SklogWiki - User contributions [en]

Talk:Hard square lattice model

2008-08-20T13:46:29Z

161.111.27.7: New page: Should it be hard square model the title of this page? where are the lattices? --~~~~

Should it be hard square model the title of this page?
where are the lattices? --[[User:161.111.27.7|161.111.27.7]] 15:46, 20 August 2008 (CEST)

Wang-Landau method

2008-07-08T09:40:30Z

161.111.27.7: some additions

{{Stub-general}}
The '''Wang-Landau method''' was proposed by F. Wang and D. P. Landau (Ref. 1) to compute the density of
states, <math> \Omega (E) </math>, of [[Potts model|Potts models]];
where <math> \Omega(E) </math> is the number of [[microstate |microstates]] of the system having energy
<math> E </math>.

The '''Wang-Landau method''' in its original version is a simulation technique designed to reach an uniform
sampling of the energies of the system in a given range.

==References==
#[http://dx.doi.org/10.1103/PhysRevE.64.056101 Fugao Wang and D. P. Landau "Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram", Physical Review E '''64''' 056101 (2001)]
#[http://dx.doi.org/10.1119/1.1707017 D. P. Landau, Shan-Ho Tsai, and M. Exler "A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling", American Journal of Physics '''72''' pp. 1294-1302 (2004)]
#[http://dx.doi.org/10.1063/1.2794042 Georg Ganzenmüller and Philip J. Camp "Applications of Wang-Landau sampling to determine phase equilibria in complex fluids", Journal of Chemical Physics '''127''' 154504 (2007)]
#[http://dx.doi.org/10.1063/1.2803061 R. E. Belardinelli and V. D. Pereyra "Wang-Landau algorithm: A theoretical analysis of the saturation of the error", Journal of Chemical Physics '''127''' 184105 (2007)]
#[http://dx.doi.org/10.1103/PhysRevE.75.046701 R. E. Belardinelli and V. D. Pereyra "Fast algorithm to calculate density of states", Physical Review E '''75''' 046701 (2007)]
[[category: Monte Carlo]]
[[category: computer simulation techniques]]

Cluster algorithms

2007-09-05T08:22:27Z

161.111.27.7: /* Probability-Changing Cluster Algorithm */

'''Cluster algorithms''' are mainly used in the simulation of [[Ising Models|Ising-like models]]. The essential feature is the use of collective motions
of ''particles (spins)'' in a single [[Monte Carlo]] step.
An interesting property of some of these application is the fact that the [[percolation analysis]] of the clusters can
be used to study [[phase transitions]].
== Swendsen-Wang algorithm ==
As an introductory example we shall discuss the Swendsen-Wang technique (Ref 1) in the simulation of [[Ising Models]].
In one [[Monte Carlo]] step of the algorithm the following recipe is used:

* Consider every pair of interacting sites (spins).

In the current configuration the [[Intermolecular pair potential |pair interaction]] can be either negative: <math> \Phi_{ij}/k_B T= -K </math> or positive <math> \Phi_{ij}/k_B T = + K </math>,
depending on the product: <math> S_{i} S_{j} </math> (See [[Ising Models]] for details on the notation)

* For pairs of interacting sites (nearest neighbors) with <math> \Phi_{ij}/k_B T < 0 </math> [[random numbers |randomly]] create a bond between the two spins with a given probability <math> p </math>.

: <math> p </math> will be chosen to be a function of <math> K </math>.

* The bonds generated in the previous step are used to build up clusters of sites (spins).

* Build up the partition of the system in the corresponding clusters of spins.

In each cluster all the spins will have the same state (either <math> S = 1 </math> or <math> S = -1 </math>).

* For each cluster, independently, choose at random with equal probabilities whether to flip (invert the value of <math> S </math>) or not to flip the whole set of spins belonging to the cluster.

The bonding probability <math> p </math> is given by:

: <math> p = 1 - \exp [ -2 K ] </math>

== Wolff algorithm ==
The procedure to create a given bond is the same as in the Swendsen-Wang algorithm. However in Wolff's method
the whole set of interacting pairs is not tested to generate (possible) bonds. In stead, a single cluster
is built. See Ref 2 for details.

* The initial cluster contains one site (selected at random)

* Possible bonds between the initial site and other sites of the system are tested:

The bonded sites are included in the cluster

* Then recursively, one checks the existence of bonds between the new members of the cluster and sites of the system to add, if bonds are generated, new sites to the ''growing'' cluster, until no more bonds are generated.

* At this point, the whole cluster is flipped (see above)

== Invaded Cluster Algorithm ==
The purpose of this algorithm is to locate [[critical points]] (critical temperature). So, in this case
the probability of bonding neighboring sites with equal spins is not set ''a priori''. (See Ref 3)
The algorithm for an Ising system with [[periodic boundary conditions]] can be implemented as follows:

Given a certain configuration of the system:

* One considers the possible bonds on the system (pairs of nearest neighbours with favourable interaction)

* Using [[random numbers]] we assign a random order to these possible bonds

* The possible bonds are being ''activated'' in the order fixed in the previous step (the cluster structure is watched during this process)

* The bond activation stops when one cluster percolates through the entire system (i.e. considering the periodic boundary conditions
the cluster becomes of infinite size)

* Then, every cluster (as in the Swendsen-Wang algorithm) is flipped with proability 1/2.

* An effective bond probability for the percolation threshold, <math> p_{per} </math> can be computed as:

: <math> p_{per} = M_{act}/M </math>

with <math> M_{act}</math> being the number of activated bonds when the first cluster ''percolates'', and <math> M </math> is the number
of possible bonds.

The value of <math> p_{per} </math> (in one realisation, or the averaged value over the simulation, See the references for the practical application) can be related with the critical coupling constant, <math> k_c </math> as:

: <math> p_{per} \approx 1 - \exp \left[ - 2 k_c \right] </math>

== Probability-Changing Cluster Algorithm ==

This method was proposed by Tomita and Okabe (See Ref 4). The procedure is intented to compute
the critical points. It applies when the simmetry of the interactions implies that the critical
temperature is that in which the clusters built using a Swendsen-Wang type algorithm reach
the percolation threshold.

The simulation proceeds by a fine tuning of the temperature (or the coupling constant)

Given a

== Beyond the Ising and Potts models ==

The methods described so far, can be used with minor changes in the simulation of [[Potts model|Potts models]].
In addition, extensions have been proposed in the literature (References will be added one of these days)
to build up very efficient cluster algorithm to simulate more complex lattice systems ([[XY model]],
[[Heisenberg model]], etc).

== Application to continuous (atomistic) models ==

It is sometimes possible (and very convenient) to include cluster algorithms in the simulation of
models with continuous translational degrees of freedom. In most cases the cluster algorithm has
to be complemented with other sampling moves to ensure [[ergodicity]]. Examples:

* Spin fluids
* [[Binary mixtures]] (with symmetry in the interactions)
* Continuous version of [[XY model]], [[Heisenberg model]], etc.

In these cases, the usual approach is to combine one-particle moves (e.g. particle translations),
with cluster procedures. In the cluster steps, multiparticle modification of -composition, orientations, etc.-
is carried out.

== Other (not so smart) applications of cluster algorithms ==
Monte Carlo simulation of atomistic systems with multiparticle moves, for example see:

*[http://dx.doi.org/10.1063/1.2759924 N. G. Almarza and E. Lomba "Cluster algorithm to perform parallel Monte Carlo simulation of atomistic systems", Journal of Chemical Physics '''127''' 084116 (2007)]

== References ==
#[http://dx.doi.org/10.1103/PhysRevLett.58.86 Robert H. Swendsen and Jian-Sheng Wang, "Nonuniversal critical dynamics in Monte Carlo simulations", Physical Review Letters '''58''' pp. 86 - 88 (1987) ]
#[http://dx.doi.org/10.1103/PhysRevLett.62.361 Ulli Wolff, "Collective Monte Carlo Updating for Spin Systems" , Physical Review Letters '''62''' pp. 361 - 364 (1989) ]
#[http://dx.doi.org/10.1103/PhysRevLett.75.2792 J. Machta, Y. S. Choi, A. Lucke, T. Schweizer, and L. V. Chayes, "Invaded Cluster Algorithm for Equilibrium Critical Points" , Physical Review Letters '''75''' pp. 2792 - 2795 (1995)]
#[http://dx.doi.org/10.1103/PhysRevLett.86.572 Yusuke Tomita and Yutaka Okabe, "Probability-Changing Cluster Algorithm for Potts Models", Physical Review Letters '''86''' pp. 572 - 575 (2001)]

Talk:Diffusion

2007-08-09T09:05:26Z

161.111.27.7: New page: hay que revisar las definiciones ----

hay que revisar las definiciones
----

Surface tension

2007-07-31T17:41:06Z

161.111.27.7:

The surface tension
<math> \gamma </math> is a measure of the work required to build up a surface ....

Canonical Ensemble: Two phases;

<math> \gamma = \frac{ \partial F (N,V,T, A)}{\partial A } </math>;

where

* <math> N </math> is the number of particles
*<math> V </math> is the volume
*<math> T </math> is the temperature
*<math> {\cal A} </math> is the surface area

... preliminary version (sketchy) to be continued soon, as soon as I find a way to distinghuis A (Helmholtz) and A (surface area)

==Computer Simulation==

* Binder procedure

* Explicit interface

Ramp model

2007-07-05T13:03:12Z

161.111.27.7:

The '''ramp model''' is described by:

:<math>
\Phi(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(r)</math> is the [[intermolecular pair potential]], <math>W_r > 0</math> and <math>W_a < 0</math>.

Graphically, one has:
[[Image:Ramp_potential.png|center]]
where the red line represents an attractive implementation of the model, and the green line a repulsive implementation.

==References==
#[http://dx.doi.org/10.1103/PhysRevLett.24.1284 P. C. Hemmer and G. Stell "Fluids with Several Phase Transitions", Physical Review Letters '''24''' pp. 1284 - 1287 (1970)]
#[http://dx.doi.org/10.1063/1.1677857 G. Stell and P. C. Hemmer, "Phase Transitions Due to Softness of the Potential Core", Journal of Chemical Physics' '''56''', pp. 4274-4286 (1972)]
#[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)]
#[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)]
[[Category:models]]
[[category:Polyamorphic systems]]

Ramp model

2007-07-05T13:02:43Z

161.111.27.7:

The '''ramp model''' is described by:
())))
:<math>
\Phi(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(r)</math> is the [[intermolecular pair potential]], <math>W_r > 0</math> and <math>W_a < 0</math>.

Graphically, one has:
[[Image:Ramp_potential.png|center]]
where the red line represents an attractive implementation of the model, and the green line a repulsive implementation.

==References==
#[http://dx.doi.org/10.1103/PhysRevLett.24.1284 P. C. Hemmer and G. Stell "Fluids with Several Phase Transitions", Physical Review Letters '''24''' pp. 1284 - 1287 (1970)]
#[http://dx.doi.org/10.1063/1.1677857 G. Stell and P. C. Hemmer, "Phase Transitions Due to Softness of the Potential Core", Journal of Chemical Physics' '''56''', pp. 4274-4286 (1972)]
#[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)]
#[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)]
[[Category:models]]
[[category:Polyamorphic systems]]

Inverse Monte Carlo

2007-06-11T10:47:51Z

161.111.27.7: /* References */

* Inverse Monte Carlo refers to the numerical techniques to solve the
so-called inverse problem in fluids.

* Given the structural information (distribution functions) the inverse Monte
Carlo technique tries to compute the corresponding interaction potential

* Some example(s) of these techniques can be found in the following reference(s)

== References ==

#[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Phys. Rev. E 68, 011202 (2003) (6 pages)]
#[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Phys. Rev. E 70, 021203 (2004) (5 pages) ]

Inverse Monte Carlo

2007-06-11T10:45:59Z

161.111.27.7: /* References */

* Inverse Monte Carlo refers to the numerical techniques to solve the
so-called inverse problem in fluids.

* Given the structural information (distribution functions) the inverse Monte
Carlo technique tries to compute the corresponding interaction potential

* Some example(s) of these techniques can be found in the following reference(s)

== References ==

#[http://dx.doi.org/10.1103/PhysRevE.68.011202 N. G. Almarza and E. Lomba, "Determination of the interaction potential from the pair distribution function: An inverse Monte Carlo technique", Phys. Rev. E 68, 011202 (2003) (6 pages)]

#[http://dx.doi.org/10.1103/PhysRevE.70.021203 N. G. Almarza, E. Lomba, and D. Molina. "Determination of effective pair interactions from the structure factor", Phys. Rev. E 70, 021203 (2004) (5 pages) ]

Inverse Monte Carlo

2007-06-08T17:35:45Z

161.111.27.7:

Configuration analysis

2007-06-08T17:34:12Z

161.111.27.7:

*[[Voronoi cells]]
*[[Percolation analysis]]

Inverse Monte Carlo

2007-06-08T17:05:55Z

161.111.27.7:

Inverse Monte Carlo

2007-06-08T17:04:57Z

161.111.27.7:

Inverse Monte Carlo

2007-06-08T17:04:11Z

161.111.27.7: New page: Inverse Monte Carlo refers to the numerical techniques to solve the so-called inverse problem in fluids. i.e. Given the structural information (distribution functions) the inverse Monte Ca...

Inverse Monte Carlo refers to the numerical techniques to solve the
so-called inverse problem in fluids.
i.e. Given the structural information (distribution functions) the inverse Monte
Carlo technique tries to compute the corresponding interaction potential

Monte Carlo methods

2007-06-08T17:01:08Z

161.111.27.7:

*[[Markov chain]]
*[[Metropolis Monte Carlo]]
*[[Monte Carlo in the microcanonical ensemble]]
*[[Monte Carlo in the grand-canonical ensemble]]
*[[Gibbs ensemble Monte Carlo]]
*[[Configurational bias Monte Carlo]]
*[[Reverse Monte Carlo]]
*[[Inverse Monte Carlo]]

Configuration analysis

2007-06-08T16:59:18Z

161.111.27.7: New page: *Voronoi cells

*[[Voronoi cells]]

Computer simulation techniques

2007-06-08T16:58:09Z

161.111.27.7:

*[[Models]]
*[[Force fields]]
*[[Ergodic hypothesis]]
*[[Monte Carlo]]
*[[Molecular dynamics]]
*[[Coarse graining]]
*[[Electrostatics]]
*[[Boundary conditions]]
*[[Finite size effects]]
*[[Wang-Landau method]]
*[[Materials modeling and computer simulation codes]]
*[[Configuration analysis]]

Lennard-Jones model

2007-03-23T11:12:49Z

161.111.27.7:

== Equation for the potential energy ==

The '''Lennard-Jones''' potential is given by

:<math> V(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math>

where:

* <math> V(r) </math> : potential energy of interaction between two particles at a distance r;

* <math> \sigma </math> : diameter (length);

* <math> \epsilon </math> : well depth (energy)

Reduced units:

* Density, <math> \rho^* \equiv \rho \sigma^3 </math>, where <math> \rho = N/V </math> (number of particles <math> N </math> divided by the volume <math> V </math>.)

* Temperature; <math> T^* \equiv k_B T/\epsilon </math>, where <math> T </math> is the absolute temperature and <math> k_B </math> is the [[Boltzmann constant]]
==Argon==
The Lennard-Jones parameters for argon are <math>\epsilon/k_B \approx</math> 119.8 K and <math>\sigma \approx</math> 0.3405 nm. (Ref. ?)
[[Image:Lennard-Jones.png|center]]
This figure was produced using [http://www.gnuplot.info/ gnuplot] with the command:

plot (4*120*((0.34/x)**12-(0.34/x)**6))

== Features ==

Special points:

* <math> V(\sigma) = 0 </math>

* Minimum value of <math> V(r) </math> at <math> r = r_{min} </math>;

: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math>

== Related potential models ==

==References==

#[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)]
[[Category:Models]]

Lennard-Jones model

2007-03-23T11:11:54Z

161.111.27.7: /* Features */

The '''Lennard-Jones''' potential is given by

:<math> V(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math>

where:

* <math> V(r) </math> : potential energy of interaction between two particles at a distance r;

* <math> \sigma </math> : diameter (length);

* <math> \epsilon </math> : well depth (energy)

Reduced units:

* Density, <math> \rho^* \equiv \rho \sigma^3 </math>, where <math> \rho = N/V </math> (number of particles <math> N </math> divided by the volume <math> V </math>.)

* Temperature; <math> T^* \equiv k_B T/\epsilon </math>, where <math> T </math> is the absolute temperature and <math> k_B </math> is the [[Boltzmann constant]]
==Argon==
The Lennard-Jones parameters for argon are <math>\epsilon/k_B \approx</math> 119.8 K and <math>\sigma \approx</math> 0.3405 nm. (Ref. ?)
[[Image:Lennard-Jones.png|center]]
This figure was produced using [http://www.gnuplot.info/ gnuplot] with the command:

plot (4*120*((0.34/x)**12-(0.34/x)**6))

== Features ==

Special points:

* <math> V(\sigma) = 0 </math>

* Minimum value of <math> V(r) </math> at <math> r = r_{min} </math>;

: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math>

== Related potential models ==

==References==

#[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)]
[[Category:Models]]

Lennard-Jones model

2007-03-23T11:10:51Z

161.111.27.7: /* Features */

The '''Lennard-Jones''' potential is given by

:<math> V(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math>

where:

* <math> V(r) </math> : potential energy of interaction between two particles at a distance r;

* <math> \sigma </math> : diameter (length);

* <math> \epsilon </math> : well depth (energy)

Reduced units:

* Density, <math> \rho^* \equiv \rho \sigma^3 </math>, where <math> \rho = N/V </math> (number of particles <math> N </math> divided by the volume <math> V </math>.)

* Temperature; <math> T^* \equiv k_B T/\epsilon </math>, where <math> T </math> is the absolute temperature and <math> k_B </math> is the [[Boltzmann constant]]
==Argon==
The Lennard-Jones parameters for argon are <math>\epsilon/k_B \approx</math> 119.8 K and <math>\sigma \approx</math> 0.3405 nm. (Ref. ?)
[[Image:Lennard-Jones.png|center]]
This figure was produced using [http://www.gnuplot.info/ gnuplot] with the command:

plot (4*120*((0.34/x)**12-(0.34/x)**6))

== Features ==

Special points:

* <math> V(\sigma) = 0 </math>

* Minimum value of <math> V(r) </math> at <math> r = r_{min} </math>;

: <math> \frac{r_{min}}{\sigma} = 2^{1/6} \simeq 1.12246 ... </math>

==References==

#[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)]
[[Category:Models]]

Lennard-Jones model

2007-03-23T11:06:41Z

161.111.27.7: /* Argon */

The '''Lennard-Jones''' potential is given by

:<math> V(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] </math>

where:

* <math> V(r) </math> : potential energy of interaction between two particles at a distance r;

* <math> \sigma </math> : diameter (length);

* <math> \epsilon </math> : well depth (energy)

Reduced units:

* Density, <math> \rho^* \equiv \rho \sigma^3 </math>, where <math> \rho = N/V </math> (number of particles <math> N </math> divided by the volume <math> V </math>.)

* Temperature; <math> T^* \equiv k_B T/\epsilon </math>, where <math> T </math> is the absolute temperature and <math> k_B </math> is the [[Boltzmann constant]]
==Argon==
The Lennard-Jones parameters for argon are <math>\epsilon/k_B \approx</math> 119.8 K and <math>\sigma \approx</math> 0.3405 nm. (Ref. ?)
[[Image:Lennard-Jones.png|center]]
This figure was produced using [http://www.gnuplot.info/ gnuplot] with the command:

plot (4*120*((0.34/x)**12-(0.34/x)**6))

== Features ==

==References==

#[http://dx.doi.org/10.1088/0959-5309/43/5/301 J. E. Lennard-Jones, "Cohesion", Proceedings of the Physical Society, '''43''' pp. 461-482 (1931)]
[[Category:Models]]

Talk:Hard sphere: virial coefficients

2007-03-21T13:14:41Z

161.111.27.7: New page: References for dimension > 3 are required (I guess) --~~~~

References for dimension > 3 are required (I guess) --[[User:161.111.27.7|161.111.27.7]] 14:14, 21 March 2007 (CET)

Building up a face centered cubic lattice

2007-03-20T10:45:27Z

161.111.27.7:

* Consider:
# a cubic simulation box whose sides are of length <math>\left. L \right. </math>
# a number of lattice positions, <math> \left. M \right. </math> given by <math> \left. M = 4 m^3 \right. </math>,
with <math> m </math> being a positive integer

* The <math> \left. M \right. </math> positions are those given by:

:<math>
\left\{ \begin{array}{l}
x_a = i_a \times (\delta l) \\
y_a = j_a \times (\delta l) \\
z_a = k_a \times (\delta l)
\end{array}
\right\}
</math>

where the indices of a given valid site are integer numbers that must fulfill the following criteria

* <math> 0 \le i_a < 2m </math>
* <math> 0 \le j_a < 2m </math>
* <math> 0 \le k_a < 2m </math>,
* the sum of <math> \left. i_a + j_a + k_a \right. </math> must be, for instance, an even number.

with
<math>
\left.
\delta l = L/(2m)
\right.
</math>

Talk:Hard sphere model

2007-03-19T16:11:45Z

161.111.27.7: New page: Please define CDOT --~~~~

Please define CDOT --[[User:161.111.27.7|161.111.27.7]] 17:11, 19 March 2007 (CET)

Semi-grand ensembles

2007-03-06T11:03:24Z

161.111.27.7: /* Fixed pressure and temperature: Semi-grand ensemble: Partition function */

== General features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In these ensembles the total number of molecules is fixed, but the composition can change.

== Canonical ensemble: fixed volume, temperature and number(s) of molecules ==

We shall consider a system consisting of ''c'' components;.
In the [[Canonical ensemble|canonical ensemble]], the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the [[absolute temperature]]
*<math> E </math> is the [[internal energy]]
*<math> p </math> is the [[pressure]]
*<math> \mu_i </math> is the [[Chemical potential|chemical potential]] of the species <math>i</math>
*<math> N_i </math> is the number of molecules of the species <math>i</math>

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we wish to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i \right. </math>;

but the composition can change, from thermodynamic considerations one can apply a [[Legendre transform]] [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

* Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>

or,

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>

where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>.
* Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> one can write:

:<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

== Fixed pressure and temperature: Semi-grand ensemble ==

Following the procedure described above one can write:

:<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,

where the ''new'' thermodynamical Potential <math> \beta \Phi </math> is given by:

:<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
</math>

== Fixed pressure and temperature: Semi-grand ensemble: Partition function ==

In the fixed composition ensemble we will have:

<math> Q_{N_i,p,T} = \frac{ \beta p }{\prod_{i=1}^c \left( \Lambda_i^{3N_i} N_i! \right) } \int_{0}^{\infty} dV e^{-\beta p V } V^N \cdots
</math>

TO BE CONTINUED SOON

Semi-grand ensembles

2007-03-06T10:28:44Z

161.111.27.7: /* Fixed pressure and temperature: Semi-grand ensemble: Partition function */

Semi-grand ensembles

2007-03-06T10:24:41Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */ elliminated the link in thermodynamic potential (everything is a thermodinamic pot.)

Semi-grand ensembles

2007-03-05T15:03:28Z

161.111.27.7: /* Fixed pressure and temperature: Semigrand esemble */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In these ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a system with "c" components;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>

where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>. Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

== Fixed pressure and temperature: Semigrand ensemble ==

Following the procedure described above we can write:

<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,
where the ''new'' thermodynamical Potential <math> \beta \Phi </math> is given by:

<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
</math>

TO BE CONTINUED ... SOON

Semi-grand ensembles

2007-03-05T15:02:21Z

161.111.27.7: /* Fixed pressure and temperature: Semigrand esemble */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In these ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a system with "c" components;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>

where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>. Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

== Fixed pressure and temperature: Semigrand esemble ==

Following the procedure described above we can write:

<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \beta \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,
where the ''new'' thermodynamical Potential <math> \beta \Phi </math> is given by:

<math> d (\beta \Phi) = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
</math>

Semi-grand ensembles

2007-03-05T15:00:44Z

161.111.27.7: /* Fixed pressure and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In these ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a system with "c" components;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>

where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>. Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

== Fixed pressure and temperature: Semigrand esemble ==

Following the procedure described above we can write:

<math> \beta G (\beta,\beta p, N_1, N_2, \cdots N_c ) \rightarrow \Phi (\beta, \beta p, N, \beta \mu_{21}, \cdots, \beta \mu_{c1} ) </math>,
where the ''new'' thermodynamical Potential <math> \Phi </math> is given by:

<math> d \Phi = d \left[ \beta G - \sum_{i=2}^c (\beta \mu_{i1} N_i ) \right] = E d \beta + V d (\beta p) + \beta \mu_1 d N
- \sum_{i=2}^c N_i d (\beta \mu_{i1} ).
</math>

Semi-grand ensembles

2007-03-05T14:52:43Z

161.111.27.7: /* General Features */

Semi-grand ensembles

2007-03-05T14:52:23Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a system with "c" components;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 \sum_{i=2}^c d N_i + \sum_{i=2}^c \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta (\mu_2-\mu_i) d N_i; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \sum_{i=2}^c \beta \mu_{i1} d N_i; </math>

where <math> \left. \mu_{i1} \equiv \mu_i - \mu_1 \right. </math>. Now considering the thermodynamical potential: <math> \beta A - \sum_{i=2}^c \left( N_i \beta \mu_{i1} \right) </math>

:<math> d \left[ \beta A - \sum_{i=2}^c ( \beta \mu_{i1} N_i ) \right] = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

Semi-grand ensembles

2007-03-05T14:46:15Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a system with "c" components;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = \sum_{i=1}^c N_i \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the variable change <math> N_1 \rightarrow N </math> i.e.: <math> \left. N_1 = N- \sum_{i=2}^c N_i \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>

where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>

:<math> d \left( \beta A - \beta \mu_{21} N_2 \right) = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

Semi-grand ensembles

2007-03-05T14:43:22Z

161.111.27.7: /* Canonical Ensemble: fixed volume, temperature and number(s) of molecules */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a system with "c" components;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^c (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = N_1 + N_2 \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> i.e.: <math> \left. N_1 = N-N_2 \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>

where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>

:<math> d \left( \beta A - \beta \mu_{21} N_2 \right) = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

Semi-grand ensembles

2007-03-05T14:42:27Z

161.111.27.7: /* Fixed pressure and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a binary system;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = N_1 + N_2 \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> i.e.: <math> \left. N_1 = N-N_2 \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>

where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>

:<math> d \left( \beta A - \beta \mu_{21} N_2 \right) = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \left( \beta \mu_i \right) d N_i </math>

where:

* <math> G </math> is the [[Gibbs energy function]]

Semi-grand ensembles

2007-03-05T14:39:20Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a binary system;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = N_1 + N_2 \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> i.e.: <math> \left. N_1 = N-N_2 \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>

where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>

:<math> d \left( \beta A - \beta \mu_{21} N_2 \right) = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

== Fixed pressure and temperature ==

In the [[Isothermal-Isobaric ensemble]]: <math> (N_1,N_2, \cdots, N_c, p, T) </math> ensemble we can write:

<math> d (\beta G) = E d \beta + V d (\beta p) + \sum_{i=1}^c \mu_i d N_i </math>

TO BE CONTINUED ... SOON

Semi-grand ensembles

2007-03-05T14:35:12Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a binary system;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = N_1 + N_2 \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> i.e.: <math> \left. N_1 = N-N_2 \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>

where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>

:<math> d \left( \beta A - \beta \mu_{21} N_2 \right) = E d \beta - \left( \beta p \right) d V + \beta \mu_{1} d N - N_2 d \left( \beta \mu_{21} \right).
</math>

TO BE CONTINUED ... SOON

Semi-grand ensembles

2007-03-05T13:03:58Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

== General Features ==
Semi-grand ensembles are used in Monte Carlo simulation of mixtures.

In this ensembles the total number of molecules is fixed, but the composition can change.

== Canonical Ensemble: fixed volume, temperature and number(s) of molecules ==

We will consider a binary system;.
In the Canonical Ensemble, the differential
equation energy for the [[Helmholtz energy function]] can be written as:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \sum_{i=1}^2 (\beta \mu_i) d N_i </math>,
where:

*<math> A </math> is the [[Helmholtz energy function]]
*<math> \beta \equiv 1/k_B T </math>
*<math> k_B</math> is the [[Boltzmann constant]]
*<math> T </math> is the absolute temperature
*<math> E </math> is the internal energy
*<math> p </math> is the pressure
*<math> \mu_i </math> is the chemical potential of the species "i"
*<math> N_i </math> is the number of molecules of the species "i"

== Semi-grand ensemble at fixed volume and temperature ==

Consider now that we want to consider a system with fixed total number of particles, <math> N </math>

: <math> \left. N = N_1 + N_2 \right. </math>;

but the composition can change, from the thermodynamics we can apply a Legendre's transform [HAVE TO CHECK ACCURACY]
to the differential equation written above in terms of <math> A (T,V,N_1,N_2) </math>.

# Consider the change <math> (N_1,N_2) \rightarrow (N,N_2) </math> i.e.: <math> \left. N_1 = N-N_2 \right. </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N - \beta \mu_1 d N_2 + \beta \mu_2 d N_2; </math>

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta (\mu_2-\mu_1) d N_2; </math>

Or:

: <math> d \left( \beta A \right) = E d \beta - (\beta p) d V + \beta \mu_1 d N + \beta \mu_{21} d N_2; </math>

where <math> \mu_{21} = \mu_2 - \mu_1 </math>. Now considering the thermodynamical potentia: <math> \beta A - N_2 \beta \mu_{21} </math>

TO BE CONTINUED ... SOON

Semi-grand ensembles

2007-03-05T12:14:54Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

Semi-grand ensembles

2007-03-05T12:09:12Z

161.111.27.7:

Semi-grand ensembles

2007-03-05T12:08:18Z

161.111.27.7: /* Semi-grand ensemble at fixed volume and temperature */

Semi-grand ensembles

2007-03-05T12:00:51Z

161.111.27.7: /* Fixed Volume and Temperature */

Semi-grand ensembles

2007-03-05T11:47:29Z

161.111.27.7: New page: == General Features == Semi-grand ensembles are used in Monte Carlo simulation of mixtures. In this ensembles the total number of molecules is fixed, but the composition can change. == F...

Microcanonical ensemble

2007-02-28T13:59:12Z

161.111.27.7: /* Partition function */ N instead of n

== Ensemble variables ==
(One component system, 3-dimensional system, ... ):

* <math> \left. N \right. </math>: Number of Particles

* <math> \left. V \right. </math>: Volume

* <math> \left. E \right. </math>: Internal energy (kinetic + potential)

== Partition function ==

:<math> Q_{NVE} = \frac{1}{h^{3N} N!} \int \int d (p)^{3N} d(q)^{3N} \delta ( H(p,q) - E).
</math>

where:

*<math> \left. h \right. </math> is the [[Planck constant]]

*<math> \left( q \right)^{3N} </math> represents the 3N Cartesian position coordinates.

*<math> \left( p \right)^{3N} </math> represents the 3N momenta.

* <math> H \left(p,q\right) </math> represent the [[Hamiltonian]], i.e. the total energy of the system as a function of coordinates and momenta.

*<math> \delta \left( x \right) </math> is the [[Dirac delta distribution]]

== Thermodynamics ==

:<math> \left. S = k_B \log Q_{NVE} \right. </math>

where:

*<math> \left. S \right. </math> is the [[Entropy|entropy]].

*<math> \left. k_B \right. </math> is the [[Boltzmann constant]]

== References ==
# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Algorithms to Applications", Academic Press
[[Category:Statistical mechanics]]

Hard sphere model

2007-02-27T17:59:47Z

161.111.27.7: /* Equations of state */

[[Image:sphere_green.png|thumb|right]]
== Interaction Potential ==
The hard sphere interaction potential is given by

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

where <math> V\left(r \right) </math> is the potential energy between two spheres at a distance <math> r </math>, and <math> \sigma </math> is the sphere diameter.

== Equations of state ==
Hard sphere fluid:
# See [[Carnahan-Starling]] (three dimensions)
# See Ref.1

== Related systems ==

[[Hard disk|Hard disks]] in a 2-dimensional space

[[Hard rods]] in one dimension

[[Widom-Rowlinson model]]

[[Polydisperse hard spheres]]

== Data ==
[[Hard sphere: virial equation of state|Virial coefficients]] of hard spheres and hard disks

==References==

#[http://dx.doi.org/10.1088/0953-8984/9/41/006 Robin J Speedy 1997 J. Phys.: Condens. Matter 9 8591-8599 doi:10.1088/0953-8984/9/41/006]

[[Category:Models]]

Isothermal-isobaric ensemble

2007-02-26T09:19:39Z

161.111.27.7:

Variables:

* N (Number of particles)
* p (Pressure)
* T (Temperature)
* V (Volume)

The classical partition function, for a one-component atomic system in 3-dimensional space, is given by

<math> Q_{NpT} = \frac{\beta p}{\Lambda^3 N!} \int_{0}^{\infty} d V V^{N} \exp \left[ - \beta p V \right] \int d ( R^*)^{3N} \exp \left[ - \beta U \left(V,(R^*)^{3N} \right) \right]
</math>

where
*<math> \beta = \frac{1}{k_B T} </math>;

*<math> \Lambda </math> is the '''de Broglie''' wavelength

*<math> \left( R^* \right)^{3N} </math> represent the reduced position coordinates of the particles; i.e. <math> \int d ( R^*)^{3N} = 1 </math>

== References ==

# D. Frenkel and B. Smit, "Understanding Molecular Simulation: From Alogrithms to Applications", Academic Press

Talk:Ideal gas assumptions

2007-02-23T09:13:05Z

161.111.27.7:

If the particles have no finite size, are there collisions? (Noe)

--If a tree falls in the forest, does it make a sound? I don't know.. --[[User:Carl McBride|Carl McBride]] 18:36, 22 February 2007 (CET)

I guess it does, --N G Almarza 18:44, 22 February 2007 (CET)

-- In an ideal gas there are no inter-particle interactions. --[[User:Carl McBride|Carl McBride]] 19:03, 22 February 2007 (CET)

-- Is a collision an interaction? --[[User:161.111.27.7|161.111.27.7]] 10:13, 23 February 2007 (CET)

Flexible molecules

2007-02-22T10:09:48Z

161.111.27.7:

Modelling of internal degrees of freedom, usual techniques:

== Bond distances ==
* Atoms linked by a chemical bond (stretching):

<math> V_{str} (r_{12}) = \frac{1}{2} K_{str} ( r_{12} - b_0 )^2 </math>

== Bond Angles ==

Bond sequence: 1-2-3:

Bond Angle: <math> \theta </math>

<math> \cos \theta = \frac{ \vec{r}_{21} \cdot \vec{r}_{23} } {|\vec{r}_{21}| |\vec{r}_{23}|}
</math>

Two typical forms are used to model the ''bending'' potential:

<math>
V_{bend}(\theta) = \frac{1}{2} k_{\theta} \left( \theta - \theta_0 \right)^2
</math>

<math>
V_{bend}(\cos \theta) = \frac{1}{2} k_{c} \left( \cos \theta - c_0 \right)^2
</math>

== Internal Rotation ==

Flexible molecules

2007-02-22T09:58:14Z

161.111.27.7: New page: Modelling of internal degrees of freedom, usual techniques: == Bond distances == * Atoms linked by a chemical bond (stretching): <math> V_{str} (r_{12}) = \frac{1}{2} K_{str} ( r_{12} -...

Models

2007-02-22T09:49:21Z

161.111.27.7:

*[[Hard rods]]
*[[Hard sphere]]
*[[Hard ellipsoids]]
*[[Hard hexagons]]
*[[Hard spherocylinders]]
*[[Hard core Yukawa]]
*[[Square well]]
*[[Square shoulder]]
*[[Square shoulder + square well]]
*[[Ramp]]
*[[Gaussian overlap]]
*[[Gay-Berne]]
*[[Lennard-Jones]]
*[[Flexible Molecules|Flexible Molecules]]
*[[Polyamorphic systems]]
*[[Confined systems]]