 2007 Rutgers University
*[http://www.ucm.es/info/molecsim/luis/workshop.htm Workshop on Theory and Computer Simulations of Inhomogenoeus Fluids] May 16-18, 2007, Universidad Complutense, Madrid.
*[http://www-thphys.physics.ox.ac.uk/users/JuliaYeomans/OxfordWorkshop/home.php Mesoscale Modelling for Complex Fluids and Flows] June 25--27, 2007 University of Oxford, UK.
*[http://www.ecns2007.org 4th European Conference on Neutron Scattering] Lund, Sweden 25-29 June 2007
*[http://www.realitygrid.org/CompSci07 Computational Science 2007] 25-26 June 2007 The Royal Society, London
*[https://www.cecam.fr/index.php?content=activities/workshop New directions in liquid state theory] CECAM workshop, 2 - 4 July, 2007 at ENS, Lyon, France.
*[http://www.cecam.fr/index.php?content=activities/workshop&action=details&wid=157 Fluid phase behaviour and critical phenomena from liquid state theories and simulations] 5-7 July 2007 CECAM workshop, Lyon, France.
*[http://www.statphys23.org/ STATPHYS 23] Genova, Italy, from July 9 to 13, 2007.
*[http://www.iupac2007.org/ 41st IUPAC World Chemistry Conference] Turin, Italy, August 5-11th
*[http://www.srcf.ucam.org/~jae1001/ccp5_2007 CCP5 Annual Conference] 29th-31st August 2007 New Hall, Cambridge, UK
*[http://ccp2007.ulb.ac.be CCP 2007] Brussels 5-8 September 2007
*[http://www.castep.org CASTEP Workshop] 17th - 21st September 2007 University of York, UK
*[http://thermo2007.ifp.fr Thermodynamics 2007] 26-28 September 2007, IFP - Rueil-Malmaison (France)
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2007] Corfu, Greece, 25-30 September 2007
*[http://www.chem.unisa.it/polnan/index.html Polymers in Nanotechnology] 27-28th September 2007, Salerno, Italy.
*[http://www.fz-juelich.de/iff/ismc2007/ International Soft Matter Conference 2007] 1 - 4 October 2007, Eurogress, Aachen (Germany)
*[http://www.escet.urjc.es/~fisica/encuentro_complejos/index.html II Meeting on Modelling of Complex Systems] Universidad Rey Juan Carlos, Mostoles (Madrid), October 25-26 2007
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/SCD/event_22663.html Structuring Colloidal Dispersions by External Fields] 21 November 2007, Institute of Physics, London, United Kingdom
*[http://www.math.rutgers.edu/events/smm/ 98th Statistical Mechanics Conference] December 16-18, 2007, Rutgers University, New Jersey, USA.

=====2008=====
*[http://www.iop.org/activity/groups/subject/lcf/Events/file_25833.pdf Recent Advances in the Understanding of Confined Fluids: from Superfluids to Oil Reservoirs] 9-11 January, Cosener's House, Abingdon UK
*[http://academic.sun.ac.za/summerschool/2008.html Workshop on Soft Condensed Matter and Physics of Biological Systems - Perspectives and topics for South Africa] 23 Jan 2008 - 1 Feb 2008 National Institute for Theoretical Physics at Stellenbosch Institute of Advanced Study, Stellenbosch, Western Cape, South Africa
*[http://bifi.unizar.es/events/bifi2008/main.htm Bifi 2008 Large Scale Simulations of Complex Systems, Condensed Matter and Fusion Plasma] 6–8 February 2008, Zaragoza, Spain
*[http://events.dechema.de/Tagungen/MolMod+Workshop.html International Workshop Molecular Modeling and Simulation in Applied Material Science] March 10-11, DECHEMA-Haus, Frankfurt am Main, Germany
*[http://www.aps.org/meetings/march/index.cfm APS March Meeting] March 10-14, 2008. New Orleans, Louisiana, USA
*[http://users.physik.tu-muenchen.de/metz/jerusalem.html Modelling anomalous diffusion and relaxation] 23–28 March 2008, Jerusalem, Israel
*[http://www.newton.cam.ac.uk/programmes/CSM/csmw02.html Markov-Chain Monte Carlo Methods] 25 March to 28 March 2008 Isaac Newton Institute for Mathematical Sciences, Cambridge, UK
*[http://www.usal.es/~fises/ XV Congreso de Física Estadística (Fises' 08)] 27-29 March 2008, Salamanca, Spain
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/CMMP08/event_26545.html Condensed Matter and Materials Physics (CMMP)] 26-28 March 2008 Royal Holloway, University of London, UK
*[http://www.icmab.es/softmatter2008/index.html SoftMatter 2008] "Workshop on Electrostatic Effects in Soft Matter: Bringing Experiments, Theory and Simulation Together" 10 – 11 April ICMAB-CSIC, Barcelona, Spain.
*[http://www.icmab.es/11emscf/index.html 11th European Meeting on Supercritical Fluids] "New Perspectives in Supercritical Fluids: Nanoscience, Materials and Processing" 4-7 May 2008, ICMAB-CSIC, Barcelona, Spain
*[http://www.math.rutgers.edu/events/smm/index.html 99th Statistical Mechanics Conference] May 11-13, 2008 Rutgers University, USA
*[http://www.fondation-pgg.org/events/degennesdays/ DeGennesDays] 15-17 May 2008 Collège de France, Paris.
*[http://www.ill.fr/Events/rktsymposium/ Surfaces and Interfaces in Soft Matter and Biology SISMB 2008] "The impact and future of neutron reflectivity - A Symposium in Honor of Robert K. Thomas" 21-23 May 2008, Institut Laue-Langevin (Grenoble, France).
*[http://www.plmmp.univ.kiev.ua/ Physics of Liquid Matter: Modern Problems] May 23-26, 2008, Kyiv National Taras Shevchenko University, Ukraine
*[http://www.ffn.ub.es/sitges/ XXI Sitges Conference] XXI Sitges Conference on Statistical Mechanics. Statistical Mechanics of Molecular Biophysics 2nd-6th June 2008, Sitges, Spain
*[http://www1.ci.uc.pt/gcpi/poly2008 Polyelectrolytes 2008] 16th-19th June 2008, Coimbra, Portugal
*[http://www-spht.cea.fr/Meetings/BegRohu2008/index.html The Beg Rohu Summer School: Manifolds in random media, random matrices and extreme value statistics] 16th - 28th June 2008, French National Sailing School, Quiberon peninsula, France.
*[http://www.chm.bris.ac.uk/cms/ Computational Molecular Science 2008] 22nd – 25th June 2008, Cirencester, UK.
*[http://www.liquids2008.se/ 7th Liquid Matter Conference] 27 June - 1 July 2008 Lund, Sweden
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School 2008] University of Sheffield, England July 6 - July 15 2008.
*Potentials Workshop: Recent developments in interatomic potentials 7-8 July 2008 Oxford, UK
*[http://www.lcc-toulouse.fr/molmat2008/ MOLMAT2008 International Symposium on Molecular Materials based on Chemistry, Solid State Physics, Theory and Nanotechnology] July 8-11th 2008 Toulouse, France
*[http://www2.polito.it/eventi/sigmaphi2008/ SigmaPhi2008] 14th -18th July 2008 Kolympari, Crete, Greece
*[http://www.cecam.org/workshop-188.html Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers] July, 16th-18th 2008 CECAM-EPFL, Lausanne, Switzerland
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/event_22243.html Molecular Dynamics for Non-Adiabatic Processes] 21 July 2008 to 22 July 2008 Institute of Physics, London
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcitaa_08/announcement.html Quantum Monte Carlo in the Apuan Alps IV] 26th July - Sat 2nd August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.cecam.org/workshop-215.html New directions in the theory and modelling of liquid crystals] July, 28th 2008 to July, 30th 2008, CECAM-EPFL, Lausanne, Switzerland
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcatcp_08/announcement.html Quantum Monte Carlo and the CASINO program III] 3rd August - Sun 10th August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.icct2008.org/ 20th International Conference on Chemical Thermodynamics] August 3-8, 2008, Warsaw, Poland.
*[http://perso.ens-lyon.fr/thierry.dauxois/LORIS/LesHouchesSummerSchool2008.html Long-Range Interacting Systems] Summer School in Les Houches (France), 4-29 August 2008
*[http://ctbp.ucsd.edu/summer_school08/apply2008.html Coarse-Grained Physical Modeling of Biological Systems: Advanced Theory and Methods] August 11-15, 2008 University of California San Diego
*[http://www.chem.ucl.ac.uk/astrosurf/SIPML.html Surface and Interface Processes at the Molecular Level] 17 - 23 August 2008, Lucca, Italy
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD141/index.asp Faraday Discussion 141: Water - From Interfaces to the Bulk] 27 - 29 August 2008 Heriot-Watt University, Edinburgh, United Kingdom
*[http://ectp18.conforganizer.net 18th European Conference on Thermophysical Properties] 31 Aug-4 Sep 2008 Pau, France
*[http://iber2008.df.fct.unl.pt/ 9th Iberian Joint Meeting on Atomic and Molecular Physics - IBER 2008] 7-9th September, Capuchos, Portugal
*[http://www.cmmp.ucl.ac.uk/~dmd/ccp5.htm CCP5 Annual Meeting: Surfaces and Interfaces] 8-10th September, London, UK
*[http://ergodic.ugr.es/cp/ 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials] September 15-19, 2008 Granada, Spain
*[http://www.cecam.org/workshop-222.html Standardisation and databasing of ab-initio and classical simulations] September, 18th 2008 to September, 19th 2008 CECAM-ETHZ, Zurich, Switzerland
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2008 (ICCMSE 2008)] 25-30 September, Crete, Greece
*[http://www.mmm2008.org/bin/view.pl/Main/WebHome Multscale Materials Modeling] 27–31 October 2008, Tallahassee, Florida USA
*[http://www.aiche.org/Conferences/AnnualMeeting/index.aspx 2008 AIChE Annual Meeting] November 16-21 2008 Philadelphia, Pennsylvania USA
*[http://www.ihp.jussieu.fr/ceb/Trimestres/T08-4/C3/index.html Statistical mechanics] Paris (France) 8-12 December 2008
*[http://complex.ffn.ub.es/bcnetworkshop BCNet Workshop] Trends and perspectives in complex networks. Barcelona (Spain) 10-12 December 2008
*[http://www.math.rutgers.edu/events/smm/ 100th Statistical Mechanics Conference] Rutgers University, (USA) 13-18 December 2008
=====2009=====
*[http://www.ccpb.ac.uk/events/conference/2009/ Biomolecular Simulation 2009] 6-8 January, Yorkshire Museum and Gardens, York, United Kingdom 2009
*[http://www.fisica.unam.mx/externos/wintermeeting/ XXXVIII edition of the Winter Meeting on Statistical Physics] Taxco, Guerrero (Mexico) 6th - 9th January, 2009.
*[http://www.ucl.ac.uk/msl/events/2009/workshop09.htm 2009 MSL Workshop: Accessing large length and time scales with accurate quantum methods] 12th - 13th January 2009 University College London (United Kingdom)
*[http://euler.us.es/%7Eopap/stochgame/index-en.html Stochastic Models in Physics, Biology, and Social Sciences] Carmona (Sevilla), Spain February 12-14 (2009)
*[http://hera.physik.uni-konstanz.de/igk/news/workshops/homepage/index.html Frontiers of Soft Condensed Matter 2009] Les Houches () 15-20 February 2009
*[http://www.mpipks-dresden.mpg.de/~mbsffe09/ Many-body systems far from equilibrium: Fluctuations, slow dynamics and long-range interactions] Max Planck Institute for the Physics of Complex Systems Dresden, Germany February 16 - 27 (2009)
*[http://www.formulation.org.uk/Conference_flyers_Sept2007_on/Flyer-sims.pdf Workshop on advances in modelling for formulations] 25th of March 2009 at GSK Waybridge (United Kingdom)
*[http://www.physik.uni-leipzig.de/~janke/meco34/ 34th Conference of the Middle European Cooperation in Statistical Physics] 30 March - 01 April 2009 Universität Leipzig (Germany)
*[http://www.ipam.ucla.edu/programs/ktws2/ Workshop II: The Boltzmann Equation: DiPerna-Lions Plus 20 Years] April 15-17 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-313.html Computational Studies of Defects in Nanoscale Carbon Materials] May 11, 2009 to May 13, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.cecam.org/workshop-320.html Modeling of Carbon and Inorganic Nanotubes and Nanostructures] May 13, 2009 to May 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ima.umn.edu/2008-2009/W5.18-22.09/ Molecular Simulations: Algorithms, Analysis, and Applications] Institute for Mathematics and its Applications, University of Minnesota (USA), May 18-22, 2009
*[http://www.ipam.ucla.edu/programs/ktws4/ Workshop IV: Asymptotic Methods for Dissipative Particle Systems] May 18-22 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-310.html Computer Simulation in Food Science: CFD meets Soft Matter] May 25, 2009 to May 27, 2009 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://complenet09.diit.unict.it CompleNet 2009] International Workshop on Complex Networks. Catania (Italy), May 26-28, 2009
*[http://go.warwick.ac.uk/maths/research/events/2008_2009/symposium/wks5/ EPSRC Symposium Workshop on Molecular Dynamics] Monday 1 – Friday 5 June (2009) Warwick (United Kingdom)
*[http://www.cecam.org/workshop-272.html Theoretical Modeling of Transport in Nanostructures] June 2, 2009 to June 5, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.mpip-mainz.mpg.de/mmsd/ Mainz Materials Simulation Days] 3-5 June 2009 Max Planck Institute for Polymer Research (Germany)
*[http://www.soms.ethz.ch/workshop2009 Coping with Crises in Complex Socio-Economic Systems] ETH Zurich (Switzerland), June 8-13, 2009
*[[MOSSNOHO Workshop 2009]] 16 of June 2009, Universidad Complutense de Madrid (Spain)
*[http://cftc.cii.fc.ul.pt/Workshops/flowNjam/index.html Flow(ers) and jam(mers): from liquid crystals to grains] Lisbon (Portugal) 17-19 June 2009
*[http://ipht.cea.fr/Meetings/BegRohu2009/ Beg Rohu Summer School: Quantum Physics Out of Equilibrium] Ecole Nationale de Voile (France) 15-27 June 2009
*[http://symp17.boulder.nist.gov Seventeenth Symposium on Thermophysical Properties] Boulder, Colorado (USA), June 21-26, 2009
*[http://www.fhi-berlin.mpg.de/th/Meetings/DFT-workshop-Berlin2009/ Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions] Berlin (Germany) June 22-July 1 2009
*[https://www.irphe.univ-mrs.fr/~fe09 Fluid and Elasticity] Carry-le-Rouet, near Marseilles (France) June 23-26, 2009
*[http://www.icmp.lviv.ua/statphys2009/ Statistical Physics: Modern Trends and Applications] June 23-25, 2009 Lviv, (Ukraine)
*[http://www.fuw.edu.pl/~wssph/ 3rd Warsaw School of Statistical Physics] Kazimierz Dolny (Poland), 27 June - 4 July, 2009
*[http://www.cecam.org/workshop-298.html Modeling and Simulation of Water at Interfaces from Ambient to Supercooled Conditions] June 29, 2009 to July 1, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School] University of Sheffield (England) July 5 - July 14 2009
*[http://math.arizona.edu/~goriely/leshouches/home.html New trends in the Physics and Mechanics of Biological Systems"] Les Houches (France), 6-31 July 2009
*[http://research.yale.edu/boulder/Boulder-2009/index.html Nonequilibrium Statistical Mechanics: Fundamental Problems and Applications] July 6-24 2009 Boulder (USA)
*[http://www.xrqtc.cat/index.php/ca/homew New trends in Computational Chemistry for Industry Applications] July 6-7 Barcelona (Spain)
*[http://www.frias.uni-freiburg.de/BFF Computational Methods for Soft Matter and Biological Systems] July 8-11, 2009, FRIAS, Freiburg (Germany)
*[http://www.cecam.org/workshop-286.html Structural Transitions in Solids: Theory, Simulations, Experiments and Visualization Techniques] July 8, 2009 to July 11, 2009 CECAM-USI, Lugano (Switzerland)
*[http://fomms.org FOMMS 2009, Fourth International Conference Foundations of Molecular Modeling and Simulation] Semiahmoo Resort, Blaine, WA (USA) 12-16 July 2009
*[http://www.cecam.org/workshop-308.html Computer Simulation Approaches to Study Self-Assembly: From Patchy Nano-Colloids to Virus Capsids] July 13, 2009 to July 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.md-net.org.uk/events/bath_jul_2009.htm EPSRC Network Mathematical Challenges of Molecular Dynamics] 13-15 July (2009) Bath, United Kingdom.
*[http://www.cecam.org/workshop-279.html New Trends in Simulating Colloids: from Models to Applications] July 15, 2009 to July 18, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD144/index.asp Faraday Discussion 144: Multiscale Modelling of Soft Matter] University of Groningen (The Netherlands) 20 - 22 July 2009
*[http://www2.yukawa.kyoto-u.ac.jp/~ykis2009/Welcome.html Frontiers in Nonequilibrium Physics: Fundamental Theory, Glassy & Granular Materials, and Computational Physics] July 21 - August 21, 2009 Kyoto (Japan)
*[http://www.cecam.org/workshop-293.html Fundamental Aspects of Deterministic Thermostats] July 27, 2009 to July 29, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program IV] 2nd-9th August 2009 Towler Institute, Tuscany, (Italy)
*[http://www.warwick.ac.uk/go/quantsim09 EPSRC Symposium Workshop on Quantum Simulations] 24th-28th August 2009 University of Warwick (United Kingdom)
*[http://itf.fys.kuleuven.be/~fpspXII/ Fundamental Problems in Statistical Physics XII] August 31 - September 11, 2009 Leuven (Belgium)
*[http://denali.phys.uniroma1.it/~idmrcs6/ 6th International Discussion Meeting on Relaxations in Complex Systems] August 31- September 5, 2009, Roma (Italy)
*[http://www.dft09.org/ International Conference on the Applications of Density Functional Theory in Chemistry and Physics] August 31st to September 4th 2009 Lyon (France)
*[http://www.dfrl.ucl.ac.uk/CCP5/ccp5.htm CCP5 Annual Meeting 2009 Structure Prediction] 7th to 9th, September 2009 London (United Kingdom)
*[http://fises.dfa.uhu.es/fises09/ XVI Congreso de Física Estadística] Huelva, 10-12 September 2009, Spain
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/ 22nd Marian Smoluchowski Symposium On Statistical Physics] 12-17 September 2009 Zakopane (Poland)
*[http://www.isis.rl.ac.uk/largescale/loq/SAS2009/SAS2009.htm SAS-2009] XIV International Conference on Small-Angle Scattering, Sunday 13 - Friday 18 September, 2009, Oxford (UK)
*[http://www.esc.sandia.gov/dsmc09/dsmc09.html Direct Simulation Monte Carlo workshop] September 13-16, 2009 Santa Fe, New Mexico, USA,
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/PHYSASPECTS_09/index.html Physical Aspects of Polymer Science] 14-16 September 2009 Bristol (United Kingdom)
*[http://www.thermodynamics2009.org/ Thermodynamics 2009] September 23-25 Imperial College London , U.K. (2009)
*[http://www.crm.cat/wkstatisticalphysics/ Techniques and Challenges from Statistical Physics] October 14 to 16, 2009 Barcelona, (Spain)
*[http://paginas.fe.up.pt/~equifase/ EQUIFASE 2009] VIII Iberoamerican Conference on Phase Equilibria and Fluid Properties for Process Design, Praia da Rocha, Portugal 17-21 October 2009
*[http://cint.lanl.gov/workshop2009/ Multiple Length Scales in Polymers and Complex Fluids] October 18-21, 2009 Bishop's Lodge, Santa Fe (USA)
*[http://www.cecam.org/workshop-290.html Classical Density Functional Theory Methods in Soft and Hard Matter] October 21, 2009 to October 23, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://seneca.fis.ucm.es/parr/ktlog2_09 kTlog2 '09: Computing Matters Workshop] 22-24 October, 2009, Toledo (Spain)
*[https://www.cecam.org/workshop-325.html Linking Systems Biology and Biomolecular Simulations] November 16-19 2009 CECAM Lausanne, (Switzerland)
*[http://www.lorentzcenter.nl/lc/web/2009/357/info.php3?wsid=357 Subdivide and Tile: Triangulating spaces for understanding the world] November 16 to 20 2009, Leiden (Netherlands)
*[http://www.dfi.uchile.cl/~granular09/Welcome.html Southern Workshop on Granular Materials 2009 - SWGM09] November 30 to December 4 2009, Viña del Mar (Chile)
*[http://www.math.rutgers.edu/events/smm/index.html 102nd Statistical Mechanics Conference] Rutgers University, December 13-15, 2009 (USA)
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP)] 15-17 December 2009 Warwick University (United Kingdom)
=====2010=====
*[http://ddays2010.northwestern.edu Dynamics Days] January 4-7, 2010 Evanston Illinois (USA)
*[http://www2.surrey.ac.uk/maths/news/events/2010/emerging_interfaces_workshop.htm Emerging Interfacial Dynamics] January 6-8 University of Surrey (UK)
*[http://www.comphys.ethz.ch/jstat/ Journées de Physique Statistique 2010] 28-29 January 2010 Paris (France)
*[http://www.theory.caltech.edu/~ooguri/CMP-HEP/CMP-HEP.htm Condensed Matter Physics Meets High Energy Physics] University of Tokyo, February 8-12, 2010 (Japan)
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/2010/CFFI10/index.html Complex Fluid-Fluid Interfaces] 25 February 2010 Institute of Physics, London (UK)
*[http://tfy.tkk.fi/soft/levi2010/ International Workshop on Coarse-Grained Biomolecular Modeling] 7-12 March, Levi, (Finland)
*[http://www.lpm.u-nancy.fr/webperso/chatelain.c/GrpPhysStat/Meco35/index.php 35th Conference of the Middle European Cooperation in Statistical Physics] March 15th - 19th, 2010 Pont-à-Mousson (France)
*[https://www.cecam.org/workshop-445.html Interatomic potentials for transition metals and their compounds] April 12-14 2010 CECAM-ETHZ, Zurich, (Switzerland)
*[https://www.cecam.org/workshop-442.html Materials Informatics: Tools for Design and Discovery] April 19-21 2010 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[https://www.cecam.org/workshop-413.html Molecular Simulation of Clathrate Hydrates] May 6-8 2010 ACAM, Dublin, (Ireland)
*[http://www.math.rutgers.edu/events/smm/index.html 103rd Statistical Mechanics Conference] 9-11 May, Rutgers, The State University of New Jersey (USA)
*[http://www.ppeppd2010.cn/ Properties and Phase Equilibria for Product and Process Design] May 16-21, 2010 Suzhou, Jiangsu (China)
*[https://www.cecam.org/workshop-458.html Dynamic coarse-graining: Towards quantitative mesoscale modeling of complex fluids] May 19-21 2010 ACAM, Dublin, (Ireland)
*[http://bifi.es/events/cel2010/ Complex Energy Landscapes: Computational and Statistical Methods for Soft Matter] June 2-4, 2010 Zaragoza (Spain)
*[https://www.cecam.org/workshop-382.html Advances in the Implementation of Polarizable Force Fields for Molecular Simulations] June 7-9, 2010 CECAM-HQ-EPFL, Lausanne,(Switzerland)
*[http://www.ffn.ub.es/sitges/index.php XXII Sitges Conference on Statistical Mechanics] 7-11 June 2010, Sitges (Spain)
*[http://liblice.icpf.cas.cz/2010/2010.php Eighth Liblice Conference on the Statistical Mechanics of Liquids] June 13-18, 2010 Brno (Czech Republic)
*[http://www.mace.manchester.ac.uk/5thspheric 5th International SPHERIC SPH Workshop] June 23-25, 2010 Manchester (UK)
*[http://ismc2010.ugr.es International Soft Matter Conference 2010] 5th-8th July 2010, Granada, Spain
*[http://www.statphys.org.au StatPhys 24: XXIV International Conference on Statistical Physics of the International Union for Pure and Applied Physics (IUPAP)] 19-23 July, (2010). Cairns (Australia)
*[https://www.cecam.org/workshop-450.html Mesoscale methods for colloidal hydrodynamics] July 19-21 2010 CECAM-EPFL, Lausanne (Switzerland)
*[https://www.cecam.org/workshop-472.html Crystallisation: from colloids to pharmaceuticals] July 22-24, 2010 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo in the Apuan Alps VI] 24th-31st July 2010, Vallico Sotto, Italy
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program V] 1st-8th August 2010, Vallico Sotto (Italy)
*[http://ipht.cea.fr/Meetings/BegRohu2010/ Concepts and Methods of Statistical Mechanics] 23 August - 4 September 2010 Saint Pierre Quiberon (France)
*[http://neptuno.unizar.es/events/constraints2010/ Constraints in molecular simulation] September 2-4, 2010 at ZCAM, Zaragoza, (Spain)

__NOTOC__
[[category: Miscellaneous]]

Carnahan-Starling equation of state

2009-11-20T15:21:24Z

Noe: /* Thermodynamic expressions */ small correction

The '''Carnahan-Starling equation of state''' is an approximate (but quite good) [[Equations of state |equation of state]] for the fluid phase of the [[hard sphere model]] in three dimensions. It is given by (Ref <ref name="CH"> [http://dx.doi.org/10.1063/1.1672048 N. F. Carnahan and K. E. Starling,"Equation of State for Nonattracting Rigid Spheres" Journal of Chemical Physics '''51''' pp. 635-636 (1969)] </ref> Eqn. 10).

: <math>
Z = \frac{ p V}{N k_B T} = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }.
</math>

where:
*<math> p </math> is the [[pressure]]
*<math> V </math> is the volume
*<math> N </math> is the number of particles
*<math> k_B </math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute [[temperature]]
*<math> \eta </math> is the [[packing fraction]]:

:<math> \eta = \frac{ \pi }{6} \frac{ N \sigma^3 }{V} </math>

*<math> \sigma </math> is the [[hard sphere model | hard sphere]] diameter.
==Virial expansion==
It is interesting to compare the [[Virial equation of state | virial coefficients]] of the Carnahan-Starling equation of state (Eq. 7 of <ref name="CH"> </ref>) with the [[Hard sphere: virial coefficients | hard sphere virial coefficients]] in three dimensions (exact up to <math>B_4</math>, and those of Clisby and McCoy <ref> [http://dx.doi.org/10.1007/s10955-005-8080-0 Nathan Clisby and Barry M. McCoy "Ninth and Tenth Order Virial Coefficients for Hard Spheres in D Dimensions", Journal of Statistical Physics '''122''' pp. 15-57 (2006)] </ref>):
{| style="width:40%; height:100px" border="1"
|-
| <math>n</math> ||Clisby and McCoy ||<math>B_n=n^2+n-2</math>
|-
| 2 || 4 || 4
|-
| 3 || 10 || 10
|-
| 4 || 18.3647684 || 18
|-
| 5 || 28.224512 || 28
|-
| 6 || 39.8151475 || 40
|-
| 7 || 53.3444198 || 54
|-
| 8 || 68.5375488 || 70
|-
| 9 || 85.8128384 || 88
|-
| 10 || 105.775104 || 108
|}

==Thermodynamic expressions==
From the Carnahan-Starling equation for the fluid phase
the following thermodynamic expressions can be derived
(Ref <ref>[http://dx.doi.org/10.1063/1.469998 Lloyd L. Lee "An accurate integral equation theory for hard spheres: Role of the zero-separation theorems in the closure relation", Journal of Chemical Physics '''103''' pp. 9388-9396 (1995)]</ref> Eqs. 2.6, 2.7 and 2.8)

[[Pressure]] (compressibility):

:<math>\frac{p^{CS}V}{N k_B T } = \frac{1+ \eta + \eta^2 - \eta^3}{(1-\eta)^3}</math>

Configurational [[chemical potential]]:

:<math>\frac{ \overline{\mu }^{CS}}{k_B T} = \frac{8\eta -9 \eta^2 + 3\eta^3}{(1-\eta)^3}</math>

Isothermal [[compressibility]]:

:<math>\chi_T -1 = \frac{1}{k_BT} \left.\frac{\partial P^{CS}}{\partial \rho}\right\vert_{T} = \frac{8\eta -2 \eta^2 }{(1-\eta)^4}</math>

where <math>\eta</math> is the [[packing fraction]].

Configurational [[Helmholtz energy function]]:

:<math> \frac{ A_{ex}^{CS}}{N k_B T} = \frac{4 \eta - 3 \eta^2 }{(1-\eta)^2}</math>

==The 'Percus-Yevick' derivation==
It is interesting to note (Ref <ref> [http://dx.doi.org/10.1063/1.1675048 G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr. "Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres", Journal of Chemical Physics '''54''' pp. 1523-1525 (1971)] </ref> Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the [[exact solution of the Percus Yevick integral equation for hard spheres]] via the compressibility route, to one third via the pressure route, i.e.

:<math>Z = \frac{ p V}{N k_B T} = \frac{2}{3} \left[ \frac{(1+\eta+\eta^2)}{(1-\eta)^3} \right] + \frac{1}{3} \left[ \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2} \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }</math>

The reason for this seems to be a slight mystery (see discussion in Ref. <ref>[http://dx.doi.org/10.1021/j100356a008 Yuhua Song, E. A. Mason, and Richard M. Stratt "Why does the Carnahan-Starling equation work so well?", Journal of Physical Chemistry '''93''' pp. 6916-6919 (1989)]</ref> ).

== See also ==
*[[Equations of state for hard spheres]]
*[[Kolafa-Labík-Malijevský equation of state]]

== References ==
<references/>
[[Category: Equations of state]]
[[category: hard sphere]]

Percolation analysis

2009-11-18T11:30:08Z

Noe: /* Percolation threshold and critical thermodynamic transitions */ minor edit

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009)]</ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001)] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices", Physical Review B '''40''', 5007 - 5014 (1989) ] </ref>
. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-11-18T11:28:59Z

Noe: /* Percolation threshold and critical thermodynamic transitions */ new reference added

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009)]</ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001)] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
<ref name=hu >[http://dx.doi.org//10.1103/PhysRevB.40.5007 Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices" Physical Review B '''40''', 5007 - 5014 (1989) </ref>
. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-11-18T11:27:31Z

Noe: /* Percolation threshold and critical thermodynamic transitions */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009)]</ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001)] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref>
<ref name=hu >
[http://dx.doi.org//10.1103/PhysRevB.40.5007
Chin-Kun Hu and Kit-Sing Ma, "Monte Carlo study of the Potts model on the square and the simple cubic lattices" Phys. Rev. B 40, 5007 - 5014 (1989) </ref>
. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Researchers and research groups

2009-11-11T17:53:28Z

Noe: /* Spain */

==Argentina==
*[http://www.iflysib.unlp.edu.ar/index.html Instituto de Física de Líquidos y Sistemas Biológicos] IFLYSIB, La Plata
*[http://linux0.unsl.edu.ar/~fisica/superfi.htm Laboratorio de Ciencias de Superficies y Medios Porosos] Universidad Nacional de San Luis
*[http://www.tandar.cnea.gov.ar/~pastorin/ Claudio Pastorino] Comisión Nacional de Energía Atómica
*[http://cabtes55.cnea.gov.ar/personales/jagla/homepage.htm Eduardo A. Jagla] Centro Atómico Bariloche, Comisión Nacional de Energía Atómica

==Australia==
*[http://rsc.anu.edu.au/~evans/index.php Denis Evans] Australian National University
*[http://chem.sci.gu.edu.au/~s384166/d_bernhardt.html Debra J. Bernhardt (née Searles)] Griffith University
*[http://www.unisanet.unisa.edu.au/staff/PhilAttard/ Phil Attard] University of South Australia
*[http://www.phys.unsw.edu.au/%7Egary/statmech.html Statistical Mechanics and Dynamical Systems Group] University of New South Wales
*[http://www.nanochemistry.curtin.edu.au/research/theoretical.cfm Nanochemistry Research Institute: Theoretical and Computational] Curtin University of Technology
*[http://www.it.swin.edu.au/centres/cms/ Centre for Molecular Simulation] Swinburne University of Technology
*[http://www.rmit.edu.au/browse;ID=xono5nykwdrm The Condensed Matter Theory Group] RMIT (Royal Melbourne Institute of Technology) University

==Austria==
*[http://tph.tuwien.ac.at/smt/index.html Soft Matter Theory] Vienna University of Technology
==Belgium==
*[http://www.ulb.ac.be/sciences/polphy/ Laboratoire de Physique des Polymères] Université Libre de Bruxelles
*[http://www.grasp.ulg.ac.be/ Group for Research and Applications in Statistical Physics (GRASP)] Université de Liège
*[http://www.ulb.ac.be/rech/inventaire/unites/ULB658.html Laboratoire de Physique de la Matière molle (LPMM)] Université Libre de Bruxelles

==Brazil==
*[http://www.cbpf.br/GrupPesq/StatisticalPhys/ Group of Statistical Physics] Centro Brasileiro de Pesquisas Físicas
*[http://www.if.ufrgs.br/fcomplex/ Complex Fluids] Universidade Federal do Rio Grande do Sul

==Canada==
*[http://www.physics.uoguelph.ca/~des/ Research on Liquid Crystals and Complex Fluids] University of Guelph
*[http://mse.mcmaster.ca/faculty/johari/ Gyan Johari] McMaster University
*[http://www.pmc.umontreal.ca/~mousseau/site_an/index.php?n=Main.Welcome Normand Mousseau] Université de Montréal
*[http://www.apmaths.uwo.ca/~mkarttu/ The Karttunen group] University of Western Ontario in London
*[http://www.theory.chem.uwo.ca/drupal5/ Styliani Constas] University of Western Ontario in London
*[http://www.chem.queensu.ca/people/faculty/cann/ Natalie M. Cann] Queen's University, Kingston, Ontario
*[http://www.chem.utoronto.ca/staff/JMS/schofield_j.html Jeremy Schofield] University of Toronto
*[http://www.chem.utoronto.ca/~swhittin/ Stu Whittington - Chemical Physics Theory Group] University of Toronto
*[http://physics.sfu.ca/people/profiles/plischke Michael Plischke] Simon Fraser University, Burnaby, British Columbia
*[http://www.sfu.ca/~boal/ Dave Boal] Simon Fraser University, Burnaby, British Columbia
*[http://homepages.ucalgary.ca/~pkusalik/ Kusalik Research Group] University of Calgary
*[http://poole.stfx.ca/ Peter H. Poole] St. Francis Xavier University

==China==
*[http://www.zhuzit.edu.cn/en/edu_res/units.asp Dr. Zhou Shiqi, Modern Statistic Mechanics Research Institute] Hunan University of Technology
==Czech Republic==
*[http://www.natur.cuni.cz/~pmc/namecard.php?id=47 Tomáš Boublík] Univerzita Karlova v Praze
*[http://www.icpf.cas.cz/theory/IvoNez.html Ivo Nezbeda] Akademie věd České republiky
*[http://www.vscht.cz/fch/en/research/theory.html Department of Physical Chemistry: Theory] (Dir.: Anatol Malijevsky) Prague Institute of Chemical Technology
==Denmark==
*[http://www.mip.sdu.dk/~jperram/ John Perram] University of Southern Denmark
*[http://glass.ruc.dk/ "Glass and Time"] DNRF Centre for Viscous Liquid Dynamics, Roskilde University, Denmark

==France==
*[http://www.lcp.u-psud.fr/ Laboratoire de Chimie Physique] CNRS/Université Paris-Sud
*[http://www.lps.ens.fr/ Laboratoire de physique statistique] Ecole Normale Superieure
*[http://www.lptl.jussieu.fr/Welcome.html Laboratoire de Physique Théorique de la Matière Condensée] (Dir. Bertrand Guillot) Université Pierre et Marie Curie/CNRS
*[http://www.msc.univ-paris7.fr/site/index.html Matière et Systèmes Complexes] (Dir.: Jean-Marc di Meglio) Université Paris 7 - Denis Diderot
*[http://w3.lcvn.univ-montp2.fr/~kob/ Walter Kob] Universite Montpellier II
*[http://www.th.u-psud.fr/rubrique.php3?id_rubrique=8 Groupe de physique statistique] Laboratoire de Physique Théorique d'Orsay, CNRS et de l'Université Paris-Sud 11
*[http://www.lpm.u-nancy.fr/activite_physique_statistique/index.php?lang=en_GB Groupe de Physique Statistique ], Institut Jean Lamour, Nancy Université

==Germany==
*[http://www.cond-mat.physik.uni-mainz.de/ Condensed Matter Theory Group KOMET 331] (Dir: Kurt Binder) Johannes Gutenberg-University, Mainz
*[http://www.icp.uni-stuttgart.de/ Institute for Computational Physics] (Dir: Christian Holm) Stuttgart University
*[http://www.mpip-mainz.mpg.de/theory/ Polymer Theory Group Prof. Dr. Kurt Kremer] Max Planck Institute for Polymer Research, Mainz
*[http://www.theorie.physik.uni-goettingen.de/forschung/mm/index.en.html Marcus Müller's research group] Georg-August-Universität Göttingen
*[http://www.physik.uni-bielefeld.de/theory/cm/ Condensed Matter Theory Group] Universität Bielefeld
*[http://www.staff.uni-marburg.de/~germano/ Computer Simulation Group] Philipps-Universität Marburg
*[http://constanze.materials.uni-wuppertal.de/ Dr. R. Hentschke] Universität Wuppertal
*[http://van-der-waals.pc.uni-koeln.de/persons/kraskaE.html Dr. Thomas Kraska] Universität zu Köln
*[http://www.uni-koeln.de/math-nat-fak/phchem/deiters/index.html Statistische Thermodynamik] (Dir. Prof. Dr. Deiters) Universität zu Köln
*[http://www2.tu-berlin.de/%7Einsi/ag_schoen/people/profile/klapp/welcome.html Emmy-Noether research group: Complex fluids in external fields] (Dir. Dr. Sabine H. L. Klapp) Stranski-Laboratorium für Physikalische und Theoretische Chemie
*[http://theorie.physik.uni-konstanz.de/lsfuchs/ Soft Matter Theory Group] Universität Konstanz
*[http://www.theo.chemie.tu-darmstadt.de/front/joomla/ Theoretical Physical Chemistry Group] (Dir.: Dr. Florian Müller-Plathe) Technische Universität Darmstadt
*[http://ganter.chemie.uni-dortmund.de/index.shtml The Simulation and Theory Group] (Dir.: Prof. Dr. Alfons Geiger) Universität Dortmund
*[http://thwww.chemietechnik.uni-dortmund.de/en/textonly/content/staff/head/Sadowski.html Prof. Dr. Gabriele Sadowski] Universität Dortmund
*[http://agknapp.chemie.fu-berlin.de/agknapp/ Macromolecular Modelling Group] (Dir.: Ernst-Walter Knapp) Freie Universität Berlin

==Greece==
*[http://mmml.chem.demokritos.gr/ Molecular Thermodynamics and Modeling of Materials Laboratory] National Research Center for Physical Sciences "Demokritos"
*[http://www.matersci.upatras.gr/SoftMat/ Soft Matter Theory and Simulations Group] University of Patras

==Hungary==
*[http://www.chem.elte.hu/departments/elmkem/baranyai/index.htm András Baranyai] Eötvös Loránd University
*[http://www.chem.elte.hu/departments/kolloid/personnel/jp/ Pál Jedlovszky] Eötvös Loránd University
*[http://www.chem.elte.hu/departments/elmkem/toth/ Gergely Tóth] Eötvös University

==India==
*[http://www.iitk.ac.in/phy/New01/phy_CMT.html Condensed Matter Theory Group] Indian Institute of Technology (Kanpur)
*[http://home.iitk.ac.in/~amalen/ Prof Amalendu Chandra] Indian Institute of Technology (Kanpur)
*[http://www.physics.iitm.ac.in/%7Elabs/cfl/index.html Complex Fluids Laboratory] Indian Institute of Technology (Madras)
*[http://web.iitd.ac.in/~charusita/ Professor Charusita Chakravarty] Indian Institute of Technology (Delhi)
*[http://www.bhu.ac.in/science/faculty/Department_of_Physics_Dr_S_Singh.htm Dr. Shri Singh] Banaras Hindu University

==Iran==
*[http://sina.sharif.edu/~chinfo/parsafarh.html Gholamabbas Parsafar] Sharif University of Technology
==Israel==
*[http://chemistry.huji.ac.il/cgi-bin/chemistry/show_page.pl?L=E&Id=11 Prof. Arieh Ben-naim] Hebrew University of Jerusalem
*[http://www.fh.huji.ac.il/~viki/ Professor Victoria Buch] Hebrew University, Jerusalem

==Italy==
*[http://www2.fci.unibo.it/~bebo/z/index.html Claudio Zannoni home page] Università di Bologna
*[http://abaddon.phys.uniroma1.it/ GCI Computational Physics Group] (Dir. Giovanni Ciccotti) Universita’ di Roma La Sapienza
*[http://glass.phys.uniroma1.it/sciortino/index.html Francesco Sciortino] Universita’ di Roma La Sapienza
*[http://www.ictp.it/pages/research/cmsp.html CMSP - Condensed Matter and Statistical Physics at ICTP] Trieste
*[http://www.pv.infn.it/~romano/ Silvano Romano] Pavia
*[http://www.unive.it/nqcontent.cfm?a_id=36652&persona=000244 Domenico Gazzillo] Università Ca' Foscari Venezia
*[http://www.sissa.it/sbp/web_2008/index.html Statistical and Biological Chemistry Sector] Scuola Internazionale Superiore di Studi Avanzati (SISSA)

==Japan==
*[http://theochem.chem.okayama-u.ac.jp/?lang=en Theoretical Chemistry Group] Okayama University
*[http://www.ifs.tohoku.ac.jp/tokuyama-lab/ Michio Tokuyama Laboratory] Tohoku University

==Mexico==
*[http://abaco.izt.uam.mx/espanol/investigacion/liquidos/index.html Física de Líquidos] Universidad Autónoma Metropolitana
*[http://abaco.izt.uam.mx/espanol/profesores/estadistica.html Mecánica Estadística] Universidad Autónoma Metropolitana
*[http://www.ifug.ugto.mx/Investigacion/MecanicaEstadistica.php Mecánica Estadística] Universidad de Guanajuato
*[http://www.iquimica.unam.mx/pizio.html Orest Pizio] Universidad Nacional Autónoma de México
*[http://xml.cie.unam.mx/xml/tc/ft/mlh/ Mariano López de Haro] Universidad Nacional Autónoma de México
*[http://quimica.izt.uam.mx/Areas/QuimCuant/datosJRA.htm Jose Alejandre] Universidad Autónoma Metropolitana

==Netherlands==
*[http://www1.phys.uu.nl/scm/default.htm Soft Condensed Matter Group] Utrecht University
*[http://fcc.chem.uu.nl/peopleindex/henk/henk.htm Henk N.W. Lekkerkerker] Debye Research Institute, Utrecht University
*[http://www.science.uva.nl/~bolhuis/ Simulation of complex fluids] University of Amsterdam
*[http://www.rug.nl/gbb/research/researchgroups/molecularDynamics/ Marrink's MD group] University of Groningen

==Norway==
*[http://home.phys.ntnu.no/instdef/personale/hjemmesider/johan.hoye/index.html Johan Skule Høye] Norwegian University of Science and Technology (NTNU)
==Poland==
*[http://ichfit.ch.pwr.wroc.pl/?q=node/10 Molecular Modelling and Quantum Chemistry] Wrocław University of Technology
*[http://poczta.umcs.lublin.pl/zmpfch/index_en.htm Department for the Modelling of Physico-Chemical Processes] Maria Curie-Skłodowska University
*[http://ichf.edu.pl/person/ciach.html Professor Alina Ciach] Instytut Chemii Fizycznej, Polskiej Akademii Nauk
*[http://th-www.if.uj.edu.pl/zfs/ Statistical Physics Division] (Dir. Prof. Lech Longa) Uniwersytet Jagielloński
*[http://www.ifmpan.poznan.pl/zp10/zp10_www.htm Nonlinear Dynamics and Computer Simulations] (Dir. Prof. Dr. Habil. K. W. Wojciechowski) Institute of Molecular Physics, Polish Academy of Sciences

==Portugal==
*[http://cftc.cii.fc.ul.pt/index.php Centro de Física Teórica e Computacional] Universidade de Lisboa
==Russia==
*[http://theor.jinr.ru/~kuzemsky/ Alexander L. Kuzemsky] Bogoliubov Laboratory of Theoretical Physics
*[http://www.ihed.ras.ru/norman Атомистическое моделирование и теория конденсированного состояния и неидеальной плазмы] Институт теплофизики экстремальных состояний

==Spain==
*[http://www.qft.iqfr.csic.es/ Theoretical Physical Chemistry Group] (Dir. Enrique Lomba García) Instituto de Química-Física "Rocasolano" (IQFR), CSIC
*[http://www.icmm.csic.es/Teoria/ Condensed Matter Theory] Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC
*[http://emoles.quim.ucm.es/ Carlos Vega Statistical Thermodynamics of Molecular Fluids Group] Universidad Complutense de Madrid
*[http://seneca.fis.ucm.es/ Group of Statistical Mechanics (GISC)] Universidad Complutense de Madrid
*[http://www.uhu.es/filico/home.html Physics of Complex Liquids Group] (Dir.Dr. Enrique de Miguel Agustino) University of Huelva
*[http://gisc.uc3m.es/~cuesta/Science/scientific.html Scientific page of José A. Cuesta] Universidad Carlos III de Madrid
*[http://valbuena.fis.ucm.es/gisc/ Grupo Interdisciplinar de Sistemas Complejos] (GISC)
*[http://www.icmm.csic.es/mossnoho/ MOSSNOHO] Madrid
*[http://www.uam.es/departamentos/ciencias/fisicateoricamateria/propia/fluidos.html Investigación en física estadística de líquidos complejos y biofísica] Universidad Autónoma de Madrid
*[http://www.uam.es/personal_pdi/ciencias/dduque/ Daniel Duque] Universidad Autónoma de Madrid
*[http://www.uam.es/personal_pdi/ciencias/gnavascu/ Guillermo Navascués] Universidad Autónoma de Madrid
*[http://www.uam.es/personal_pdi/ciencias/evelasco/ Enrique Velasco] Universidad Autónoma de Madrid
*[http://www.uned.es/dpto-fisicoquimica/personas/lorna.htm Lorna Bailey Chapman] Universidad Nacional de Educación a Distancia (UNED)
*[http://www.uned.es/dpto-fisicoquimica/personas/luis.htm Luis M. Sesé Sánchez] Universidad Nacional de Educación a Distancia (UNED)
*[http://oboe.quim.ucm.es/jfg.html Juan J. Freire] Universidad Nacional de Educación a Distancia (UNED)
*[http://www.fisfun.uned.es/~pep/ Pep Español] Universidad Nacional de Educación a Distancia (UNED)
*[http://oboe.quim.ucm.es/ Simulation of Chain Molecules] Universidad Complutense de Madrid
*[http://www.upo.es/depa/webdex/quimfis/slagara/slagara.htm Santiago Lago Aranda] Universidad Pablo De Olavide
*[http://www.upo.es/depa/webdex/quimfis/miembros/Web_Sofia/Sofia_archivos/Group.htm Sofía Calero Materials Computational Group] Universidad Pablo De Olavide
*[http://www.grupo.us.es/gmecest/ Group of Statistical Mechanics] University of Seville
*[http://www.icmab.es/molsim/ Lourdes F. Vega Molecular Simulation Group] Instituto de Ciencia de Materiales de Barcelona (ICMAB), CSIC
*[http://grupos.unican.es/GTFE/ Grupo de Termodinámica y Física Estadística] Universidad de Cantabria
*[http://complex.ffn.ub.es/ Physics of Complex Systems Group] Universitat de Barcelona
*[http://simcon.upc.edu/ Computer Simulation in Condensed Matter Group SIMCON] Universitat Politècnica de Catalunya
*[http://www-fen.upc.es/cscmcs/index.html Complex Systems. Computer Simulation of Materials and Biological Systems] Universitat Politècnica de Catalunya
*[http://web.ffn.ub.es/node/6&id=1061 Grupo de Física Estadística] Universitat de Barcelona
*[http://www.unex.es/eweb/fisteor/index_eng.html Statistical Physics Group at the University of Extremadura (SPHINX)] University of Extremadura
*[http://www.ual.es/GruposInv/FQM-230/componentes/jcaballe.htm José Baldomero Caballero Moraleda] Universidad de Almería
*[http://www.ual.es/GruposInv/FQM-230/home_nestcape_e.htm Group of Complex Fluids] Universidad de Almería
*[http://ergodic.ugr.es/ Statistical Physics Group] University of Granada
*[http://www.etseq.urv.es/COMPLEX/index_cs.htm Complex Systems] (Dir.: Allan Mackie) Universitat Rovira i Virgili

==Sweden==
*[http://folding.bmc.uu.se/ David van der Spoel] Uppsala University
*[http://www.fos.su.se/page.php?pid=155&id=407 Alexander Lyubartsev] Stockholms universitet

==Switzerland==
*[http://www.igc.ethz.ch/ The van Gunsteren group's home page] ETH Hönggerberg, HCI
*[http://www.chemie.unibas.ch/~huber/index.html Prof. Dr. Hanspeter Huber] University of Basel
*[http://www.rgp.ethz.ch/ Professor Michele Parrinello's Research Group] ETH Zurich (Swiss Federal Institute of Technology Zurich)
*[https://www.cecam.org/ CECAM] Centre Européen de Calcul Atomique et Moléculaire, Lausanne, Switzerland
==Ukraine==
*[http://ph.icmp.lviv.ua/~lyuda/ Department for Statistical Theory of Condensed Systems] Ukrainian National Academy of Sciences

==United Kingdom==
*[http://www.md-net.org.uk/ MD network]
*[http://www.dur.ac.uk/mark.wilson/ The Wilson Group] Durham University
*[http://cmt.dur.ac.uk/ Condensed Matter Theory] (Dir.: Professor Richard Abram) University of Durham
*[http://www.shu.ac.uk/research/meri/mmg/ Materials Modelling group] Sheffield Hallam University
*[http://www2.warwick.ac.uk/fac/sci/physics/theory/research/simulation/ Molecular Simulation Group] (Dir.: Dr. M. Allen) University of Warwick
*[http://www2.warwick.ac.uk/fac/sci/chemistry/research/molsaw/ MOLecular Simulations At Warwick] University of Warwick
*[http://www.ceas.manchester.ac.uk/research/groups/multiscale/ Multi-scale and Multi-phase Systems] University of Manchester
*[http://www.sci-eng.mmu.ac.uk/facstaffdetails/mneal/default.htm Maureen P. Neal] Manchester Metropolitan University
*[http://www3.imperial.ac.uk/ceMMT/ Molecular modelling and thermodynamics] Imperial College London
*[http://www.ch.ic.ac.uk/quirke/ Computational Physical Chemistry Group] (Dir.: Nick Quirke) Imperial College London
*[http://www.ch.ic.ac.uk/bresme/ Dr. Fernando Bresme Group] Imperial College London
*[http://www.csec.ed.ac.uk/main.html Center for Science at Extreme Conditions] University of Edinburgh
*[http://www.ph.ed.ac.uk/cmatter/ Condensed Matter Group] University of Edinburgh
*[http://www.homepages.ed.ac.uk/pjc01/ Philip J. Camp] University of Edinburgh
*[http://www.chem.ucl.ac.uk/people/catlow/ Richard Catlow FRS] University College London
*[http://www.chem.ucl.ac.uk/people/coveney/ Professor Peter V. Coveney] University College London
*[http://titus.phy.qub.ac.uk/ Atomistic Simulation Centre] Queen's University Belfast
*[http://www.ucl.ac.uk/msl Thomas Young Centre at UCL] University College London
*[http://www-theor.ch.cam.ac.uk/people/jphgroup/ Research group of Professor Hansen] Cambridge University
*[http://www.ch.cam.ac.uk/staff/df.html Professor Daan Frenkel] University of Cambridge
*[http://www-theor.ch.cam.ac.uk/people/sprikgroup/ Sprik Group] University of Cambridge
*[http://www-wales.ch.cam.ac.uk/ Wales group home page] University of Cambridge
*[http://www.bath.ac.uk/physics/research/theory/ Condensed Matter Theory] University of Bath
*[http://staff.bath.ac.uk/chsscp/ Computational Solid State Chemistry Group] University of Bath
*[http://www.irc.leeds.ac.uk/~phy6pdo/ Peter D. Olmsted] University of Leeds
*[http://wheatley.chem.nottingham.ac.uk/ Dr. Richard Wheatley] University of Nottingham
*[http://www.strings.ph.qmul.ac.uk/~cmsmg/ Condensed Matter and Statistical Mechanics Group] (Dir.: Dr. Bob Jones) Queen Mary, University of London
*[http://www-thphys.physics.ox.ac.uk/user/ArdLouis/ Ard Louis research group] University of Oxford
*[http://physchem.ox.ac.uk/%7Edoye/index.html Jonathan Doye's Research Group] University of Oxford
*[http://www-thphys.physics.ox.ac.uk/user/JuliaYeomans/ Julia Yeomans' research group] University of Oxford
*[http://sbcb.bioch.ox.ac.uk/ Structural Bioinformatics and Computational Biochemistry] (Dir.: Prof. Mark S. P. Sansom) University of Oxford
*[http://www-thphys.physics.ox.ac.uk/user/SoftBio/ Theory of Soft and Biological Matter] University of Oxford
*[http://www.strath.ac.uk/chemeng/research/groupdetails/drmartinsweatman-seniorlecturer/ Dr. Martin Sweatman] University of Strathclyde
*[http://www.chm.bris.ac.uk/pt/jeroen/jsvdhome.html Jeroen van Duijneveldt's research group] University of Bristol
*[http://www.umi.surrey.ac.uk/research/scm Soft Condensed Matter Physics Group] University of Surrey
*[http://www.chm.bris.ac.uk/pt/allan/Research/ Computational Materials Chemistry] (Dir.: Professor Neil L. Allan) University of Bristol
*[http://www.stp.dias.ie/~dorlas/ Professor Teunis C. Dorlas] Dublin Institute for Advanced Studies
*[http://www.york.ac.uk/chemistry/staff/academic/a-c/mbates/ Dr Martin Bates] University of York
*[http://www.physics.leeds.ac.uk/pages/JRHenderson J. R. Henderson] University of Leeds
*[http://www.physics.leeds.ac.uk/pages/TCBMcLeish T. C. B. McLeish] University of Leeds
*[http://www.mth.kcl.ac.uk/~tcoolen/nnds/nnds.html Disordered Systems Group] King's College, University of London
*[http://www.shef.ac.uk/materials/staff/kptravis.html Dr Karl P. Travis] University of Sheffield
*[http://www.ma.hw.ac.uk/~oliver/ Professor Oliver Penrose] Heriot-Watt University
*[http://www.ccp5.ac.uk/ Collaborative Computational Project 5 - The Computer Simulation of Condensed Phases]
*[http://www.uwethiele.de/ Uwe Thiele] Loughborough University

== United States of America ==
*[http://faculty.ucmerced.edu/lhirst/index.html The Hirst Group] University of California, Merced
*[http://www.ecs.umass.edu/che/NSF_WWW/ Nanoscale Interdisciplinary Research Team (NIRT)] University of Massachusetts Amherst and University of Delaware
*[http://www.sas.upenn.edu/chem/groups/klein/klein.html The Klein Group] University of Pennsylvania
*[http://chumba.che.ncsu.edu/ Keith E. Gubbins' Research Group] North Carolina State University
*[http://turbo.che.ncsu.edu/index.html Carol K. Hall's Research Group] North Carolina State University
*[http://www.physics.ncsu.edu/people/faculty_lado.html Fred Lado] North Carolina State University
*[http://dl9s6.chem.unc.edu/ Polymer Theory Group] (Dir.: Michael Rubinstein) University of North Carolina at Chapel Hill
*[http://www.che.vanderbilt.edu/cummings1.htm Peter T. Cummings] Vanderbilt University and Oak Ridge National Laboratory
*[http://people.vanderbilt.edu/~c.mccabe/ McCabe Group] Vanderbilt University
*[http://www.cbe.buffalo.edu/kofke.htm David A. Kofke] University at Buffalo
*[http://www.chemical.buffalo.edu/ Errington Research Group] University at Buffalo
*[http://www.sunysb.edu/chemistry/faculty/gstell.htm George Stell] Stony Brook University
*[http://inka.mssm.edu/~mezei/ Mihaly Mezei] Mount Sinai School of Medicine, New York
*[http://pablonet.princeton.edu/ Pablo Gaston Debenedetti Group] Princeton University
*[http://cherrypit.princeton.edu/index.html Complex Materials Theory Group] (Dir.: Salvatore Torquato) Princeton University
*[http://www.princeton.edu/che/people/faculty/panagiotopoulos/group/ Panagiotopoulos Group Homepage] Princeton University
*[http://www.princeton.edu/~cargroup/ The Car Group] (Dir. Dr. Roberto Car) Princeton University
*[http://polymer.bu.edu/hes/ H. Eugene Stanley] Boston University
*[http://physics.bu.edu/people/show/161 Nicolas Giovambattista] Boston University
*[http://spider.pas.rochester.edu/mainFrame/people/pages/Shapir_Yonathan.html Yonathan Shapir] University of Rochester
*[http://bly.colorado.edu/index.html Liquid Crystal Physics Group] (Dir.: Noel Clark) University of Colorado at Boulder
*[http://www.columbia.edu/cu/chemistry/groups/berne/ Berne Group] Columbia University in the City of New York
*[http://people.chem.byu.edu/doug Douglas J. Henderson] Brigham Young University
*[http://www.chem.umn.edu/groups/siepmann/index.html Siepmann Group] University of Minnesota
*[http://www.engr.wisc.edu/groups/mtsm/ Molecular Thermodynamics and Statistical Mechanics Research Group] (Dir.: Juan J. de Pablo) University of Wisconsin-Madison
*[http://ising.phys.cwru.edu/ Soft Condensed Matter Theory Group of Professor Philip Taylor] Case Western Reserve University
*[http://liq-xtal.cwru.edu/ Case Liquid Crystal and Complex Fluids Group] (Dir.: Charles Rosenblatt) Case Western Reserve University
*[http://www.cchem.berkeley.edu/jmpgrp/index.htm John M. Prausnitz] University of California, Berkeley
*[http://cheme.berkeley.edu/people/faculty/smit/smit.html Berend Smit] University of California Berkeley
*[http://gold.cchem.berkeley.edu:8080/index.html The Chandler Group] University of California, Berkeley
*[http://europa.chem.uga.edu/ Allinger's Molecular Mechanics Research Lab] University of Georgia
*[http://zarbi.chem.yale.edu/ William L. Jorgensen Research Group] Yale University
*[http://www.inl.gov/cams/ Center for Advanced Modeling and Simulation] Idaho National Laboratory
*[http://www.mwdeem.rice.edu/djearl/index.html David J. Earl group] University of Pittsburgh
*[http://www.pitt.edu/~jordan/index.html Ken Jordan Theoretical and Computational Chemistry] University of Pittsburgh
*[http://www.wag.caltech.edu/ Materials and Process Simulation Center] (Dir.: Dr. William A. Goddard III) California Institute of Technology
*[http://dasher.wustl.edu/ Jay Ponder Lab] Washington University School of Medicine
*[http://www.chemistry.wustl.edu/~gelb/ Lev David Gelb Research Group] Washington University in St. Louis
*[http://www.glue.umd.edu/~xpectnil/ Michael E. Fisher] University of Maryland
*[http://www.glue.umd.edu/~jdw/ John D. Weeks] University of Maryland
*[http://www.chem.wisc.edu/~yethiraj/ The Yethiraj group] University of Wisconsin
*[http://www.ksu.edu/chem/people/faculty/smith.html Dr. Paul E. Smith] Kansas State University
*[http://www.chem.unl.edu/faculty/eachfaculty/zeng.shtml Xiao Cheng Zeng] University of Nebraska-Lincoln
*[http://www.chm.colostate.edu/bl/ Branka M. Ladanyi] Colorado State University
*[http://www.engr.ucr.edu/~jwu/ Jianzhong Wu] University of California, Riverside
*[http://tigger.uic.edu/~mansoori/TRL_html Thermodynamics Research Laboratory] (Dir.: Dr. G. Ali Mansoori) University of Illinois at Chicago
*[http://www.chem.cornell.edu/faculty/index.asp?fac=45 Professor Benjamin Widom] Cornell University
*[https://engineering.purdue.edu/ChE/Directory/Faculty/Corti.html David S. Corti] Purdue University
*[http://boltzmann.rockefeller.edu/ E. G. D. Cohen Laboratory] The Rockefeller University
*[http://www.phys.washington.edu/users/thouless/cmt.html Condensed Matter Theory group] University of Washington
*[http://www.math.rutgers.edu/~lebowitz/ Joel L. Lebowitz] Rutgers University
*[http://www.physics.rutgers.edu/cmt/group-cmt.html Theoretical Condensed Matter Physics] Rutgers University
*[http://www.science.duq.edu/faculty/talbot.html Julian Talbot] Duquesne University
*[http://cbme.ou.edu/faculty/lee.htm Lloyd L. Lee] University of Oklahoma
*[http://www.public.asu.edu/~caangell/ C. Austen Angell] Arizona State University
*[http://www.ruf.rice.edu/~saft/ Walter G. Chapman] Rice University
*[http://www.dartmouth.edu/~chem/faculty/JEGL.html Prof. Jane E. G. Lipson] Dartmouth College
*[http://thglab.lbl.gov/ Teresa Head-Gordon's Lab] Lawrence Berkeley National Laboratory
*[http://www.chm.tcu.edu/faculty/huckaby/ Dale A. Huckaby] Texas Christian University
*[http://www.engin.umich.edu/dept/che/research/glotzer/index.html Glotzer group] University of Michigan
*[http://www.nd.edu/~gezelter/Main/ Gezelter Lab] University of Notre Dame
*[http://www.cbms.utah.edu/ Voth Group] University of Utah
*[http://williamhoover.info/ Herr Professor Doctor William Graham Hoover] UC Davis, University of California
*[http://www.egr.msu.edu/~priezjev/ Nikolai Priezjev] Michigan State University
*[http://chemistry.uchicago.edu/fac/rice.shtml Prof. Stuart A. Rice] University of Chicago

[[category: Miscellaneous]]

Ramp model

2009-10-20T14:59:43Z

Noe: Internal link added

The '''ramp model''', proposed by Jagla
<ref>[http://dx.doi.org/10.1063/1.480241 E. A. Jagla "Core-softened potentials and the anomalous properties of water", Journal of Chemical Physics' '''111''' pp. 8980-8986 (1999)]</ref>
and sometimes known as the '''Jagla model''', is described by:

:<math>
\Phi_{12}(r) = \left\{
\begin{array}{ll}
\infty & {\rm if} \; r < \sigma \\
W_r - (W_r-W_a) \frac{r-\sigma}{d_a-\sigma} & {\rm if} \; \sigma \leq r \leq d_a \\
W_a - W_a \frac{r-d_a}{d_c-d_a} & {\rm if} \; d_a < r \leq d_c \\
0 & {\rm if} \; r > d_c
\end{array} \right.
</math>

where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]], <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, <math>W_r > 0</math> and <math>W_a < 0</math>.

Graphically, one has:
[[Image:Ramp_potential.png|350px|center]]
where the red line represents an attractive implementation of the model, and the green line a repulsive implementation.
==Critical points==
For the particular case <math> W_r^*=3.5; W_a^*=-1.0, d_a^*=1.72, d_c^*=3.0 </math>,
the liquid-vapour critical point is located at
<ref name="lomba">
[http://dx.doi.org/10.1063/1.2748043 E. Lomba, N. G. Almarza, C. Martin, C. McBride "Phase behaviour of attractive and repulsive ramp fluids: integral equation and computer simulation studies", Journal of Chemical Physics '''126''' 244510 (2007)]
</ref>:

:<math>T_c^* = 1.487 \pm 0.003</math>
:<math>\rho_c \sigma^3 = 0.103 \pm 0.001</math>
:<math>p_c^* \simeq 0.042</math>

and the [[Polyamorphic systems |liquid-liquid]] critical point:

:<math>T_c^* \simeq 0.378 \pm 0.003</math>
:<math>\rho_c \sigma^3 \simeq 0.380 \pm 0.002</math>
:<math>p_c^*/T_c^* \simeq 0.49 \pm 0.01</math>

== Repulsive Ramp Model ==
In the repulsive ramp case, where <math> W_a = 0 </math>, neither liquid-vapor nor liquid-liquid stable equilibria occur
<ref name="lomba"> reference to Lomba paper</ref>.
However, for this model a low density crystalline phase has been found.
This solid phase presents re-entrant melting, i.e. this solid melts into the fluid phase as the pressure is increased.

==== Lattice gas version ====
Recently, similar behaviour has been found in a three-dimensional Repulsive
Ramp [[lattice gas|Lattice Gas]] model
<ref>
[http://dx.doi.org/10.1080/00268970902729269 Johan Skule Hoye, Enrique Lomba, and Noe Garcia Almarza, "One- and three-dimensional lattice models with two repulsive ranges: simple systems with complex phase behaviour", Molecular Physics '''107''', 321-330 (2009)]
</ref>
The system is defined on a simple cubic lattice. The interaction is that of a [[lattice hard spheres|lattice
hard sphere]] model with exclusion of nearest neighbours of occupied positions plus a repulsive interaction
with next-to-nearest neighbours.
The total potential energy of the system is then given by:

:<math>
U = \epsilon \sum_{[ij]} S_i S_j
</math>

where <math> \epsilon > 0 </math> ; <math> [ij] </math> refers to all the pairs of sites that are
second neighbors, and <math> S_k </math> indicates the occupation of site <math> k </math>
(0 indicates an empty site, 1 indicates an occupied site).

==See also==
*[[Triangular lattice ramp model]]
*[[Polyamorphism: Ramp model]]
==References==
<references />
'''Related literature'''
*[http://dx.doi.org/10.1103/PhysRevE.74.031108 Limei Xu, Sergey V. Buldyrev, C. Austen Angell, and H. Eugene Stanley "Thermodynamics and dynamics of the two-scale spherically symmetric Jagla ramp model of anomalous liquids", Physical Review E '''74''' 031108 (2006)]
*[http://dx.doi.org/10.1063/1.3043665 Limei Xu, Sergey V. Buldyrev, Nicolas Giovambattista, C. Austen Angell, and H. Eugene Stanley "A monatomic system with a liquid-liquid critical point and two distinct glassy states", Journal of Chemical Physics '''130''' 054505 (2009)]
[[Category:models]]
[[category:Polyamorphic systems]]

Talk:Clausius equation of state

2009-10-19T17:53:25Z

Noe:

I think that an EOS like: <math> p (V-Nb) = N R T </math> is also known as Clausius equation of state. (?)

----
--Noe 11:54, 14 October 2009 (CEST)
:I am not sure. I do not have access to the original article, but Equation 2 in a [http://hal.archives-ouvertes.fr/jpa-00237812/en/ (rather badly scanned) paper by J. Violle dated 1881] has a very similar form to the version here on SklogWiki. The best thing would be to find a paper copy of the original by [[Rudolf Julius Emanuel Clausius | Clausius]]. --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 17:24, 19 October 2009 (CEST)

---
Yes, I have also seen that paper, what I mean is that sometimes the same name is used for the e.o.s. quoted above. If I find
some reliable source I will add that information on the page,
----
--Noe 19:53, 19 October 2009 (CEST)

Talk:Clausius equation of state

2009-10-19T17:53:08Z

Noe:

Wohl equation of state

2009-10-19T14:37:57Z

Noe: spelling consistency

The '''Wohl''' [[equations of state| equation of state]] is given by<ref>A. Wohl "Investigation of the condition equation", Zeitschrift für Physikalische Chemie (Leipzig) '''87''' pp. 1-39 (1914) </ref>
:<math>\left[ p + \frac{a}{Tv(v-b)} - \frac{c}{T^2v^3} \right] (v-b) = RT</math>

where

:<math>\left. a \right.= 6P_cT_cv_c^2</math>

:<math>b= \frac{v_c}{4}</math>

and

:<math>\left. c \right. = 4P_cT_c^2v_c^3</math>

where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points | critical]] temperature and <math>P_c</math> is the pressure at the critical point.
==References==
<references/>
[[category: equations of state]]

Equations of state for hard spheres

2009-10-16T17:58:50Z

Noe: link added

The following is a list of [[equations of state]] designed for the [[hard sphere model]]:
*[[Carnahan-Starling equation of state]]
*[[Equations of state for crystals of hard spheres]]
*[[Hard hypersphere equation of state |Hard hyperspheres]]
*[[Kolafa-Labík-Malijevský equation of state]]
*[[Santos-Lopez de Haro hard sphere equation of state]]
*[[Equations of state for crystals of hard spheres]]
==See also==
*[[Exact solution of the Percus Yevick integral equation for hard spheres]]
*[[Equations of state for hard sphere mixtures]]
*[[Equations of state for hard disks]]
==Related reading==
*[http://dx.doi.org/10.1007/978-3-540-78767-9_3 A. Mulero, C.A. Galán, M.I. Parra and F. Cuadros "Equations of State for Hard Spheres and Hard Disks", Lecture Notes in Physics '''753''' Chapter 3 pp.37-109 (2008)]
[[category: equations of state]]

Statistical geometry

2009-10-16T17:53:49Z

Noe: link has been added

*[[Alpha_shape | <math>\alpha</math> shapes]]
*[[Delaunay simplexes]]
*[[Scaled-particle theory]]
*[[Voronoi cells]]
*[[Percolation analysis]]
==References==
#J. D. Bernal "", Proceedings of the Royal Institution of Great Britain '''37''' pp. 355- (1959)
#[http://dx.doi.org/10.1038/183141a0 J. D. Bernal "A Geometrical Approach to the Structure Of Liquids", Nature '''183''' pp. 141-147 (1959)]
#[http://dx.doi.org/10.1098/rspa.1964.0147 J. D. Bernal "The Bakerian Lecture, 1962. The Structure of Liquids", Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences '''280''' pp. 299-322 (1964)]
#[http://dx.doi.org/10.1039/F29777300714 Robin J. Speedy "Accurate theory of the hard sphere fluid", Journal of the Chemical Society, Faraday Transactions 2 '''73''' pp. 714-721 (1977)]
#[http://dx.doi.org/10.1039/F29817700329 Robin J. Speedy "Cavities and free volume in hard-disc and hard-sphere systems", Journal of the Chemical Society, Faraday Transactions 2 '''77''' pp. 329 - 335 (1981)]
# G. Stell "Mayer-Montroll equations (and some variants) through history for fun and profit" in THE WONDERFUL WORLD OF STOCHASTICS A Tribute to Elliott W. Montroll Eds. M.F. Shlesinger G.H. Weiss Studies in Statistical Mechanics, 12 Elsevier (1985)
#[http://dx.doi.org/10.1007/BF00753819 N. N. Medvedev and Yu. I. Naberukhin "Study of the structure of simple liquids and amorphous solids by statistical geometry methods", Journal of Structural Chemistry '''28''' pp. 433-446 (1987)]
#[http://dx.doi.org/10.1021/j100191a005 Howard Reiss "Statistical geometry in the study of fluids and porous media", Journal of Physical Chemistry '''96''' pp. 4736 - 4747 (1992)]
#[http://dx.doi.org/10.1103/PhysRevE.73.010402 J. R. Henderson "Geometric thermodynamic fields and the generalised ensemble in colloidal physics", Physical Review E '''73''' 010402(R) (4 pages) (2006)]
#[http://dx.doi.org/10.1098/rsta.1980.0150 M. O'Keeffe and B. G. Hyde "Plane Nets in Crystal Chemistry", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences '''295''' pp. 553-618 (1980)]
[[category: Statistical geometry]]

Clausius equation of state

2009-10-14T15:12:58Z

Noe:

The '''Clausius equation of state''' is given by

:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT</math>

where

:<math>a = \frac{27R^2T_c^2}{64P_c}</math>

:<math>b= v_c - \frac{RT_c}{4P_c}</math>

and

:<math>c= \frac{3RT_c}{8P_c}-v_c</math>

where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points | critical]] temperature and <math>P_c</math> is the [[pressure]] at the critical point, and <math> v_c </math> is the critical volume per mol.
==References==
#[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Ueber das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]
[[category: equations of state]]

Clausius equation of state

2009-10-14T15:12:43Z

Noe: minor edit

The '''Clausius equation of state''' is given by

:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT</math>

where

:<math>a = \frac{27R^2T_c^2}{64P_c}</math>

:<math>b= v_c - \frac{RT_c}{4P_c}</math>

and

:<math>c= \frac{3RT_c}{8P_c}-v_c</math>

where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points | critical]] temperature and <math>P_c</math> is the [[pressure]] at the critical point, and <math> v_c <math> is the critical volume per mol.
==References==
#[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Ueber das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]
[[category: equations of state]]

Clausius equation of state

2009-10-14T15:11:07Z

Noe: symbol explanations added

The '''Clausius equation of state''' is given by

:<math>\left[ p + \frac{a}{T(v+c)^2}\right] (v-b) =RT</math>

where

:<math>a = \frac{27R^2T_c^2}{64P_c}</math>

:<math>b= v_c - \frac{RT_c}{4P_c}</math>

and

:<math>c= \frac{3RT_c}{8P_c}-v_c</math>

where <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]], <math> v </math> is the volume per mol, and <math>R</math> is the [[molar gas constant]]. <math>T_c</math> is the [[critical points | critical]] temperature and <math>P_c</math> is the [[pressure]] at the critical point. The variables with subindex "c" refer to the critical values.
==References==
#[http://dx.doi.org/10.1002/andp.18802450302 R. Clausius "Ueber das Verhalten der Kohlensäure in Bezug auf Druck, Volumen und Temperatur", Annalen der Physik und Chemie '''9''' pp. 337-357 (1880)]
[[category: equations of state]]

Talk:Clausius equation of state

2009-10-14T09:54:29Z

Noe: New page: I think that an EOS like: <math> p (V-Nb) = N R T </math> is also known as Clausius equation of state. (?) ---- --~~~~

I think that an EOS like: <math> p (V-Nb) = N R T </math> is also known as Clausius equation of state. (?)

----
--Noe 11:54, 14 October 2009 (CEST)

Wikis and Science 2.0

2009-10-07T15:27:07Z

Noe: Undo revision 9103 by Sophiegreen (Talk)

Here is some interesting reading pertaining to wiki, with particular emphasis on their relation to science (in chronological order):
==2005==
*[http://dx.doi.org/10.1038/438548a Declan Butler "Science in the web age: Joint efforts", Nature '''438''' pp. 548-549 1 December (2005)]
<blockquote>"Yet scientists are largely being left behind in this second revolution, as they are proving slow to adopt many of the latest technologies that could help them communicate online more rapidly and collaboratively than they do now."</blockquote>
==2007==
*[http://physicsworld.com/cws/article/print/26711 Martin Griffiths "Talking physics in the social Web", Physics World '''20''' January pp. 24-28 (2007)]
*[http://arxiv.org/abs/cs.DL/0702140 Dennis M. Wilkinson, Bernardo A. Huberman "Assessing the Value of Coooperation in Wikipedia", arxiv.org/abs/cs.DL/0702140]
*[http://dx.doi.org/10.1038/news070226-6 Philip Ball "The more, the wikier", Nature news 27 February 2007]
*[http://dx.doi.org/10.1038/nm0307-231 Brandon Keim "WikiMedia", Nature Medicine '''13''' pp. 231-233 (2007)]
<blockquote>"Uneasy with information websites policed by people with little expertise, scientists are creating their own online encyclopedias"</blockquote>
*[http://www.businessweek.com/print/innovate/content/mar2007/id20070302_219704.htm Don Tapscott and Anthony D. Williams "The New Science of Sharing", BusinessWeek March 2 (2007)]
<blockquote>"...the same technological and demographic forces that are turning the Web into a massive collaborative work space are helping to transform the realm of science into an increasingly open and collaborative endeavor. Yes, the Web was, in fact, invented as a way for scientists to share information. But advances in storage, bandwidth, software, and computing power are pushing collaboration to the next level. Call it Science 2.0."</blockquote>
and
<blockquote>"Leading scientific observers already expect more change in the next 50 years of science than in the last 400 years of inquiry combined. As the pace of science quickens, there will be less value in stashing new scientific ideas, methods, and results in subscription-only journals and databases, and more value in wide-open collaborative-knowledge platforms that are refreshed with each new discovery. These changes will enhance the ability of scientists to find, retrieve, sort, evaluate, and filter the wealth of human knowledge, and, of course, to continue to enlarge and improve it."</blockquote>
*[http://www.theregister.co.uk/2007/03/11/sxsw_science_web_2/ Chris Williams "Scientists shun Web 2.0", The Register 11th March (2007)]
<blockquote>"Science publishers' efforts to have the research community sup the Web 2.0 Kool-Aid have failed, and scientists have given a resounding thumbs down to a gamut of crowd-tapping initiatives, showgoers at SXSW heard on Saturday.
A panel of science web publishers said scientists had consistently shunned wikis, tagging, and social networks, and have even proven reticent to leave comments on web pages."</blockquote>
*[http://www.ctwatch.org/quarterly/articles/2007/08/web-20-in-science/ Timo Hannay "Web 2.0 in Science", CTWatch Quarterly, Volume 3, Number 3, August (2007)]

==2008==
*[http://radar.oreilly.com/archives/2008/01/wikipedia-community-publishing.html Tim O’Reilly "Wikipedia: A community of editors or a community of authors?", January 3 2008]
<blockquote>"This is why publishers should be studying Wikipedia (and YouTube, and Google) -- because they are all showing us the new face of publishing. At their heart, they involve new means of content creation yes, but more profoundly, they involve new means of curation. Wikipedia creates a context within which authors can exercise their skills, displaying their knowledge and their passion. Yes, it allows for collaborative creation, and that's good."</blockquote>
*[http://www.sciam.com/article.cfm?id=science-2-point-0-great-new-tool-or-great-risk&page=1 M. Mitchell Waldrop "Science 2.0: Great New Tool, or Great Risk?", Scientific American January 9 (2008)] (see also: Scientific American May Vol. 298 Issue 5 pp. 68-73 (2008))
<blockquote>"...Web-based "Science 2.0" is not only more collegial than the traditional variety, but considerably more productive."</blockquote>
<blockquote>"Web 2.0 fits so perfectly with the way science works, it's not whether the transition will happen but how fast".</blockquote>
*[http://www.madrimasd.org/informacionidi/noticias/noticia.asp?id=33045 "Los hijos de la Wikipedia", ABC Periódico Electrónico S.A. / notiweb madri+d 28th January (2008)]
*[http://www.newscientist.com/channel/opinion/mg19726473.300-physicists-slam-publishers-over-wikipedia-ban.html?feedId=online-news_rss20 "Physicists slam publishers over Wikipedia ban", NewScientist.com news service 16th March (2008)]
*[http://www.damtp.cam.ac.uk/user/jono/item/toc.html Jonathan Oppenheim "Traditional journals and copyright transfer", 16th March (2008)]
*[http://meta.wikimedia.org/wiki/Wikimedia_press_releases/Wikimedia_Foundation_Supports_Efforts_By_Scientists_to_Use_Free_Licenses "Wikimedia Foundation Supports Efforts By Scientists to Use Free Licenses", 15 May (2008)]
**'''APS "Replies": ([http://publish.aps.org/copyrightFAQ.html#wiki link])
:As the author of an APS-published article, can I post my article or a portion of my article on a web resource like wikipedia or quantiki?
:Sites like wikipedia and quantiki are strict about permissions and require that authors hold copyright to articles that they post there. In order to allow authors to comply with this requirement, APS permits authors to hold copyright to a "derived work" based on an article published in an APS journal as long as the work contains at least 10% new material not covered by APS's copyright and does not contain more than 50% of the text (including equations) of the original article.
*[http://www.publico.es/ciencias/investigacion/131032/auge/internet/impulsa/nueva/ciencia/20 "El auge de Internet impulsa la nueva Ciencia 2.0", Publico.es 1 July (2008)]
*[http://dx.doi.org/10.1038/455273a "Data on display", Nature News 15 September (2008)]
*[http://arxiv.org/abs/0809.3030v1 Bernardo A. Huberman, Daniel M. Romero, Fang Wu "Crowdsourcing, Attention and Productivity", arXiv:0809.3030v1 17 September (2008)]
*[http://dx.doi.org/10.1038/news.2008.1312 Declan Butler "Publish in Wikipedia or perish", Nature News 16 December (2008)]
:"Anyone submitting to a section of the journal RNA Biology will, in the future, be required to also submit a Wikipedia page that summarizes the work. The journal will then peer review the page before publishing it in Wikipedia."
*[http://ways.org/en/blogs/2008/dec/28/the_journal_scope_in_focus_putting_scholarly_communication_in_context Daniel Mietchen "The journal scope in focus -- putting scholarly communication in context", daniel's blog 28th December (2008)]

==2009==
*[http://seedmagazine.com/content/print/scientific_truth_in_the_age_of_wikipedia/ T. J. Kelleher "Does the radical egalitarianism of the wiki undermine traditional notions of scientific authority and consensus?", SEEDMAGAZINE.COM February 9 (2009)]
<blockquote>"...authority and peer review are concepts built into the core of science wikis."</blockquote>
*[http://arXiv.org/abs/0902.3439 Carl McBride "wikiFactor: a measure of the importance of a wiki site", arXiv:0902.3439 19 Feb (2009)]
*[http://dx.doi.org/10.1038/nphys1238 Michael Nielsen "Information awakening", Nature Physics '''5''' pp. 238-240 (2009)]
<blockquote>"Blogs, wikis, open notebooks, InnoCentive and the like are just the beginning of online innovation."</blockquote>
*[http://physicsworld.com/cws/article/print/38904 Michael Nielsen "Doing science in the open", physicsworld.com May 1, (2009)]
<blockquote>"Online networking tools are pervasive, but why have scientists been so slow to adopt many of them? Michael Nielsen explains how we can build a better culture of online collaboration"</blockquote>
*[http://libresoft.es/Members/jfelipe/phd-thesis Felipe Ortega "Wikipedia: A Quantiative Analysis", PhD Thesis, Universidad Rey Juan Carlos (2009).]
[[category: miscellaneous]]

Conferences

2009-10-07T09:58:01Z

Noe: /* September */

The following is a chronological list of conferences, seminars, or meetings related to thermodynamics, statistical mechanics, soft condensed matter, many-body problems, complex fluids etc.
===2009===
====October====
*[http://www.crm.cat/wkstatisticalphysics/ Techniques and Challenges from Statistical Physics] October 14 to 16, 2009 Barcelona, (Spain)
*[http://paginas.fe.up.pt/~equifase/ EQUIFASE 2009] VIII Iberoamerican Conference on Phase Equilibria and Fluid Properties for Process Design, Praia da Rocha, Portugal 17-21 October 2009
*[http://cint.lanl.gov/workshop2009/ Multiple Length Scales in Polymers and Complex Fluids] October 18-21, 2009 Bishop's Lodge, Santa Fe (USA)
*[http://www.cecam.org/workshop-290.html Classical Density Functional Theory Methods in Soft and Hard Matter] October 21, 2009 to October 23, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://seneca.fis.ucm.es/parr/ktlog2_09 kTlog2 '09: Computing Matters Workshop] 22-24 October, 2009, Toledo (Spain)

====November====
*[http://www.dfi.uchile.cl/~granular09/Welcome.html Southern Workshop on Granular Materials 2009 - SWGM09] November 30 to December 4 2009, Viña del Mar (Chile)
*[http://www.lorentzcenter.nl/lc/web/2009/357/info.php3?wsid=357 Subdivide and Tile: Triangulating spaces for understanding the world] Nov 16 to 20 Nov 20 2009, Leiden (Netherlands)

====December====
*[http://www.math.rutgers.edu/events/smm/index.html 102nd Statistical Mechanics Conference] Rutgers University, December 13-15, 2009 (USA)
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP)] 15-17 December 2009 Warwick University (United Kingdom)

===2010===
====January====
*[http://ddays2010.northwestern.edu Dynamics Days] January 4-7, 2010 Evanston Illinois (USA)
====May====
*[http://www.ppeppd2010.cn/ Properties and Phase Equilibria for Product and Process Design] May 16-21, 2010 Suzhou, Jiangsu (China)
====June====
*[http://liblice.icpf.cas.cz/2010/2010.php Eighth Liblice Conference on the Statistical Mechanics of Liquids] June 13-18, 2010 Brno (Czech Republic)
====July====
*[http://ismc2010.ugr.es International Soft Matter Conference 2010] 5th-8th July 2010, Granada, Spain
*[http://www.statphys.org.au StatPhys 24: XXIV International Conference on Statistical Physics of the International Union for Pure and Applied Physics (IUPAP)] 19-23 July, (2010). Cairns (Australia)

====September====
*[http://ergodic.ugr.es/cp 11th Granada Seminar: Foundations of Nonequilibrium Statistical Physics — From Basic Science to Future Challenges] 13-17 September, (2010). La Herradura, Granada (Spain)

==Previous conferences==
=====2007=====
*[http://www.simbioma.cecam.org/ Simulation of Hard Bodies] April 16 to April 19 2007 in Lyon.
*[http://www.math.rutgers.edu/events/smm/ 97th Statistical Mechanics Conference] May 6-8, 2007 Rutgers University
*[http://www.ucm.es/info/molecsim/luis/workshop.htm Workshop on Theory and Computer Simulations of Inhomogenoeus Fluids] May 16-18, 2007, Universidad Complutense, Madrid.
*[http://www-thphys.physics.ox.ac.uk/users/JuliaYeomans/OxfordWorkshop/home.php Mesoscale Modelling for Complex Fluids and Flows] June 25--27, 2007 University of Oxford, UK.
*[http://www.ecns2007.org 4th European Conference on Neutron Scattering] Lund, Sweden 25-29 June 2007
*[http://www.realitygrid.org/CompSci07 Computational Science 2007] 25-26 June 2007 The Royal Society, London
*[https://www.cecam.fr/index.php?content=activities/workshop New directions in liquid state theory] CECAM workshop, 2 - 4 July, 2007 at ENS, Lyon, France.
*[http://www.cecam.fr/index.php?content=activities/workshop&action=details&wid=157 Fluid phase behaviour and critical phenomena from liquid state theories and simulations] 5-7 July 2007 CECAM workshop, Lyon, France.
*[http://www.statphys23.org/ STATPHYS 23] Genova, Italy, from July 9 to 13, 2007.
*[http://www.iupac2007.org/ 41st IUPAC World Chemistry Conference] Turin, Italy, August 5-11th
*[http://www.srcf.ucam.org/~jae1001/ccp5_2007 CCP5 Annual Conference] 29th-31st August 2007 New Hall, Cambridge, UK
*[http://ccp2007.ulb.ac.be CCP 2007] Brussels 5-8 September 2007
*[http://www.castep.org CASTEP Workshop] 17th - 21st September 2007 University of York, UK
*[http://thermo2007.ifp.fr Thermodynamics 2007] 26-28 September 2007, IFP - Rueil-Malmaison (France)
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2007] Corfu, Greece, 25-30 September 2007
*[http://www.chem.unisa.it/polnan/index.html Polymers in Nanotechnology] 27-28th September 2007, Salerno, Italy.
*[http://www.fz-juelich.de/iff/ismc2007/ International Soft Matter Conference 2007] 1 - 4 October 2007, Eurogress, Aachen (Germany)
*[http://www.escet.urjc.es/~fisica/encuentro_complejos/index.html II Meeting on Modelling of Complex Systems] Universidad Rey Juan Carlos, Mostoles (Madrid), October 25-26 2007
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/SCD/event_22663.html Structuring Colloidal Dispersions by External Fields] 21 November 2007, Institute of Physics, London, United Kingdom
*[http://www.math.rutgers.edu/events/smm/ 98th Statistical Mechanics Conference] December 16-18, 2007, Rutgers University, New Jersey, USA.
=====2008=====
*[http://www.iop.org/activity/groups/subject/lcf/Events/file_25833.pdf Recent Advances in the Understanding of Confined Fluids: from Superfluids to Oil Reservoirs] 9-11 January, Cosener's House, Abingdon UK
*[http://academic.sun.ac.za/summerschool/2008.html Workshop on Soft Condensed Matter and Physics of Biological Systems - Perspectives and topics for South Africa] 23 Jan 2008 - 1 Feb 2008 National Institute for Theoretical Physics at Stellenbosch Institute of Advanced Study, Stellenbosch, Western Cape, South Africa
*[http://bifi.unizar.es/events/bifi2008/main.htm Bifi 2008 Large Scale Simulations of Complex Systems, Condensed Matter and Fusion Plasma] 6–8 February 2008, Zaragoza, Spain
*[http://events.dechema.de/Tagungen/MolMod+Workshop.html International Workshop Molecular Modeling and Simulation in Applied Material Science] March 10-11, DECHEMA-Haus, Frankfurt am Main, Germany
*[http://www.aps.org/meetings/march/index.cfm APS March Meeting] March 10-14, 2008. New Orleans, Louisiana, USA
*[http://users.physik.tu-muenchen.de/metz/jerusalem.html Modelling anomalous diffusion and relaxation] 23–28 March 2008, Jerusalem, Israel
*[http://www.newton.cam.ac.uk/programmes/CSM/csmw02.html Markov-Chain Monte Carlo Methods] 25 March to 28 March 2008 Isaac Newton Institute for Mathematical Sciences, Cambridge, UK
*[http://www.usal.es/~fises/ XV Congreso de Física Estadística (Fises' 08)] 27-29 March 2008, Salamanca, Spain
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/CMMP08/event_26545.html Condensed Matter and Materials Physics (CMMP)] 26-28 March 2008 Royal Holloway, University of London, UK
*[http://www.icmab.es/softmatter2008/index.html SoftMatter 2008] "Workshop on Electrostatic Effects in Soft Matter: Bringing Experiments, Theory and Simulation Together" 10 – 11 April ICMAB-CSIC, Barcelona, Spain.
*[http://www.icmab.es/11emscf/index.html 11th European Meeting on Supercritical Fluids] "New Perspectives in Supercritical Fluids: Nanoscience, Materials and Processing" 4-7 May 2008, ICMAB-CSIC, Barcelona, Spain
*[http://www.math.rutgers.edu/events/smm/index.html 99th Statistical Mechanics Conference] May 11-13, 2008 Rutgers University, USA
*[http://www.fondation-pgg.org/events/degennesdays/ DeGennesDays] 15-17 May 2008 Collège de France, Paris.
*[http://www.ill.fr/Events/rktsymposium/ Surfaces and Interfaces in Soft Matter and Biology SISMB 2008] "The impact and future of neutron reflectivity - A Symposium in Honor of Robert K. Thomas" 21-23 May 2008, Institut Laue-Langevin (Grenoble, France).
*[http://www.plmmp.univ.kiev.ua/ Physics of Liquid Matter: Modern Problems] May 23-26, 2008, Kyiv National Taras Shevchenko University, Ukraine
*[http://www.ffn.ub.es/sitges/ XXI Sitges Conference] XXI Sitges Conference on Statistical Mechanics. Statistical Mechanics of Molecular Biophysics 2nd-6th June 2008, Sitges, Spain
*[http://www1.ci.uc.pt/gcpi/poly2008 Polyelectrolytes 2008] 16th-19th June 2008, Coimbra, Portugal
*[http://www-spht.cea.fr/Meetings/BegRohu2008/index.html The Beg Rohu Summer School: Manifolds in random media, random matrices and extreme value statistics] 16th - 28th June 2008, French National Sailing School, Quiberon peninsula, France.
*[http://www.chm.bris.ac.uk/cms/ Computational Molecular Science 2008] 22nd – 25th June 2008, Cirencester, UK.
*[http://www.liquids2008.se/ 7th Liquid Matter Conference] 27 June - 1 July 2008 Lund, Sweden
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School 2008] University of Sheffield, England July 6 - July 15 2008.
*Potentials Workshop: Recent developments in interatomic potentials 7-8 July 2008 Oxford, UK
*[http://www.lcc-toulouse.fr/molmat2008/ MOLMAT2008 International Symposium on Molecular Materials based on Chemistry, Solid State Physics, Theory and Nanotechnology] July 8-11th 2008 Toulouse, France
*[http://www2.polito.it/eventi/sigmaphi2008/ SigmaPhi2008] 14th -18th July 2008 Kolympari, Crete, Greece
*[http://www.cecam.org/workshop-188.html Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers] July, 16th-18th 2008 CECAM-EPFL, Lausanne, Switzerland
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/event_22243.html Molecular Dynamics for Non-Adiabatic Processes] 21 July 2008 to 22 July 2008 Institute of Physics, London
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcitaa_08/announcement.html Quantum Monte Carlo in the Apuan Alps IV] 26th July - Sat 2nd August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.cecam.org/workshop-215.html New directions in the theory and modelling of liquid crystals] July, 28th 2008 to July, 30th 2008, CECAM-EPFL, Lausanne, Switzerland
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcatcp_08/announcement.html Quantum Monte Carlo and the CASINO program III] 3rd August - Sun 10th August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.icct2008.org/ 20th International Conference on Chemical Thermodynamics] August 3-8, 2008, Warsaw, Poland.
*[http://perso.ens-lyon.fr/thierry.dauxois/LORIS/LesHouchesSummerSchool2008.html Long-Range Interacting Systems] Summer School in Les Houches (France), 4-29 August 2008
*[http://ctbp.ucsd.edu/summer_school08/apply2008.html Coarse-Grained Physical Modeling of Biological Systems: Advanced Theory and Methods] August 11-15, 2008 University of California San Diego
*[http://www.chem.ucl.ac.uk/astrosurf/SIPML.html Surface and Interface Processes at the Molecular Level] 17 - 23 August 2008, Lucca, Italy
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD141/index.asp Faraday Discussion 141: Water - From Interfaces to the Bulk] 27 - 29 August 2008 Heriot-Watt University, Edinburgh, United Kingdom
*[http://ectp18.conforganizer.net 18th European Conference on Thermophysical Properties] 31 Aug-4 Sep 2008 Pau, France
*[http://iber2008.df.fct.unl.pt/ 9th Iberian Joint Meeting on Atomic and Molecular Physics - IBER 2008] 7-9th September, Capuchos, Portugal
*[http://www.cmmp.ucl.ac.uk/~dmd/ccp5.htm CCP5 Annual Meeting: Surfaces and Interfaces] 8-10th September, London, UK
*[http://ergodic.ugr.es/cp/ 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials] September 15-19, 2008 Granada, Spain
*[http://www.cecam.org/workshop-222.html Standardisation and databasing of ab-initio and classical simulations] September, 18th 2008 to September, 19th 2008 CECAM-ETHZ, Zurich, Switzerland
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2008 (ICCMSE 2008)] 25-30 September, Crete, Greece
*[http://www.mmm2008.org/bin/view.pl/Main/WebHome Multscale Materials Modeling] 27–31 October 2008, Tallahassee, Florida USA
*[http://www.aiche.org/Conferences/AnnualMeeting/index.aspx 2008 AIChE Annual Meeting] November 16-21 2008 Philadelphia, Pennsylvania USA
*[http://www.ihp.jussieu.fr/ceb/Trimestres/T08-4/C3/index.html Statistical mechanics] Paris (France) 8-12 December 2008
*[http://complex.ffn.ub.es/bcnetworkshop BCNet Workshop] Trends and perspectives in complex networks. Barcelona (Spain) 10-12 December 2008
*[http://www.math.rutgers.edu/events/smm/ 100th Statistical Mechanics Conference] Rutgers University, (USA) 13-18 December 2008
=====2009=====
*[http://www.ccpb.ac.uk/events/conference/2009/ Biomolecular Simulation 2009] 6-8 January, Yorkshire Museum and Gardens, York, United Kingdom 2009
*[http://www.fisica.unam.mx/externos/wintermeeting/ XXXVIII edition of the Winter Meeting on Statistical Physics] Taxco, Guerrero (Mexico) 6th - 9th January, 2009.
*[http://www.ucl.ac.uk/msl/events/2009/workshop09.htm 2009 MSL Workshop: Accessing large length and time scales with accurate quantum methods] 12th - 13th January 2009 University College London (United Kingdom)
*[http://euler.us.es/%7Eopap/stochgame/index-en.html Stochastic Models in Physics, Biology, and Social Sciences] Carmona (Sevilla), Spain February 12-14 (2009)
*[http://hera.physik.uni-konstanz.de/igk/news/workshops/homepage/index.html Frontiers of Soft Condensed Matter 2009] Les Houches () 15-20 February 2009
*[http://www.mpipks-dresden.mpg.de/~mbsffe09/ Many-body systems far from equilibrium: Fluctuations, slow dynamics and long-range interactions] Max Planck Institute for the Physics of Complex Systems Dresden, Germany February 16 - 27 (2009)
*[http://www.formulation.org.uk/Conference_flyers_Sept2007_on/Flyer-sims.pdf Workshop on advances in modelling for formulations] 25th of March 2009 at GSK Waybridge (United Kingdom)
*[http://www.physik.uni-leipzig.de/~janke/meco34/ 34th Conference of the Middle European Cooperation in Statistical Physics] 30 March - 01 April 2009 Universität Leipzig (Germany)
*[http://www.ipam.ucla.edu/programs/ktws2/ Workshop II: The Boltzmann Equation: DiPerna-Lions Plus 20 Years] April 15-17 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-313.html Computational Studies of Defects in Nanoscale Carbon Materials] May 11, 2009 to May 13, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.cecam.org/workshop-320.html Modeling of Carbon and Inorganic Nanotubes and Nanostructures] May 13, 2009 to May 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ima.umn.edu/2008-2009/W5.18-22.09/ Molecular Simulations: Algorithms, Analysis, and Applications] Institute for Mathematics and its Applications, University of Minnesota (USA), May 18-22, 2009
*[http://www.ipam.ucla.edu/programs/ktws4/ Workshop IV: Asymptotic Methods for Dissipative Particle Systems] May 18-22 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-310.html Computer Simulation in Food Science: CFD meets Soft Matter] May 25, 2009 to May 27, 2009 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://complenet09.diit.unict.it CompleNet 2009] International Workshop on Complex Networks. Catania (Italy), May 26-28, 2009
*[http://go.warwick.ac.uk/maths/research/events/2008_2009/symposium/wks5/ EPSRC Symposium Workshop on Molecular Dynamics] Monday 1 – Friday 5 June (2009) Warwick (United Kingdom)
*[http://www.cecam.org/workshop-272.html Theoretical Modeling of Transport in Nanostructures] June 2, 2009 to June 5, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.mpip-mainz.mpg.de/mmsd/ Mainz Materials Simulation Days] 3-5 June 2009 Max Planck Institute for Polymer Research (Germany)
*[http://www.soms.ethz.ch/workshop2009 Coping with Crises in Complex Socio-Economic Systems] ETH Zurich (Switzerland), June 8-13, 2009
*[[MOSSNOHO Workshop 2009]] 16 of June 2009, Universidad Complutense de Madrid (Spain)
*[http://cftc.cii.fc.ul.pt/Workshops/flowNjam/index.html Flow(ers) and jam(mers): from liquid crystals to grains] Lisbon (Portugal) 17-19 June 2009
*[http://ipht.cea.fr/Meetings/BegRohu2009/ Beg Rohu Summer School: Quantum Physics Out of Equilibrium] Ecole Nationale de Voile (France) 15-27 June 2009
*[http://symp17.boulder.nist.gov Seventeenth Symposium on Thermophysical Properties] Boulder, Colorado (USA), June 21-26, 2009
*[http://www.fhi-berlin.mpg.de/th/Meetings/DFT-workshop-Berlin2009/ Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions] Berlin (Germany) June 22-July 1 2009
*[https://www.irphe.univ-mrs.fr/~fe09 Fluid and Elasticity] Carry-le-Rouet, near Marseilles (France) June 23-26, 2009
*[http://www.icmp.lviv.ua/statphys2009/ Statistical Physics: Modern Trends and Applications] June 23-25, 2009 Lviv, (Ukraine)
*[http://www.fuw.edu.pl/~wssph/ 3rd Warsaw School of Statistical Physics] Kazimierz Dolny (Poland), 27 June - 4 July, 2009
*[http://www.cecam.org/workshop-298.html Modeling and Simulation of Water at Interfaces from Ambient to Supercooled Conditions] June 29, 2009 to July 1, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School] University of Sheffield (England) July 5 - July 14 2009
*[http://math.arizona.edu/~goriely/leshouches/home.html New trends in the Physics and Mechanics of Biological Systems"] Les Houches (France), 6-31 July 2009
*[http://research.yale.edu/boulder/Boulder-2009/index.html Nonequilibrium Statistical Mechanics: Fundamental Problems and Applications] July 6-24 2009 Boulder (USA)
*[http://www.xrqtc.cat/index.php/ca/homew New trends in Computational Chemistry for Industry Applications] July 6-7 Barcelona (Spain)
*[http://www.frias.uni-freiburg.de/BFF Computational Methods for Soft Matter and Biological Systems] July 8-11, 2009, FRIAS, Freiburg (Germany)
*[http://www.cecam.org/workshop-286.html Structural Transitions in Solids: Theory, Simulations, Experiments and Visualization Techniques] July 8, 2009 to July 11, 2009 CECAM-USI, Lugano (Switzerland)
*[http://fomms.org FOMMS 2009, Fourth International Conference Foundations of Molecular Modeling and Simulation] Semiahmoo Resort, Blaine, WA (USA) 12-16 July 2009
*[http://www.cecam.org/workshop-308.html Computer Simulation Approaches to Study Self-Assembly: From Patchy Nano-Colloids to Virus Capsids] July 13, 2009 to July 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.md-net.org.uk/events/bath_jul_2009.htm EPSRC Network Mathematical Challenges of Molecular Dynamics] 13-15 July (2009) Bath, United Kingdom.
*[http://www.cecam.org/workshop-279.html New Trends in Simulating Colloids: from Models to Applications] July 15, 2009 to July 18, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD144/index.asp Faraday Discussion 144: Multiscale Modelling of Soft Matter] University of Groningen (The Netherlands) 20 - 22 July 2009
*[http://www2.yukawa.kyoto-u.ac.jp/~ykis2009/Welcome.html Frontiers in Nonequilibrium Physics: Fundamental Theory, Glassy & Granular Materials, and Computational Physics] July 21 - August 21, 2009 Kyoto (Japan)
*[http://www.cecam.org/workshop-293.html Fundamental Aspects of Deterministic Thermostats] July 27, 2009 to July 29, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program IV] 2nd-9th August 2009 Towler Institute, Tuscany, (Italy)
*[http://www.warwick.ac.uk/go/quantsim09 EPSRC Symposium Workshop on Quantum Simulations] 24th-28th August 2009 University of Warwick (United Kingdom)
*[http://itf.fys.kuleuven.be/~fpspXII/ Fundamental Problems in Statistical Physics XII] August 31 - September 11, 2009 Leuven (Belgium)
*[http://denali.phys.uniroma1.it/~idmrcs6/ 6th International Discussion Meeting on Relaxations in Complex Systems] August 31- September 5, 2009, Roma (Italy)
*[http://www.dft09.org/ International Conference on the Applications of Density Functional Theory in Chemistry and Physics] August 31st to September 4th 2009 Lyon (France)
*[http://www.dfrl.ucl.ac.uk/CCP5/ccp5.htm CCP5 Annual Meeting 2009 Structure Prediction] 7th to 9th, September 2009 London (United Kingdom)
*[http://fises.dfa.uhu.es/fises09/ XVI Congreso de Física Estadística] Huelva, 10-12 September 2009, Spain
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/ 22nd Marian Smoluchowski Symposium On Statistical Physics] 12-17 September 2009 Zakopane (Poland)
*[http://www.isis.rl.ac.uk/largescale/loq/SAS2009/SAS2009.htm SAS-2009] XIV International Conference on Small-Angle Scattering, Sunday 13 - Friday 18 September, 2009, Oxford (UK)
*[http://www.esc.sandia.gov/dsmc09/dsmc09.html Direct Simulation Monte Carlo workshop] September 13-16, 2009 Santa Fe, New Mexico, USA,
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/PHYSASPECTS_09/index.html Physical Aspects of Polymer Science] 14-16 September 2009 Bristol (United Kingdom)
*[http://www.thermodynamics2009.org/ Thermodynamics 2009] September 23-25 Imperial College London , U.K. (2009)

__NOTOC__
[[category: Miscellaneous]]

Conferences

2009-10-07T09:57:10Z

Noe: /* July */ granada seminar, sept 2010 added

The following is a chronological list of conferences, seminars, or meetings related to thermodynamics, statistical mechanics, soft condensed matter, many-body problems, complex fluids etc.
===2009===
====October====
*[http://www.crm.cat/wkstatisticalphysics/ Techniques and Challenges from Statistical Physics] October 14 to 16, 2009 Barcelona, (Spain)
*[http://paginas.fe.up.pt/~equifase/ EQUIFASE 2009] VIII Iberoamerican Conference on Phase Equilibria and Fluid Properties for Process Design, Praia da Rocha, Portugal 17-21 October 2009
*[http://cint.lanl.gov/workshop2009/ Multiple Length Scales in Polymers and Complex Fluids] October 18-21, 2009 Bishop's Lodge, Santa Fe (USA)
*[http://www.cecam.org/workshop-290.html Classical Density Functional Theory Methods in Soft and Hard Matter] October 21, 2009 to October 23, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://seneca.fis.ucm.es/parr/ktlog2_09 kTlog2 '09: Computing Matters Workshop] 22-24 October, 2009, Toledo (Spain)

====November====
*[http://www.dfi.uchile.cl/~granular09/Welcome.html Southern Workshop on Granular Materials 2009 - SWGM09] November 30 to December 4 2009, Viña del Mar (Chile)
*[http://www.lorentzcenter.nl/lc/web/2009/357/info.php3?wsid=357 Subdivide and Tile: Triangulating spaces for understanding the world] Nov 16 to 20 Nov 20 2009, Leiden (Netherlands)

====December====
*[http://www.math.rutgers.edu/events/smm/index.html 102nd Statistical Mechanics Conference] Rutgers University, December 13-15, 2009 (USA)
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP)] 15-17 December 2009 Warwick University (United Kingdom)

===2010===
====January====
*[http://ddays2010.northwestern.edu Dynamics Days] January 4-7, 2010 Evanston Illinois (USA)
====May====
*[http://www.ppeppd2010.cn/ Properties and Phase Equilibria for Product and Process Design] May 16-21, 2010 Suzhou, Jiangsu (China)
====June====
*[http://liblice.icpf.cas.cz/2010/2010.php Eighth Liblice Conference on the Statistical Mechanics of Liquids] June 13-18, 2010 Brno (Czech Republic)
====July====
*[http://ismc2010.ugr.es International Soft Matter Conference 2010] 5th-8th July 2010, Granada, Spain
*[http://www.statphys.org.au StatPhys 24: XXIV International Conference on Statistical Physics of the International Union for Pure and Applied Physics (IUPAP)] 19-23 July, (2010). Cairns (Australia)

====September====
*[http://ergodic.ugr.es/cp Granada Seminar: Foundations of Nonequilibrium Statistical Physics — From Basic Science to Future Challenges] 13-17 September, (2010). La Herradura, Granada (Spain)

==Previous conferences==
=====2007=====
*[http://www.simbioma.cecam.org/ Simulation of Hard Bodies] April 16 to April 19 2007 in Lyon.
*[http://www.math.rutgers.edu/events/smm/ 97th Statistical Mechanics Conference] May 6-8, 2007 Rutgers University
*[http://www.ucm.es/info/molecsim/luis/workshop.htm Workshop on Theory and Computer Simulations of Inhomogenoeus Fluids] May 16-18, 2007, Universidad Complutense, Madrid.
*[http://www-thphys.physics.ox.ac.uk/users/JuliaYeomans/OxfordWorkshop/home.php Mesoscale Modelling for Complex Fluids and Flows] June 25--27, 2007 University of Oxford, UK.
*[http://www.ecns2007.org 4th European Conference on Neutron Scattering] Lund, Sweden 25-29 June 2007
*[http://www.realitygrid.org/CompSci07 Computational Science 2007] 25-26 June 2007 The Royal Society, London
*[https://www.cecam.fr/index.php?content=activities/workshop New directions in liquid state theory] CECAM workshop, 2 - 4 July, 2007 at ENS, Lyon, France.
*[http://www.cecam.fr/index.php?content=activities/workshop&action=details&wid=157 Fluid phase behaviour and critical phenomena from liquid state theories and simulations] 5-7 July 2007 CECAM workshop, Lyon, France.
*[http://www.statphys23.org/ STATPHYS 23] Genova, Italy, from July 9 to 13, 2007.
*[http://www.iupac2007.org/ 41st IUPAC World Chemistry Conference] Turin, Italy, August 5-11th
*[http://www.srcf.ucam.org/~jae1001/ccp5_2007 CCP5 Annual Conference] 29th-31st August 2007 New Hall, Cambridge, UK
*[http://ccp2007.ulb.ac.be CCP 2007] Brussels 5-8 September 2007
*[http://www.castep.org CASTEP Workshop] 17th - 21st September 2007 University of York, UK
*[http://thermo2007.ifp.fr Thermodynamics 2007] 26-28 September 2007, IFP - Rueil-Malmaison (France)
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2007] Corfu, Greece, 25-30 September 2007
*[http://www.chem.unisa.it/polnan/index.html Polymers in Nanotechnology] 27-28th September 2007, Salerno, Italy.
*[http://www.fz-juelich.de/iff/ismc2007/ International Soft Matter Conference 2007] 1 - 4 October 2007, Eurogress, Aachen (Germany)
*[http://www.escet.urjc.es/~fisica/encuentro_complejos/index.html II Meeting on Modelling of Complex Systems] Universidad Rey Juan Carlos, Mostoles (Madrid), October 25-26 2007
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/SCD/event_22663.html Structuring Colloidal Dispersions by External Fields] 21 November 2007, Institute of Physics, London, United Kingdom
*[http://www.math.rutgers.edu/events/smm/ 98th Statistical Mechanics Conference] December 16-18, 2007, Rutgers University, New Jersey, USA.
=====2008=====
*[http://www.iop.org/activity/groups/subject/lcf/Events/file_25833.pdf Recent Advances in the Understanding of Confined Fluids: from Superfluids to Oil Reservoirs] 9-11 January, Cosener's House, Abingdon UK
*[http://academic.sun.ac.za/summerschool/2008.html Workshop on Soft Condensed Matter and Physics of Biological Systems - Perspectives and topics for South Africa] 23 Jan 2008 - 1 Feb 2008 National Institute for Theoretical Physics at Stellenbosch Institute of Advanced Study, Stellenbosch, Western Cape, South Africa
*[http://bifi.unizar.es/events/bifi2008/main.htm Bifi 2008 Large Scale Simulations of Complex Systems, Condensed Matter and Fusion Plasma] 6–8 February 2008, Zaragoza, Spain
*[http://events.dechema.de/Tagungen/MolMod+Workshop.html International Workshop Molecular Modeling and Simulation in Applied Material Science] March 10-11, DECHEMA-Haus, Frankfurt am Main, Germany
*[http://www.aps.org/meetings/march/index.cfm APS March Meeting] March 10-14, 2008. New Orleans, Louisiana, USA
*[http://users.physik.tu-muenchen.de/metz/jerusalem.html Modelling anomalous diffusion and relaxation] 23–28 March 2008, Jerusalem, Israel
*[http://www.newton.cam.ac.uk/programmes/CSM/csmw02.html Markov-Chain Monte Carlo Methods] 25 March to 28 March 2008 Isaac Newton Institute for Mathematical Sciences, Cambridge, UK
*[http://www.usal.es/~fises/ XV Congreso de Física Estadística (Fises' 08)] 27-29 March 2008, Salamanca, Spain
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/CMMP08/event_26545.html Condensed Matter and Materials Physics (CMMP)] 26-28 March 2008 Royal Holloway, University of London, UK
*[http://www.icmab.es/softmatter2008/index.html SoftMatter 2008] "Workshop on Electrostatic Effects in Soft Matter: Bringing Experiments, Theory and Simulation Together" 10 – 11 April ICMAB-CSIC, Barcelona, Spain.
*[http://www.icmab.es/11emscf/index.html 11th European Meeting on Supercritical Fluids] "New Perspectives in Supercritical Fluids: Nanoscience, Materials and Processing" 4-7 May 2008, ICMAB-CSIC, Barcelona, Spain
*[http://www.math.rutgers.edu/events/smm/index.html 99th Statistical Mechanics Conference] May 11-13, 2008 Rutgers University, USA
*[http://www.fondation-pgg.org/events/degennesdays/ DeGennesDays] 15-17 May 2008 Collège de France, Paris.
*[http://www.ill.fr/Events/rktsymposium/ Surfaces and Interfaces in Soft Matter and Biology SISMB 2008] "The impact and future of neutron reflectivity - A Symposium in Honor of Robert K. Thomas" 21-23 May 2008, Institut Laue-Langevin (Grenoble, France).
*[http://www.plmmp.univ.kiev.ua/ Physics of Liquid Matter: Modern Problems] May 23-26, 2008, Kyiv National Taras Shevchenko University, Ukraine
*[http://www.ffn.ub.es/sitges/ XXI Sitges Conference] XXI Sitges Conference on Statistical Mechanics. Statistical Mechanics of Molecular Biophysics 2nd-6th June 2008, Sitges, Spain
*[http://www1.ci.uc.pt/gcpi/poly2008 Polyelectrolytes 2008] 16th-19th June 2008, Coimbra, Portugal
*[http://www-spht.cea.fr/Meetings/BegRohu2008/index.html The Beg Rohu Summer School: Manifolds in random media, random matrices and extreme value statistics] 16th - 28th June 2008, French National Sailing School, Quiberon peninsula, France.
*[http://www.chm.bris.ac.uk/cms/ Computational Molecular Science 2008] 22nd – 25th June 2008, Cirencester, UK.
*[http://www.liquids2008.se/ 7th Liquid Matter Conference] 27 June - 1 July 2008 Lund, Sweden
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School 2008] University of Sheffield, England July 6 - July 15 2008.
*Potentials Workshop: Recent developments in interatomic potentials 7-8 July 2008 Oxford, UK
*[http://www.lcc-toulouse.fr/molmat2008/ MOLMAT2008 International Symposium on Molecular Materials based on Chemistry, Solid State Physics, Theory and Nanotechnology] July 8-11th 2008 Toulouse, France
*[http://www2.polito.it/eventi/sigmaphi2008/ SigmaPhi2008] 14th -18th July 2008 Kolympari, Crete, Greece
*[http://www.cecam.org/workshop-188.html Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers] July, 16th-18th 2008 CECAM-EPFL, Lausanne, Switzerland
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/event_22243.html Molecular Dynamics for Non-Adiabatic Processes] 21 July 2008 to 22 July 2008 Institute of Physics, London
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcitaa_08/announcement.html Quantum Monte Carlo in the Apuan Alps IV] 26th July - Sat 2nd August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.cecam.org/workshop-215.html New directions in the theory and modelling of liquid crystals] July, 28th 2008 to July, 30th 2008, CECAM-EPFL, Lausanne, Switzerland
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcatcp_08/announcement.html Quantum Monte Carlo and the CASINO program III] 3rd August - Sun 10th August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.icct2008.org/ 20th International Conference on Chemical Thermodynamics] August 3-8, 2008, Warsaw, Poland.
*[http://perso.ens-lyon.fr/thierry.dauxois/LORIS/LesHouchesSummerSchool2008.html Long-Range Interacting Systems] Summer School in Les Houches (France), 4-29 August 2008
*[http://ctbp.ucsd.edu/summer_school08/apply2008.html Coarse-Grained Physical Modeling of Biological Systems: Advanced Theory and Methods] August 11-15, 2008 University of California San Diego
*[http://www.chem.ucl.ac.uk/astrosurf/SIPML.html Surface and Interface Processes at the Molecular Level] 17 - 23 August 2008, Lucca, Italy
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD141/index.asp Faraday Discussion 141: Water - From Interfaces to the Bulk] 27 - 29 August 2008 Heriot-Watt University, Edinburgh, United Kingdom
*[http://ectp18.conforganizer.net 18th European Conference on Thermophysical Properties] 31 Aug-4 Sep 2008 Pau, France
*[http://iber2008.df.fct.unl.pt/ 9th Iberian Joint Meeting on Atomic and Molecular Physics - IBER 2008] 7-9th September, Capuchos, Portugal
*[http://www.cmmp.ucl.ac.uk/~dmd/ccp5.htm CCP5 Annual Meeting: Surfaces and Interfaces] 8-10th September, London, UK
*[http://ergodic.ugr.es/cp/ 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials] September 15-19, 2008 Granada, Spain
*[http://www.cecam.org/workshop-222.html Standardisation and databasing of ab-initio and classical simulations] September, 18th 2008 to September, 19th 2008 CECAM-ETHZ, Zurich, Switzerland
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2008 (ICCMSE 2008)] 25-30 September, Crete, Greece
*[http://www.mmm2008.org/bin/view.pl/Main/WebHome Multscale Materials Modeling] 27–31 October 2008, Tallahassee, Florida USA
*[http://www.aiche.org/Conferences/AnnualMeeting/index.aspx 2008 AIChE Annual Meeting] November 16-21 2008 Philadelphia, Pennsylvania USA
*[http://www.ihp.jussieu.fr/ceb/Trimestres/T08-4/C3/index.html Statistical mechanics] Paris (France) 8-12 December 2008
*[http://complex.ffn.ub.es/bcnetworkshop BCNet Workshop] Trends and perspectives in complex networks. Barcelona (Spain) 10-12 December 2008
*[http://www.math.rutgers.edu/events/smm/ 100th Statistical Mechanics Conference] Rutgers University, (USA) 13-18 December 2008
=====2009=====
*[http://www.ccpb.ac.uk/events/conference/2009/ Biomolecular Simulation 2009] 6-8 January, Yorkshire Museum and Gardens, York, United Kingdom 2009
*[http://www.fisica.unam.mx/externos/wintermeeting/ XXXVIII edition of the Winter Meeting on Statistical Physics] Taxco, Guerrero (Mexico) 6th - 9th January, 2009.
*[http://www.ucl.ac.uk/msl/events/2009/workshop09.htm 2009 MSL Workshop: Accessing large length and time scales with accurate quantum methods] 12th - 13th January 2009 University College London (United Kingdom)
*[http://euler.us.es/%7Eopap/stochgame/index-en.html Stochastic Models in Physics, Biology, and Social Sciences] Carmona (Sevilla), Spain February 12-14 (2009)
*[http://hera.physik.uni-konstanz.de/igk/news/workshops/homepage/index.html Frontiers of Soft Condensed Matter 2009] Les Houches () 15-20 February 2009
*[http://www.mpipks-dresden.mpg.de/~mbsffe09/ Many-body systems far from equilibrium: Fluctuations, slow dynamics and long-range interactions] Max Planck Institute for the Physics of Complex Systems Dresden, Germany February 16 - 27 (2009)
*[http://www.formulation.org.uk/Conference_flyers_Sept2007_on/Flyer-sims.pdf Workshop on advances in modelling for formulations] 25th of March 2009 at GSK Waybridge (United Kingdom)
*[http://www.physik.uni-leipzig.de/~janke/meco34/ 34th Conference of the Middle European Cooperation in Statistical Physics] 30 March - 01 April 2009 Universität Leipzig (Germany)
*[http://www.ipam.ucla.edu/programs/ktws2/ Workshop II: The Boltzmann Equation: DiPerna-Lions Plus 20 Years] April 15-17 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-313.html Computational Studies of Defects in Nanoscale Carbon Materials] May 11, 2009 to May 13, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.cecam.org/workshop-320.html Modeling of Carbon and Inorganic Nanotubes and Nanostructures] May 13, 2009 to May 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ima.umn.edu/2008-2009/W5.18-22.09/ Molecular Simulations: Algorithms, Analysis, and Applications] Institute for Mathematics and its Applications, University of Minnesota (USA), May 18-22, 2009
*[http://www.ipam.ucla.edu/programs/ktws4/ Workshop IV: Asymptotic Methods for Dissipative Particle Systems] May 18-22 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-310.html Computer Simulation in Food Science: CFD meets Soft Matter] May 25, 2009 to May 27, 2009 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://complenet09.diit.unict.it CompleNet 2009] International Workshop on Complex Networks. Catania (Italy), May 26-28, 2009
*[http://go.warwick.ac.uk/maths/research/events/2008_2009/symposium/wks5/ EPSRC Symposium Workshop on Molecular Dynamics] Monday 1 – Friday 5 June (2009) Warwick (United Kingdom)
*[http://www.cecam.org/workshop-272.html Theoretical Modeling of Transport in Nanostructures] June 2, 2009 to June 5, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.mpip-mainz.mpg.de/mmsd/ Mainz Materials Simulation Days] 3-5 June 2009 Max Planck Institute for Polymer Research (Germany)
*[http://www.soms.ethz.ch/workshop2009 Coping with Crises in Complex Socio-Economic Systems] ETH Zurich (Switzerland), June 8-13, 2009
*[[MOSSNOHO Workshop 2009]] 16 of June 2009, Universidad Complutense de Madrid (Spain)
*[http://cftc.cii.fc.ul.pt/Workshops/flowNjam/index.html Flow(ers) and jam(mers): from liquid crystals to grains] Lisbon (Portugal) 17-19 June 2009
*[http://ipht.cea.fr/Meetings/BegRohu2009/ Beg Rohu Summer School: Quantum Physics Out of Equilibrium] Ecole Nationale de Voile (France) 15-27 June 2009
*[http://symp17.boulder.nist.gov Seventeenth Symposium on Thermophysical Properties] Boulder, Colorado (USA), June 21-26, 2009
*[http://www.fhi-berlin.mpg.de/th/Meetings/DFT-workshop-Berlin2009/ Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions] Berlin (Germany) June 22-July 1 2009
*[https://www.irphe.univ-mrs.fr/~fe09 Fluid and Elasticity] Carry-le-Rouet, near Marseilles (France) June 23-26, 2009
*[http://www.icmp.lviv.ua/statphys2009/ Statistical Physics: Modern Trends and Applications] June 23-25, 2009 Lviv, (Ukraine)
*[http://www.fuw.edu.pl/~wssph/ 3rd Warsaw School of Statistical Physics] Kazimierz Dolny (Poland), 27 June - 4 July, 2009
*[http://www.cecam.org/workshop-298.html Modeling and Simulation of Water at Interfaces from Ambient to Supercooled Conditions] June 29, 2009 to July 1, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School] University of Sheffield (England) July 5 - July 14 2009
*[http://math.arizona.edu/~goriely/leshouches/home.html New trends in the Physics and Mechanics of Biological Systems"] Les Houches (France), 6-31 July 2009
*[http://research.yale.edu/boulder/Boulder-2009/index.html Nonequilibrium Statistical Mechanics: Fundamental Problems and Applications] July 6-24 2009 Boulder (USA)
*[http://www.xrqtc.cat/index.php/ca/homew New trends in Computational Chemistry for Industry Applications] July 6-7 Barcelona (Spain)
*[http://www.frias.uni-freiburg.de/BFF Computational Methods for Soft Matter and Biological Systems] July 8-11, 2009, FRIAS, Freiburg (Germany)
*[http://www.cecam.org/workshop-286.html Structural Transitions in Solids: Theory, Simulations, Experiments and Visualization Techniques] July 8, 2009 to July 11, 2009 CECAM-USI, Lugano (Switzerland)
*[http://fomms.org FOMMS 2009, Fourth International Conference Foundations of Molecular Modeling and Simulation] Semiahmoo Resort, Blaine, WA (USA) 12-16 July 2009
*[http://www.cecam.org/workshop-308.html Computer Simulation Approaches to Study Self-Assembly: From Patchy Nano-Colloids to Virus Capsids] July 13, 2009 to July 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.md-net.org.uk/events/bath_jul_2009.htm EPSRC Network Mathematical Challenges of Molecular Dynamics] 13-15 July (2009) Bath, United Kingdom.
*[http://www.cecam.org/workshop-279.html New Trends in Simulating Colloids: from Models to Applications] July 15, 2009 to July 18, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD144/index.asp Faraday Discussion 144: Multiscale Modelling of Soft Matter] University of Groningen (The Netherlands) 20 - 22 July 2009
*[http://www2.yukawa.kyoto-u.ac.jp/~ykis2009/Welcome.html Frontiers in Nonequilibrium Physics: Fundamental Theory, Glassy & Granular Materials, and Computational Physics] July 21 - August 21, 2009 Kyoto (Japan)
*[http://www.cecam.org/workshop-293.html Fundamental Aspects of Deterministic Thermostats] July 27, 2009 to July 29, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program IV] 2nd-9th August 2009 Towler Institute, Tuscany, (Italy)
*[http://www.warwick.ac.uk/go/quantsim09 EPSRC Symposium Workshop on Quantum Simulations] 24th-28th August 2009 University of Warwick (United Kingdom)
*[http://itf.fys.kuleuven.be/~fpspXII/ Fundamental Problems in Statistical Physics XII] August 31 - September 11, 2009 Leuven (Belgium)
*[http://denali.phys.uniroma1.it/~idmrcs6/ 6th International Discussion Meeting on Relaxations in Complex Systems] August 31- September 5, 2009, Roma (Italy)
*[http://www.dft09.org/ International Conference on the Applications of Density Functional Theory in Chemistry and Physics] August 31st to September 4th 2009 Lyon (France)
*[http://www.dfrl.ucl.ac.uk/CCP5/ccp5.htm CCP5 Annual Meeting 2009 Structure Prediction] 7th to 9th, September 2009 London (United Kingdom)
*[http://fises.dfa.uhu.es/fises09/ XVI Congreso de Física Estadística] Huelva, 10-12 September 2009, Spain
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/ 22nd Marian Smoluchowski Symposium On Statistical Physics] 12-17 September 2009 Zakopane (Poland)
*[http://www.isis.rl.ac.uk/largescale/loq/SAS2009/SAS2009.htm SAS-2009] XIV International Conference on Small-Angle Scattering, Sunday 13 - Friday 18 September, 2009, Oxford (UK)
*[http://www.esc.sandia.gov/dsmc09/dsmc09.html Direct Simulation Monte Carlo workshop] September 13-16, 2009 Santa Fe, New Mexico, USA,
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/PHYSASPECTS_09/index.html Physical Aspects of Polymer Science] 14-16 September 2009 Bristol (United Kingdom)
*[http://www.thermodynamics2009.org/ Thermodynamics 2009] September 23-25 Imperial College London , U.K. (2009)

__NOTOC__
[[category: Miscellaneous]]

Percolation analysis

2009-10-02T13:06:28Z

Noe: /* Lattice and continuum (off-lattice) models */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Physical Review E '''80''', 041101 (2009) [9 pages]] </ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-10-02T13:05:27Z

Noe: /* Lattice and continuum (off-lattice) models */ new ref

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref> <ref name=becker > [http://dx.doi.org/10.1103/PhysRevE.80.041101 Adam M. Becker and Robert M. Ziff, "Percolation thresholds on two-dimensional Voronoi networks and Delaunay triangulations", Phys. Rev. E 80, 041101 (2009) [9 pages]] </ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-10-01T17:18:26Z

Noe: /* Lattice and continuum (off-lattice) models */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more ''realistic'' than lattice models to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-10-01T17:16:57Z

Noe:

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications they are expected to be more realistic to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411 Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results", Journal of Chemical Physics '''89''', 6427 (1988)] </ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-10-01T17:14:35Z

Noe: new reference regarding off-lattice models

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications are expected to be more realistic to capture the physics of a number of real systems <ref name=lee > [http://dx.doi.org/10.1063/1.455411
Sang Bub Lee and S. Torquato, "Pair connectedness and mean cluster size for continuum-percolation models: Computer-simulation results",
Journal of Chemical Physics '''89''', 6427 (1988) ]</ref>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-10-01T17:10:01Z

Noe: /* Sites, bonds, and clusters */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Lattice and continuum (off-lattice) models ==

Attending to the spatial distribution of the sites, one can classify the models into lattice models and continuum (or off-lattice) models.
Off-lattice models are more difficult to deal with from the numerical point of view, but in many applications are expected to be more realistic to capture the physics of a number of real systems <ref name=lee>

== Connectivity rules ==
The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be either deterministic or probabilistic.
In [[statistical mechanics]] applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of [[Ising model]]s using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Triangular lattice ramp model

2009-09-25T09:33:43Z

Noe: updating Refs

{{Stub-general}}
The '''triangular lattice ramp model'''
<ref>[http://arxiv.org/abs/0907.3447 Noe G. Almarza, Jose A. Capitan, Jose A. Cuesta, Enrique Lomba "Phase diagram of a two-dimensional lattice gas model of a ramp system", arXiv:0907.3447v1 20 July (2009)].</ref>
<ref> [http://dx.doi.org/10.1063/1.3223999 Noé G. Almarza, José A. Capitán, José A. Cuesta, and Enrique Lomba, "Phase diagram of a two-dimensional lattice gas model of a ramp system", Journal of Chemical Physics '''131''', 124506 (2009)] .</ref> is a [[Idealised_models#Lattice_models| lattice model]]
that consists of an exclusion of the six nearest neighbour sites of an occupied node, as well as a repulsion for next-nearest neighbours. As such it can be seen as a two dimensional analogue of the [[ramp model]]. In the high [[temperature]] limit this model becomes the [[hard hexagon lattice model]].
==See also==
*[[Building up a triangular lattice]]
==References==
<references/>
[[Category:Models]]

Percolation analysis

2009-09-24T17:04:47Z

Noe: /* Connectivity rules */ bond-percolation

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Probabilistic criteria: Two sites located at a certain distance <math> r </math> are bonded with a given probability <math> b(r) </math>, with
: <math> 0 \le b(r) < 1 </math>. This is the case of the so-called bond-percolation models.

* Energetic criteria (probabilistic): As an example, in the simulation of Ising models using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-24T17:00:06Z

Noe: /* Connectivity rules */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>. This criterium is used in the so-called site percolation models (see below).

* Energetic criteria: As an example, in the simulation of Ising models using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-24T11:42:55Z

Noe: /* Connectivity rules */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: As an example, in the simulation of Ising models using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, have a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-24T09:14:29Z

Noe: /* Connectivity rules */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: As an example, in the simulation of Ising models using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; where <math> u_{ij} </math> is the interaction energy between sites, and <math> \beta = 1/ k_BT</math> (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-24T09:11:00Z

Noe: /* Connectivity rules */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: As an example, in the simulation of Ising models using [[Cluster algorithms|cluster algorithms]]; two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-24T09:10:32Z

Noe: /* Connectivity rules */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: As an example, in the simulation of Ising modeles using [[Cluster algorithms|cluster algorithms]]; rwo sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ 2 \beta u_{ij}(r_{ij}) \right] \right\} </math>; (See [[Cluster algorithms|cluster algorithms]] for details).

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-24T09:06:59Z

Noe: /* Connectivity rules */ change in the energetic bonding criteria

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = 1- \min \left\{ 1, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>; (See [[Cluster algorithms|cluster algorithms]])

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Talk:Percolation analysis

2009-09-24T09:01:50Z

Noe:

Muy interesante.

There must be something wrong with the energetic definition of bonds
since max(0,exp(u)) = exp(u). Posiblemente b=min(1,exp(-u)) ?

lgmac

----
I will check, thanks --Noe 11:01, 24 September 2009 (CEST)

Percolation analysis

2009-09-23T17:02:12Z

Noe: /* Computation of the percolation threshold */ new reference

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref>
<ref name=Newmann> [http://dx.doi.org/10.1103/PhysRevE.64.016706 M. E. J. Newman and R. M. Ziff, "Fast Monte Carlo algorithm for site or bond percolation", Physical Review E '''64''', 016706 (2001) [16 pages] ] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-23T15:55:21Z

Noe: /* Computation of the percolation threshold */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> and <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Physical Review E '''58''', 1521 - 1527 (1998)] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-23T15:54:34Z

Noe: /* Computation of the percolation threshold */ new reference

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Refs. <ref name='deng' > </ref> and <ref name=lin> [http://dx.doi.org/10.1103/PhysRevE.58.1521 Chai-Yu Lin and Chin-Kun Hu, "Universal finite-size scaling functions for percolation on three-dimensional lattices", Phys. Rev. E 58, 1521 - 1527 (1998)] </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-22T14:33:05Z

Noe: /* Example */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example: Site-percolation on a square lattice ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Ref. <ref name='deng' > </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-22T14:30:12Z

Noe: /* Example */ tidying up a bit

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions", Physical Review E '''72''' 016126 (2005)]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty.
* The probability of occupancy of each site is <math> \left. x \right. </math>, with <math> 0 < \left. x \right. < 1 </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Ref. <ref name='deng' > </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Physical Review B ''' 67''' 014102 (2003)] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories",Journal of Physics A: Mathematical and Theoretical '''36''' pp. 4269-4281 (2002)] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Conferences

2009-09-21T16:47:30Z

Noe: /* December */ update of http

The following is a chronological list of conferences, seminars, or meetings related to thermodynamics, statistical mechanics, soft condensed matter, many-body problems, complex fluids etc.
===2009===
====September====
*[http://www.thermodynamics2009.org/ Thermodynamics 2009] September 23-25 Imperial College London , U.K. (2009)
====October====
*[http://paginas.fe.up.pt/~equifase/ EQUIFASE 2009] VIII Iberoamerican Conference on Phase Equilibria and Fluid Properties for Process Design, Praia da Rocha, Portugal 17-21 October 2009
*[http://cint.lanl.gov/workshop2009/ Multiple Length Scales in Polymers and Complex Fluids] October 18- 21, 2009 Bishop's Lodge, Santa Fe (USA)
*[http://www.cecam.org/workshop-290.html Classical Density Functional Theory Methods in Soft and Hard Matter] October 21, 2009 to October 23, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://seneca.fis.ucm.es/parr/ktlog2_09 kTlog2 '09: Computing Matters Workshop] 22-24 October, 2009, Toledo (Spain)
====November====
*[http://www.dfi.uchile.cl/~granular09/Welcome.html Southern Workshop on Granular Materials 2009 - SWGM09] November 30 to December 4 2009, Viña del Mar (Chile)
====December====
*[http://www.math.rutgers.edu/events/smm/index.html 102nd Statistical Mechanics Conference] Rutgers University, December 13-15, 2009 (USA)
*[http://www.cmmp.org.uk/ Condensed Matter and Materials Physics (CMMP)] 15-17 December 2009 Warwick University (United Kingdom)

===2010===
====May====
*[http://www.ppeppd2010.cn/ Properties and Phase Equilibria for Product and Process Design] May 16-21, 2010 Suzhou, Jiangsu (China)
====June====
*[http://liblice.icpf.cas.cz/2010/2010.php Eighth Liblice Conference on the Statistical Mechanics of Liquids] June 13-18, 2010 Brno (Czech Republic)

====July====
*[http://ismc2010.ugr.es International Soft Matter Conference 2010] 5th-8th July 2010, Granada, Spain
*[http://www.statphys.org.au StatPhys 24: XXIV International Conference on Statistical Physics of the International Union for Pure and Applied Physics (IUPAP)] 19-23 July, (2010). Cairns (Australia)

==Previous conferences==
=====2007=====
*[http://www.simbioma.cecam.org/ Simulation of Hard Bodies] April 16 to April 19 2007 in Lyon.
*[http://www.math.rutgers.edu/events/smm/ 97th Statistical Mechanics Conference] May 6-8, 2007 Rutgers University
*[http://www.ucm.es/info/molecsim/luis/workshop.htm Workshop on Theory and Computer Simulations of Inhomogenoeus Fluids] May 16-18, 2007, Universidad Complutense, Madrid.
*[http://www-thphys.physics.ox.ac.uk/users/JuliaYeomans/OxfordWorkshop/home.php Mesoscale Modelling for Complex Fluids and Flows] June 25--27, 2007 University of Oxford, UK.
*[http://www.ecns2007.org 4th European Conference on Neutron Scattering] Lund, Sweden 25-29 June 2007
*[http://www.realitygrid.org/CompSci07 Computational Science 2007] 25-26 June 2007 The Royal Society, London
*[https://www.cecam.fr/index.php?content=activities/workshop New directions in liquid state theory] CECAM workshop, 2 - 4 July, 2007 at ENS, Lyon, France.
*[http://www.cecam.fr/index.php?content=activities/workshop&action=details&wid=157 Fluid phase behaviour and critical phenomena from liquid state theories and simulations] 5-7 July 2007 CECAM workshop, Lyon, France.
*[http://www.statphys23.org/ STATPHYS 23] Genova, Italy, from July 9 to 13, 2007.
*[http://www.iupac2007.org/ 41st IUPAC World Chemistry Conference] Turin, Italy, August 5-11th
*[http://www.srcf.ucam.org/~jae1001/ccp5_2007 CCP5 Annual Conference] 29th-31st August 2007 New Hall, Cambridge, UK
*[http://ccp2007.ulb.ac.be CCP 2007] Brussels 5-8 September 2007
*[http://www.castep.org CASTEP Workshop] 17th - 21st September 2007 University of York, UK
*[http://thermo2007.ifp.fr Thermodynamics 2007] 26-28 September 2007, IFP - Rueil-Malmaison (France)
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2007] Corfu, Greece, 25-30 September 2007
*[http://www.chem.unisa.it/polnan/index.html Polymers in Nanotechnology] 27-28th September 2007, Salerno, Italy.
*[http://www.fz-juelich.de/iff/ismc2007/ International Soft Matter Conference 2007] 1 - 4 October 2007, Eurogress, Aachen (Germany)
*[http://www.escet.urjc.es/~fisica/encuentro_complejos/index.html II Meeting on Modelling of Complex Systems] Universidad Rey Juan Carlos, Mostoles (Madrid), October 25-26 2007
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/SCD/event_22663.html Structuring Colloidal Dispersions by External Fields] 21 November 2007, Institute of Physics, London, United Kingdom
*[http://www.math.rutgers.edu/events/smm/ 98th Statistical Mechanics Conference] December 16-18, 2007, Rutgers University, New Jersey, USA.
=====2008=====
*[http://www.iop.org/activity/groups/subject/lcf/Events/file_25833.pdf Recent Advances in the Understanding of Confined Fluids: from Superfluids to Oil Reservoirs] 9-11 January, Cosener's House, Abingdon UK
*[http://academic.sun.ac.za/summerschool/2008.html Workshop on Soft Condensed Matter and Physics of Biological Systems - Perspectives and topics for South Africa] 23 Jan 2008 - 1 Feb 2008 National Institute for Theoretical Physics at Stellenbosch Institute of Advanced Study, Stellenbosch, Western Cape, South Africa
*[http://bifi.unizar.es/events/bifi2008/main.htm Bifi 2008 Large Scale Simulations of Complex Systems, Condensed Matter and Fusion Plasma] 6–8 February 2008, Zaragoza, Spain
*[http://events.dechema.de/Tagungen/MolMod+Workshop.html International Workshop Molecular Modeling and Simulation in Applied Material Science] March 10-11, DECHEMA-Haus, Frankfurt am Main, Germany
*[http://www.aps.org/meetings/march/index.cfm APS March Meeting] March 10-14, 2008. New Orleans, Louisiana, USA
*[http://users.physik.tu-muenchen.de/metz/jerusalem.html Modelling anomalous diffusion and relaxation] 23–28 March 2008, Jerusalem, Israel
*[http://www.newton.cam.ac.uk/programmes/CSM/csmw02.html Markov-Chain Monte Carlo Methods] 25 March to 28 March 2008 Isaac Newton Institute for Mathematical Sciences, Cambridge, UK
*[http://www.usal.es/~fises/ XV Congreso de Física Estadística (Fises' 08)] 27-29 March 2008, Salamanca, Spain
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/CMMP08/event_26545.html Condensed Matter and Materials Physics (CMMP)] 26-28 March 2008 Royal Holloway, University of London, UK
*[http://www.icmab.es/softmatter2008/index.html SoftMatter 2008] "Workshop on Electrostatic Effects in Soft Matter: Bringing Experiments, Theory and Simulation Together" 10 – 11 April ICMAB-CSIC, Barcelona, Spain.
*[http://www.icmab.es/11emscf/index.html 11th European Meeting on Supercritical Fluids] "New Perspectives in Supercritical Fluids: Nanoscience, Materials and Processing" 4-7 May 2008, ICMAB-CSIC, Barcelona, Spain
*[http://www.math.rutgers.edu/events/smm/index.html 99th Statistical Mechanics Conference] May 11-13, 2008 Rutgers University, USA
*[http://www.fondation-pgg.org/events/degennesdays/ DeGennesDays] 15-17 May 2008 Collège de France, Paris.
*[http://www.ill.fr/Events/rktsymposium/ Surfaces and Interfaces in Soft Matter and Biology SISMB 2008] "The impact and future of neutron reflectivity - A Symposium in Honor of Robert K. Thomas" 21-23 May 2008, Institut Laue-Langevin (Grenoble, France).
*[http://www.plmmp.univ.kiev.ua/ Physics of Liquid Matter: Modern Problems] May 23-26, 2008, Kyiv National Taras Shevchenko University, Ukraine
*[http://www.ffn.ub.es/sitges/ XXI Sitges Conference] XXI Sitges Conference on Statistical Mechanics. Statistical Mechanics of Molecular Biophysics 2nd-6th June 2008, Sitges, Spain
*[http://www1.ci.uc.pt/gcpi/poly2008 Polyelectrolytes 2008] 16th-19th June 2008, Coimbra, Portugal
*[http://www-spht.cea.fr/Meetings/BegRohu2008/index.html The Beg Rohu Summer School: Manifolds in random media, random matrices and extreme value statistics] 16th - 28th June 2008, French National Sailing School, Quiberon peninsula, France.
*[http://www.chm.bris.ac.uk/cms/ Computational Molecular Science 2008] 22nd – 25th June 2008, Cirencester, UK.
*[http://www.liquids2008.se/ 7th Liquid Matter Conference] 27 June - 1 July 2008 Lund, Sweden
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School 2008] University of Sheffield, England July 6 - July 15 2008.
*Potentials Workshop: Recent developments in interatomic potentials 7-8 July 2008 Oxford, UK
*[http://www.lcc-toulouse.fr/molmat2008/ MOLMAT2008 International Symposium on Molecular Materials based on Chemistry, Solid State Physics, Theory and Nanotechnology] July 8-11th 2008 Toulouse, France
*[http://www2.polito.it/eventi/sigmaphi2008/ SigmaPhi2008] 14th -18th July 2008 Kolympari, Crete, Greece
*[http://www.cecam.org/workshop-188.html Dissipative Particle Dynamics: Addressing deficiencies and establishing new frontiers] July, 16th-18th 2008 CECAM-EPFL, Lausanne, Switzerland
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/event_22243.html Molecular Dynamics for Non-Adiabatic Processes] 21 July 2008 to 22 July 2008 Institute of Physics, London
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcitaa_08/announcement.html Quantum Monte Carlo in the Apuan Alps IV] 26th July - Sat 2nd August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.cecam.org/workshop-215.html New directions in the theory and modelling of liquid crystals] July, 28th 2008 to July, 30th 2008, CECAM-EPFL, Lausanne, Switzerland
*[http://www.vallico.net/tti/master.html?http://www.vallico.net/tti/qmcatcp_08/announcement.html Quantum Monte Carlo and the CASINO program III] 3rd August - Sun 10th August The Towler Institute, Vallico Sotto, Tuscany, Italy
*[http://www.icct2008.org/ 20th International Conference on Chemical Thermodynamics] August 3-8, 2008, Warsaw, Poland.
*[http://perso.ens-lyon.fr/thierry.dauxois/LORIS/LesHouchesSummerSchool2008.html Long-Range Interacting Systems] Summer School in Les Houches (France), 4-29 August 2008
*[http://ctbp.ucsd.edu/summer_school08/apply2008.html Coarse-Grained Physical Modeling of Biological Systems: Advanced Theory and Methods] August 11-15, 2008 University of California San Diego
*[http://www.chem.ucl.ac.uk/astrosurf/SIPML.html Surface and Interface Processes at the Molecular Level] 17 - 23 August 2008, Lucca, Italy
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD141/index.asp Faraday Discussion 141: Water - From Interfaces to the Bulk] 27 - 29 August 2008 Heriot-Watt University, Edinburgh, United Kingdom
*[http://ectp18.conforganizer.net 18th European Conference on Thermophysical Properties] 31 Aug-4 Sep 2008 Pau, France
*[http://iber2008.df.fct.unl.pt/ 9th Iberian Joint Meeting on Atomic and Molecular Physics - IBER 2008] 7-9th September, Capuchos, Portugal
*[http://www.cmmp.ucl.ac.uk/~dmd/ccp5.htm CCP5 Annual Meeting: Surfaces and Interfaces] 8-10th September, London, UK
*[http://ergodic.ugr.es/cp/ 10th Granada Seminar on Computational and Statistical Physics: Modeling and Simulation of New Materials] September 15-19, 2008 Granada, Spain
*[http://www.cecam.org/workshop-222.html Standardisation and databasing of ab-initio and classical simulations] September, 18th 2008 to September, 19th 2008 CECAM-ETHZ, Zurich, Switzerland
*[http://www.iccmse.org/ International Conference of Computational Methods in Sciences and Engineering 2008 (ICCMSE 2008)] 25-30 September, Crete, Greece
*[http://www.mmm2008.org/bin/view.pl/Main/WebHome Multscale Materials Modeling] 27–31 October 2008, Tallahassee, Florida USA
*[http://www.aiche.org/Conferences/AnnualMeeting/index.aspx 2008 AIChE Annual Meeting] November 16-21 2008 Philadelphia, Pennsylvania USA
*[http://www.ihp.jussieu.fr/ceb/Trimestres/T08-4/C3/index.html Statistical mechanics] Paris (France) 8-12 December 2008
*[http://complex.ffn.ub.es/bcnetworkshop BCNet Workshop] Trends and perspectives in complex networks. Barcelona (Spain) 10-12 December 2008
*[http://www.math.rutgers.edu/events/smm/ 100th Statistical Mechanics Conference] Rutgers University, (USA) 13-18 December 2008
=====2009=====
*[http://www.ccpb.ac.uk/events/conference/2009/ Biomolecular Simulation 2009] 6-8 January, Yorkshire Museum and Gardens, York, United Kingdom 2009
*[http://www.fisica.unam.mx/externos/wintermeeting/ XXXVIII edition of the Winter Meeting on Statistical Physics] Taxco, Guerrero (Mexico) 6th - 9th January, 2009.
*[http://www.ucl.ac.uk/msl/events/2009/workshop09.htm 2009 MSL Workshop: Accessing large length and time scales with accurate quantum methods] 12th - 13th January 2009 University College London (United Kingdom)
*[http://euler.us.es/%7Eopap/stochgame/index-en.html Stochastic Models in Physics, Biology, and Social Sciences] Carmona (Sevilla), Spain February 12-14 (2009)
*[http://hera.physik.uni-konstanz.de/igk/news/workshops/homepage/index.html Frontiers of Soft Condensed Matter 2009] Les Houches () 15-20 February 2009
*[http://www.mpipks-dresden.mpg.de/~mbsffe09/ Many-body systems far from equilibrium: Fluctuations, slow dynamics and long-range interactions] Max Planck Institute for the Physics of Complex Systems Dresden, Germany February 16 - 27 (2009)
*[http://www.formulation.org.uk/Conference_flyers_Sept2007_on/Flyer-sims.pdf Workshop on advances in modelling for formulations] 25th of March 2009 at GSK Waybridge (United Kingdom)
*[http://www.physik.uni-leipzig.de/~janke/meco34/ 34th Conference of the Middle European Cooperation in Statistical Physics] 30 March - 01 April 2009 Universität Leipzig (Germany)
*[http://www.ipam.ucla.edu/programs/ktws2/ Workshop II: The Boltzmann Equation: DiPerna-Lions Plus 20 Years] April 15-17 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-313.html Computational Studies of Defects in Nanoscale Carbon Materials] May 11, 2009 to May 13, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.cecam.org/workshop-320.html Modeling of Carbon and Inorganic Nanotubes and Nanostructures] May 13, 2009 to May 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ima.umn.edu/2008-2009/W5.18-22.09/ Molecular Simulations: Algorithms, Analysis, and Applications] Institute for Mathematics and its Applications, University of Minnesota (USA), May 18-22, 2009
*[http://www.ipam.ucla.edu/programs/ktws4/ Workshop IV: Asymptotic Methods for Dissipative Particle Systems] May 18-22 (2009) Los Angeles (USA)
*[http://www.cecam.org/workshop-310.html Computer Simulation in Food Science: CFD meets Soft Matter] May 25, 2009 to May 27, 2009 CECAM-HQ-EPFL, Lausanne, (Switzerland)
*[http://complenet09.diit.unict.it CompleNet 2009] International Workshop on Complex Networks. Catania (Italy), May 26-28, 2009
*[http://go.warwick.ac.uk/maths/research/events/2008_2009/symposium/wks5/ EPSRC Symposium Workshop on Molecular Dynamics] Monday 1 – Friday 5 June (2009) Warwick (United Kingdom)
*[http://www.cecam.org/workshop-272.html Theoretical Modeling of Transport in Nanostructures] June 2, 2009 to June 5, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.mpip-mainz.mpg.de/mmsd/ Mainz Materials Simulation Days] 3-5 June 2009 Max Planck Institute for Polymer Research (Germany)
*[http://www.soms.ethz.ch/workshop2009 Coping with Crises in Complex Socio-Economic Systems] ETH Zurich (Switzerland), June 8-13, 2009
*[[MOSSNOHO Workshop 2009]] 16 of June 2009, Universidad Complutense de Madrid (Spain)
*[http://cftc.cii.fc.ul.pt/Workshops/flowNjam/index.html Flow(ers) and jam(mers): from liquid crystals to grains] Lisbon (Portugal) 17-19 June 2009
*[http://ipht.cea.fr/Meetings/BegRohu2009/ Beg Rohu Summer School: Quantum Physics Out of Equilibrium] Ecole Nationale de Voile (France) 15-27 June 2009
*[http://symp17.boulder.nist.gov Seventeenth Symposium on Thermophysical Properties] Boulder, Colorado (USA), June 21-26, 2009
*[http://www.fhi-berlin.mpg.de/th/Meetings/DFT-workshop-Berlin2009/ Hands-on Tutorial on Ab Initio Molecular Simulations: Toward a First-Principles Understanding of Materials Properties and Functions] Berlin (Germany) June 22-July 1 2009
*[https://www.irphe.univ-mrs.fr/~fe09 Fluid and Elasticity] Carry-le-Rouet, near Marseilles (France) June 23-26, 2009
*[http://www.icmp.lviv.ua/statphys2009/ Statistical Physics: Modern Trends and Applications] June 23-25, 2009 Lviv, (Ukraine)
*[http://www.fuw.edu.pl/~wssph/ 3rd Warsaw School of Statistical Physics] Kazimierz Dolny (Poland), 27 June - 4 July, 2009
*[http://www.cecam.org/workshop-298.html Modeling and Simulation of Water at Interfaces from Ambient to Supercooled Conditions] June 29, 2009 to July 1, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.ccp5.ac.uk/SSCCP5/main.html CCP5 Methods in Molecular Simulation Summer School] University of Sheffield (England) July 5 - July 14 2009
*[http://math.arizona.edu/~goriely/leshouches/home.html New trends in the Physics and Mechanics of Biological Systems"] Les Houches (France), 6-31 July 2009
*[http://research.yale.edu/boulder/Boulder-2009/index.html Nonequilibrium Statistical Mechanics: Fundamental Problems and Applications] July 6-24 2009 Boulder (USA)
*[http://www.xrqtc.cat/index.php/ca/homew New trends in Computational Chemistry for Industry Applications] July 6-7 Barcelona (Spain)
*[http://www.frias.uni-freiburg.de/BFF Computational Methods for Soft Matter and Biological Systems] July 8-11, 2009, FRIAS, Freiburg (Germany)
*[http://www.cecam.org/workshop-286.html Structural Transitions in Solids: Theory, Simulations, Experiments and Visualization Techniques] July 8, 2009 to July 11, 2009 CECAM-USI, Lugano (Switzerland)
*[http://fomms.org FOMMS 2009, Fourth International Conference Foundations of Molecular Modeling and Simulation] Semiahmoo Resort, Blaine, WA (USA) 12-16 July 2009
*[http://www.cecam.org/workshop-308.html Computer Simulation Approaches to Study Self-Assembly: From Patchy Nano-Colloids to Virus Capsids] July 13, 2009 to July 15, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.md-net.org.uk/events/bath_jul_2009.htm EPSRC Network Mathematical Challenges of Molecular Dynamics] 13-15 July (2009) Bath, United Kingdom.
*[http://www.cecam.org/workshop-279.html New Trends in Simulating Colloids: from Models to Applications] July 15, 2009 to July 18, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.rsc.org/ConferencesAndEvents/RSCConferences/FD144/index.asp Faraday Discussion 144: Multiscale Modelling of Soft Matter] University of Groningen (The Netherlands) 20 - 22 July 2009
*[http://www2.yukawa.kyoto-u.ac.jp/~ykis2009/Welcome.html Frontiers in Nonequilibrium Physics: Fundamental Theory, Glassy & Granular Materials, and Computational Physics] July 21 - August 21, 2009 Kyoto (Japan)
*[http://www.cecam.org/workshop-293.html Fundamental Aspects of Deterministic Thermostats] July 27, 2009 to July 29, 2009 CECAM-HQ-EPFL, Lausanne (Switzerland)
*[http://www.vallico.net/tti/tti.html Quantum Monte Carlo and the CASINO program IV] 2nd-9th August 2009 Towler Institute, Tuscany, (Italy)
*[http://www.warwick.ac.uk/go/quantsim09 EPSRC Symposium Workshop on Quantum Simulations] 24th-28th August 2009 University of Warwick (United Kingdom)
*[http://itf.fys.kuleuven.be/~fpspXII/ Fundamental Problems in Statistical Physics XII] August 31 - September 11, 2009 Leuven (Belgium)
*[http://denali.phys.uniroma1.it/~idmrcs6/ 6th International Discussion Meeting on Relaxations in Complex Systems] August 31- September 5, 2009, Roma (Italy)
*[http://www.dft09.org/ International Conference on the Applications of Density Functional Theory in Chemistry and Physics] August 31st to September 4th 2009 Lyon (France)
*[http://www.dfrl.ucl.ac.uk/CCP5/ccp5.htm CCP5 Annual Meeting 2009 Structure Prediction] 7th to 9th, September 2009 London (United Kingdom)
*[http://fises.dfa.uhu.es/fises09/ XVI Congreso de Física Estadística] Huelva, 10-12 September 2009, Spain
*[http://th-www.if.uj.edu.pl/zfs/smoluchowski/ 22nd Marian Smoluchowski Symposium On Statistical Physics] 12-17 September 2009 Zakopane (Poland)
*[http://www.isis.rl.ac.uk/largescale/loq/SAS2009/SAS2009.htm SAS-2009] XIV International Conference on Small-Angle Scattering, Sunday 13 - Friday 18 September, 2009, Oxford (UK)
*[http://www.esc.sandia.gov/dsmc09/dsmc09.html Direct Simulation Monte Carlo workshop] September 13-16, 2009 Santa Fe, New Mexico, USA,
*[http://www.iop.org/Conferences/Forthcoming_Institute_Conferences/PHYSASPECTS_09/index.html Physical Aspects of Polymer Science] 14-16 September 2009 Bristol (United Kingdom)

__NOTOC__
[[category: Miscellaneous]]

Percolation analysis

2009-09-21T09:30:27Z

Noe: /* Percolation threshold and critical thermodynamic transitions */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Ref. <ref name='deng' > </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Phys. Rev. B''' 67''', 014102 (2003) [5 pages] ] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories", J. Phys. A: Math. Gen. '''36''', 4269-428 (2002) ] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-21T09:29:50Z

Noe: /* Percolation threshold and critical thermodynamic transitions */ new ref.

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Ref. <ref name='deng' > </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Phys. Rev. B''' 67''', 014102 (2003) [5 pages] ] </ref>
<ref name=fortunato_2> [http://dx.doi.org/10.1088/0305-4470/36/15/304 Santo Fortunato, "Cluster percolation and critical behaviour in spin models and SU(N) gauge theories", J. Phys. A: Math. Gen. 36, 4269-428 (2002) ] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-21T09:18:56Z

Noe: new reference

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Ref. <ref name='deng' > </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition <ref name=fortunato > [http://dx.doi.org/10.1103/PhysRevB.67.014102 Santo Fortunato, "Critical droplets and phase transitions in two dimensions", Phys. Rev. B''' 67''', 014102 (2003) [5 pages] ] </ref> . In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-18T14:36:21Z

Noe: /* Computation of the percolation threshold */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See Ref. <ref name='deng' > </ref> for details).

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-18T14:35:26Z

Noe: /* Computation of the percolation threshold */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See <ref name='deng' > </ref>)

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-18T14:32:01Z

Noe: /* Computation of the percolation threshold */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>. More sophisticated methods can be found in the literature (See <ref name='deng'>)

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-18T14:29:36Z

Noe:

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref name=deng > [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x, ( x \rightarrow 0 )</math> one will have <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (that is to say, in the [[thermodynamic limit]]).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

:<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

:<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs. <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-18T13:51:51Z

Noe: /* Example */ small correction

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref> [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physical Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.

=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x </math>, (<math> x \rightarrow 0 </math>) we will get <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (so to say, in the thermodynamic limit).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]

Percolation analysis

2009-09-18T13:50:37Z

Noe: /* Percolation and finite-size scaling analysis */

This entry focuses on the application of '''percolation analysis''' to problems in [[statistical mechanics]]. For a general discussion see Refs. <ref name="Stauffer"> Dietrich Stauffer and Ammon Aharony "Introduction to Percolation Theory", CRC Press (1994) ISBN 9780748402533</ref> <ref name="Torquato"> Salvatore Torquato "Random Heterogeneous Materials, Microscopic and Macroscopic Properties", Springer, New York (2002) ISBN 9780387951676</ref>

==Sites, bonds, and clusters ==

This topic concerns the analysis of connectivity of elements (sites) distributed in different positions of a given large system.
Using some connectivity rules it is possible to define ''bonds'' between pairs of sites. These bonds can be used to build up
clusters of sites. Two sites in a cluster can be connected directly by a bond between them or indirectly by one or more
sequences of bonds between pairs of sites.
The sites of the system can belong to different types (species in the chemistry language).
Bonds are usually permitted only between near sites.

== Connectivity rules ==

The connectivity rules (bonding criteria) that permit the creation of bonds can be of different nature: distance between sites, energetic
interaction, types of sites, ''etc''.
In addition the bonding criteria can be deterministic or probabilistic.
In the statistical mechanics applications one can find different bonding criteria, for example:

* Geometric distance: Two sites, <math>i</math>, <math>j</math>, are bonded if the distance between then satisfies: <math> r_{ij} < R_p </math>.

* Energetic criteria: Two sites <math>i</math>, <math>j</math>, has a bonding probability given by <math> b(r_{ij}) = \max \left\{ 0, \exp \left[ u_{ij}(r_{ij}) \right] \right\} </math>

==Percolation threshold==

The sizes of the clusters of a given system depend on different ''control parameters'': [[density]] and distribution of sites, bonding criteria (which could include the effect of [[temperature]] and energy interactions), etc.

Let us consider some initial conditions of the control parameters, in
which, the bonding criteria leads to the formation of small clusters, i.e. all the cluster contain a small number of particles and
the cluster size is much smaller than the linear dimension of the system. Now, if one varies gradually some control parameter(s) to increase
the number of bonds in the system, then the number of clusters is expected to decrease, the number of sites per cluster and the cluster size will increase; and, eventually, the largest cluster size(s) (in one or several directions) will be similar to the overall system size (the system reaches the '''percolation threshold''' of the '''percolation transition''').

=== Percolation and boundary conditions ===

There are different possible criteria to consider that a cluster has percolated. The choice of percolation criteria usually depends on computational convenience, type of boundary conditions, dimensionality of the space, ''etc''. In the particular case of considering [[periodic boundary conditions]], a cluster realization is usually considered as percolating when, at least, one of the clusters
becomes of infinite size (length) in, at least, one direction. This infinite size occurs, obviously, via the replication of the system that
appears due to the periodic boundary conditions.

== Percolation and finite-size scaling analysis ==
=== Example ===
Let us consider a standard example of percolation theory, <ref> [http://dx.doi.org/10.1103/PhysRevE.72.016126 Youjin Deng and Henk W. J. Blöte, "Monte Carlo study of the site-percolation model in two and three dimensions" Physucal Review E '''72''', 016126 (2005) [10 pages]]</ref>
a two-dimensional [[building up a square lattice|square lattice]] in which:
* Each site of the lattice can be occupied (by one ''particle'') or empty, and
* The probability of occupancy of each site is <math> x </math>.
* Two sites are considered to be bonded if and only if:
** They are nearest neighbours and
** Both sites are occupied.
=== Fraction of percolating realizations ===
On such a system, it is possible to perform simulations considering different system sizes (with <math> L \times L </math> sites), using
periodic boundary conditions. In such simulations one can generate different system realizations for given values of <math> x </math>, and compute
the fraction, <math> X_{\rm per}(x,L) </math>, of realizations with percolating clusters. For low values of <math> x </math>, (<math> x \rightarrow 0 </math>) we will get <math> X_{\rm per}(x,L) \approx 0 </math>, whereas when <math> x \rightarrow 1 </math>, then <math> X_{\rm per}(x,L) \approx 1</math>. Considering the behavior of <math> X_{\rm per} </math> as a function of <math> x </math>, for different
values of <math> L </math> the transition between <math> X_{\rm per} \approx 0 </math> and <math> X_{\rm per} \approx 1 </math> occurs more
abruptly as <math> L </math> increases. In addition, it is possible to compute the value of the occupancy probability <math> x_{c} </math>
at which the transition would take place for an infinite system (so to say, in the thermodynamic limit).

=== Finite-size scaling ===

Considering the functions <math> X_{\rm per}(x,L) </math> the percolation theory predicts for large system sizes:

*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 0 ; {\rm for} \; \; x < x_c </math>
*<math> \lim_{L \rightarrow \infty} X_{\rm per}(x,L) = 1 ; {\rm for} \; \; x > x_c </math>

In addition, at <math> x = x_c </math>, it is expected that the fraction of percolating realizations do not
depend on the system size:

*<math> X_{\rm per}(x_c,L) \approx X_{\rm per}^{(c)} </math> ; for large values of <math> L </math>.

== Computation of the percolation threshold ==
A couple of simple procedures to estimate the percolation threshold (<math> x_c </math> in the example introduced above) are described here.
These procedures are similar to those used in the analysis of critical thermodynamic transitions<ref>[http://dx.doi.org/10.2277/0521842387 David P. Landau and Kurt Binder "A Guide to Monte Carlo Simulations in Statistical Physics", Cambridge University Press (2005)] </ref>

=== Crossing of the <math> X_{\rm per}(x,L) </math> for different system sizes ===

In practice, one has to compute the fraction of percolating realizations for different values of the control parameter <math> \left. x \right. </math> and different system sizes <math> \left. L \right. </math>. The critical value <math> x_c </math> is then estimated by plotting
<math> \left. X_{\rm per}(x) \right. </math> as a function of <math> \left. x \right. </math> for several values of <math> L </math>. The crossing of
the curves with different values of <math> L </math> provide estimates of both <math> x_c </math> and <math> X_{\rm per}^{(c)} </math>.

=== Computation of pseudo-critical parameters <math> x_c(L) </math> and extrapolation ===

Given the results of <math> X_{\rm per}(x,L) </math> for a given system size <math> L </math>, a pseudo-critical size dependent variable
<math> x_c(L)=x_c^{(L)} </math> is computed by matching <math> X_{\rm per}(x_c^{(L)},L) = X_{\rm per}^{(c)}</math>.

If the ''universal'' value <math> X_{per}^{(c)} </math> value is unknown for the type of transition considered, an alternative definition
for <math> x_c\left(L \right) </math> can be taken, for instance:

<math> X_{\rm per}(x_c^{(L)},L) = 1/2</math>.

The percolation theory predicts that the pseudo-critical values <math> x_c(L) </math> will scale as:

<math> x_c \left( L \right) = x_c \left( \infty \right) + a L^{- b} </math>

where <math> b </math> is a [[Critical exponents |critical exponent]] (See Refs <ref name="Stauffer"> </ref> <ref name="Torquato"> </ref> for details). Therefore, by fitting the results of <math> x_c(L) </math> it is
possible to estimate the percolation transition location: <math> x_c = x_c ( \infty ) </math>.

== Percolation threshold and critical thermodynamic transitions ==

In some systems, with an appropriate definition of bonding criteria, the percolation transition occurs at the same value of the control parameter (density, temperature, [[chemical potential]]) as the thermodynamic transition. In these case [[cluster algorithms|cluster algorithms]] become very efficient, and moreover, the percolation analysis can be useful to develop algorithms to locate the transition (see the [[cluster algorithms|cluster algorithms]] page for more details).

==References==
<references/>
[[Category: Confined systems]]