Hard disk model

2022-09-29T21:12:11Z

90.92.206.1: /* Phase transitions */

'''Hard disks''' are [[Hard sphere model |hard spheres]] in two dimensions. The hard disk [[intermolecular pair potential]] is given by<ref>[http://dx.doi.org/10.1063/1.1699114 Nicholas Metropolis, Arianna W. Rosenbluth, Marshall N. Rosenbluth, Augusta H. Teller and Edward Teller, "Equation of State Calculations by Fast Computing Machines", Journal of Chemical Physics '''21''' pp.1087-1092 (1953)]</ref>
<ref>[http://lib-www.lanl.gov/cgi-bin/getfile?00371200.pdf W. W. Wood "Monte Carlo calculations of the equation of state of systems of 12 and 48 hard circles", Los Alamos Scientific Laboratory Report '''LA-2827''' (1963)]</ref>

: <math>
\Phi_{12}\left( r \right) = \left\{ \begin{array}{lll}
\infty & ; & r < \sigma \\
0 & ; & r \ge \sigma \end{array} \right.
</math>

where <math> \Phi_{12}\left(r \right) </math> is the [[intermolecular pair potential]] between two disks at a distance <math>r := |\mathbf{r}_1 - \mathbf{r}_2|</math>, and <math> \sigma </math> is the diameter of the disk. This page treats hard disks in a two-dimensional space, for three dimensions see the page [[hard disks in a three dimensional space]].
==Phase transitions==
Despite the apparent simplicity of this model/system, the phase behaviour and the nature of the phase transitions remains an area of active study ever since the early work of Alder and Wainwright <ref>[http://dx.doi.org/10.1103/PhysRev.127.359 B. J. Alder and T. E. Wainwright "Phase Transition in Elastic Disks", Physical Review '''127''' pp. 359-361 (1962)]</ref>. Recent works show a phase diagram containing an isotropic, a hexatic, and a solid phase <ref>[http://dx.doi.org/10.1103/PhysRevE.73.065104 C. H. Mak "Large-scale simulations of the two-dimensional melting of hard disks", Physical Review E '''73''' 065104(R) (2006)]</ref>. Highly efficient event-chain Monte Carlo simulations of over 1 million hard disks by Bernard and Krauth have solidified this picture, with a first-order phase transition between the fluid at packing fraction <math>\eta = 0.700</math> and the hexatic phase at <math>\eta = 0.716</math>, and a continuous transition between the hexatic and solid phases at <math>\eta = 0.720</math> <ref>[https://doi.org/10.1103/PhysRevLett.107.155704 E. P. Bernard and W. Krauth "Two-Step Melting in Two Dimensions: First-Order Liquid-Hexatic Transition", Physical Review Letters '''107''' 155704 (2011)]</ref>. Note that the maximum possible packing fraction is given by <math>\eta = \pi / \sqrt{12} \approx 0.906899...</math> <ref>[http://dx.doi.org/10.1007/BF01181430 L. Fejes Tóth "Über einen geometrischen Satz." Mathematische Zeitschrift '''46''' pp. 83-85 (1940)]</ref>. This scenario has since been confirmed using a variety of simulation methods <ref>[https://doi.org/10.1103/PhysRevLett.107.155704 M. Engel, J. A. Anderson, S. C. Glotzer, M. Tsobe, E. P. Bernard, and W. Krauth "Hard-disk equation of state: First-order liquid-hexatic transition in two dimensions with three simulation methods", Physical Review E '''87''' 042134 (2013)]</ref>.

Similar results have been found using the [[BBGKY hierarchy]] <ref>[http://dx.doi.org/10.1063/1.3491039 Jarosław Piasecki, Piotr Szymczak, and John J. Kozak "Prediction of a structural transition in the hard disk fluid", Journal of Chemical Physics '''133''' 164507 (2010)]</ref> and by studying tessellations (the hexatic region: <math>0.680 < \eta < 0.729</math>) <ref>[http://dx.doi.org/10.1021/jp806287e John J. Kozak, Jack Brzezinski and Stuart A. Rice "A Conjecture Concerning the Symmetries of Planar Nets and the Hard disk Freezing Transition", Journal of Physical Chemistry B '''112''' pp. 16059-16069 (2008)]</ref>. Also studied via [[integral equations]] <ref>[https://doi.org/10.1063/1.5026496 Luis Mier-y-Terán, Brian Ignacio Machorro-Martínez, Gustavo A. Chapela, and Fernando del Río "Study of the hard-disk system at high densities: the fluid-hexatic phase transition", Journal of Chemical Physics '''148''' 234502 (2018)]</ref>.
Experimental results <ref>[http://dx.doi.org/10.1103/PhysRevLett.118.158001 Alice L. Thorneywork, Joshua L. Abbott, Dirk G. A. L. Aarts, and Roel P. A. Dullens "Two-Dimensional Melting of Colloidal Hard Spheres", Physical Review Letters '''118''' 158001 (2017)]</ref>.

==Equations of state==
:''Main article: [[Equations of state for hard disks]]''
==Virial coefficients==
:''Main article: [[Hard sphere: virial coefficients]]''
==See also==
*[[Binary hard-disk mixtures]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1070/RM1970v025n02ABEH003794 Ya G Sinai "Dynamical systems with elastic reflections", Russian Mathematical Surveys '''25''' pp. 137-189 (1970)]
*[http://dx.doi.org/10.1103/PhysRevB.30.2755 Katherine J. Strandburg, John A. Zollweg, and G. V. Chester "Bond-angular order in two-dimensional Lennard-Jones and hard-disk systems", Physical Review B '''30''' pp. 2755 - 2759 (1984)]
*[http://dx.doi.org/10.1007/s00222-003-0304-9 Nándor Simányi "Proof of the Boltzmann-Sinai ergodic hypothesis for typical hard disk systems", Inventiones Mathematicae '''154''' pp. 123-178 (2003)]
*[http://dx.doi.org/10.1063/1.3687921 Roland Roth, Klaus Mecke, and Martin Oettel "Communication: Fundamental measure theory for hard disks: Fluid and solid", Journal of Chemical Physics '''136''' 081101 (2012)]

==External links==
*[http://www.smac.lps.ens.fr/index.php/Programs_Chapter_2:_Hard_disks_and_spheres Hard disks and spheres] computer code on SMAC-wiki.
[[Category: Models]]

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Hard disk model