http://www.sklogwiki.org/SklogWiki/api.php?action=feedcontributions&user=Lgmac&feedformat=atomSklogWiki - User contributions [en]2024-03-28T13:26:16ZUser contributionsMediaWiki 1.41.0http://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=20264Grand canonical Monte Carlo2019-12-11T02:20:40Z<p>Lgmac: Corrected typo (L subindex inserted) ;)</p>
<hr />
<div>'''Grand-canonical ensemble Monte Carlo''' (GCEMC or GCMC) is a very versatile and powerful [[Monte Carlo]] technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, then one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from an original state (<math>o</math>) to a new state (<math>n</math>) is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} \qquad\qquad\text{(1)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
<br />
As noted by Norman and Filinov <ref>G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)</ref>, evaluation of the proper acceptance rules requires very careful interpretation of the (classical) grand canonical probability density: <br />
<br />
:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math> <br />
<br />
where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the [[De Broglie thermal wavelength|de Broglie thermal wavelength]].<br />
<br />
The sub-index ''L'' makes emphasis on a particular definition of [[microstate]] in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in parenthesis indicate that the positions are not attributed to any particular choice of labelling and the sum runs over all possible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f_U(N+1)}{f_U(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}\qquad\qquad\text{(3)} </math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math>. The <math> 1/2 </math> factor accounts for the probability of attempting an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/(N+1) </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N+1 \rightarrow N )}{\alpha( N \rightarrow N+1)} = \frac{V}{N+1} \qquad\qquad\text{(4)}</math><br />
<br />
Substitution of Eq.(3) and Eq.(4) into Eq.(1) yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq.(2) but taking into account that there are then <math>N+1</math> labelled microstates leading to the original <math>N</math> particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
==See also==<br />
*[[Mass-stat]]<br />
== References ==<br />
<references/> <br />
'''Related reading'''<br />
*[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
*[http://dx.doi.org/10.1080/00268977500100221 D. J. Adams "Grand canonical ensemble Monte Carlo for a Lennard-Jones fluid", Molecular Physics '''29''' pp. 307-311 (1975)]<br />
*[http://dx.doi.org/10.1063/1.2839302 Attila Malasics, Dirk Gillespie, and Dezső Boda "Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms", Journal of Chemical Physics '''128''' 124102 (2008)]<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Phase_diagrams:_Pressure-temperature_plane&diff=11491Phase diagrams: Pressure-temperature plane2011-06-15T13:56:29Z<p>Lgmac: </p>
<hr />
<div>{{Stub-general}}<br />
The following is a schematic phase diagram of a monatomic substance in the temperature-pressure plane. <br />
It shows the vapour, fluid and liquid phases, as well as the crystalline solid phase.<br />
The [[critical points |critical point]] is highlighted by a red spot, and the green spots represent the <br />
[[triple point]]. This is not meant to say that all substances are necessarily green at their triple point,<br />
however (this being the case occasionally).<br />
<br />
[[Image:press_temp.png|center|450px]]<br />
==See also==<br />
*[[Density-temperature]]<br />
*[[Binary phase diagrams]]<br />
[[category:phase diagrams]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8152Wandering interface method2009-05-10T21:59:37Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by employing a [[canonical ensemble]] where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[metropolis Monte Carlo]] scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:<br />
:<math><br />
\exp(W(A)) = <br />
\left \{<br />
\begin{array}{cc}<br />
0 & A < A_{min} \\<br />
1 & A_{min} < A < A_{max} \\<br />
0 & A > A_{max}<br />
\end{array}<br />
\right .<br />
</math><br />
<br />
Several other choices are possible and have been tested since then.<br />
<br />
<br />
==References==<br />
#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8151Wandering interface method2009-05-10T21:53:52Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor [[surface tension]] and solid-fluid interfacial tension alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by employing a [[canonical ensemble]] where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual [[metropolis Monte Carlo]] scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:<br />
:<math><br />
\exp(W(A)) = <br />
\left \{<br />
\begin{array}{cc}<br />
0 & A < A_{min} \\<br />
1 & A_{min} < A < A_{max} \\<br />
0 & A > A_{max}<br />
\end{array}<br />
\right .<br />
</math><br />
<br />
Several other choices are possible and have been tested since then.<br />
<br />
<br />
==References==<br />
#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Surface_tension&diff=8150Surface tension2009-05-10T21:49:55Z<p>Lgmac: added link to wim</p>
<hr />
<div>The '''surface tension''',<br />
<math> \gamma </math>, is a measure of the [[work]] required to create an [[interface]] between<br />
two bulk phases.<br />
<br />
== Thermodynamics == <br />
In the [[Canonical ensemble]] the surface tension is formally given as:<br />
<br />
:<math> \gamma = \frac{ \partial A (N,V,T, {\mathcal A} )}{\partial {\mathcal A} } </math>;<br />
<br />
where<br />
<br />
*<math>A</math> is the [[Helmholtz energy function]]<br />
* <math> N </math> is the number of particles<br />
*<math> V </math> is the volume<br />
*<math> T </math> is the [[temperature]]<br />
*<math> {\mathcal A} </math> is the surface area<br />
<br />
==Computer Simulation==<br />
Different techniques may be used to compute this quantity, such as the traditional [[stress | stress tensor]] route. More recently, several methods have been proposed which avoid the some times difficult calculation of the stress tensor, e.g., the [[Test Area Method]] and [[Wandering Interface Method]]. A review can be found in the paper by Gloor ''et al.'' (Ref. 1).<br />
<br />
==Liquid-Vapour Interfaces of one component systems ==<br />
=== Binder procedure===<br />
Here, only an outline of the procedure is presented, more details can be found in Reference 2.<br />
For given conditions of volume and temperature, the [[Helmholtz energy function]] is computed as a function of the number of molecules, <math> A(N;V,T)</math>. The calculation is usually carried out using [[Monte Carlo]] simulation using [[boundary conditions |periodic boundary conditions]] <br />
If liquid-vapour equilibrium occurs, a plot of the [[chemical potential]], <math> \mu \equiv (\partial A/\partial N)_{V,T} </math>, <br />
as a function of <math> N </math> shows a loop.<br />
Using basic thermodynamic procedures ([[Maxwell's equal area construction]]) it is possible<br />
to compute the densities of the two phases; <math> \rho_v, \rho_l </math> at liquid-vapour equilibrium.<br />
Considering the thermodynamic limit for densities <math> \rho </math> with <math> \rho_v < \rho < \rho_l </math> the <br />
[[Helmholtz energy function]] will be:<br />
<br />
:<math> A(N) = - p_{eq} V + \mu_{eq} N + \gamma {\mathcal A}(N) </math><br />
<br />
where the quantities with the subindex "eq" are those corresponding to the fluid-phase equilibrium situation.<br />
From the previous equation one can write<br />
<br />
:<math> \Omega (N) \equiv A(N) - \mu_{eq} N = - p_{eq} V + \gamma {\mathcal A}(N) </math>.<br />
<br />
For appropriate values of <math> N </math> one can estimate the value of the surface area, <math> {\mathcal A} </math> (See MacDowell ''et al.'', Ref. 3), and compute <math> \gamma </math> directly as:<br />
<br />
:<math> \gamma = \frac{ \Omega(N) + p_{eq} V } { {\mathcal A}(N) } = \frac{ \Omega(N) - \frac{1}{2}(\Omega(N_l)+\Omega(N_v)) }{{\mathcal A}(N)} </math><br />
<br />
where <math> N_l </math> and <math> N_v </math> are given by: <math> N_l = V \cdot \rho_l </math> and <math> N_v = V \cdot \rho_v </math><br />
<br />
=== Explicit interfaces ===<br />
In these methods one performs a direct simulation of the two-phase system. [[boundary conditions |Periodic boundary conditions]] are usually employed.<br />
Simulation boxes are elongated in one direction, and the interfaces are built (and expected to stay) perpendicular to<br />
such a direction.<br />
Taking into account the [[canonical ensemble]] definition (see above), one computes the change in the [[Helmholtz energy function]] when a small (differential)<br />
change of the surface area is performed at constant <math> V, T, </math> and <math> N </math>.<br />
The explicit equations can be written in terms of the diagonal components of the [[pressure]] tensor of the system.<br />
Mechanical arguments can also be invoked to arrive at equivalent conclusions (see Ref 1 for a detailed discussion of these issues).<br />
<br />
=== System-size analysis ===<br />
<br />
The [[Finite size effects |system-size dependence]] of the results for <math> \gamma </math> have to be taken into account in order to obtain accurate results for [[Models |model systems]].<br />
Spurious effects that occur due to small system sizes can appear in the ''explicit interface'' methods. (See P.Orea ''et al.'' Ref. 4).<br />
<br />
== Mixtures ==<br />
<br />
Different ensembles can be used to compute the surface tension between two phases in the case of mixtures (See for example see Y. Zhang ''et al.'' Ref 5).<br />
The simulation techniques are essentially the same as those for one-component systems, but different ensembles can be more adequate.<br />
For instance, for binary mixtures (with components 1 and 2), the [[isothermal-isobaric ensemble]], <math> N_1,N_2,p.T </math> is a ''more natural'' ensemble to compute<br />
<math> \gamma </math> using explicit interface techniques (See Ref. 6 as an example). <br />
In the case of the Binder technique<br />
the analysis can be carried out by fixing the total number of particles: <math> N \equiv N_1 + N_2 </math>, the [[pressure]], <math> p </math>, and the [[temperature]] <math> T </math>. Then<br />
one will have to compute the variation of the adequate thermodynamic potential as a function of the composition, e.g. <math> x_1 = N_1/N </math>.<br />
<br />
==See also==<br />
*[[Line tension]]<br />
*[[Boundary tension]]<br />
*[[Droplets]]<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.2038827 Guy J. Gloor, George Jackson, Felipe J. Blas and Enrique de Miguel "Test-area simulation method for the direct determination of the interfacial tension of systems with continuous or discontinuous potentials", Journal of Chemical Physics '''123''' 134703 (2005)]<br />
#[http://dx.doi.org/10.1103/PhysRevA.25.1699 K. Binder "Monte Carlo calculation of the surface tension for two- and three-dimensional lattice-gas models", Physical Review A '''25''' pp. 1699 - 1709 (1982)]<br />
#[http://dx.doi.org/10.1063/1.2218845 L. G. MacDowell, V. K. Shen, and J. R. Errington "Nucleation and cavitation of spherical, cylindrical, and slablike droplets and bubbles in small systems", Journal of Chemical Physics '''125''' 034705 (2006)]<br />
#[http://dx.doi.org/10.1063/1.2018640 Pedro Orea, Jorge López-Lemus, and José Alejandre, "Oscillatory surface tension due to finite-size effects", Journal of Chemical Physics '''123''' 114702 (6 pages) (2005)]<br />
#[http://dx.doi.org/10.1063/1.469927 Yuhong Zhang, Scott E. Feller, Bernard R. Brooks, and Richard W. Pastor, "Computer simulation of liquid/liquid interfaces. I. Theory and application to octane/water", Journal of Chemical Physics, '''103''', pp. 10252-10266 (1995)]<br />
#[http://dx.doi.org/10.1063/1.2751153 E. de Miguel, N. G. Almarza, and G. Jackson, "Surface tension of the Widom-Rowlinson model", Journal of Chemical Physics, '''127''', 034707 (10 pages) (2007)]<br />
#[http://dx.doi.org/10.1063/1.1747248 John G. Kirkwood and Frank P. Buff "The Statistical Mechanical Theory of Surface Tension", Journal of Chemical Physics '''17''' pp. 338-343 (1949)]<br />
#[http://dx.doi.org/10.1016/0021-9991(76)90078-4 Charles H. Bennett "Efficient estimation of free energy differences from Monte Carlo data", Journal of Computational Physics '''22''' pp. 245-268 (1976)]<br />
#[http://dx.doi.org/10.1063/1.432627 J. Miyazaki, J. A. Barker and G. M. Pound "A new Monte Carlo method for calculating surface tension", Journal of Chemical Physics '''64''' pp. 3364-3369 (1976)]<br />
[[category: statistical mechanics]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8149Wandering interface method2009-05-10T21:41:28Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike. It does not require the explicit evaluation of the virial and may be employed for both continuous and discontinuous model potentials.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by employing a canonical ensemble where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:<br />
:<math><br />
\exp(W(A)) = <br />
\left \{<br />
\begin{array}{cc}<br />
0 & A < A_{min} \\<br />
1 & A_{min} < A < A_{max} \\<br />
0 & A > A_{max}<br />
\end{array}<br />
\right .<br />
</math><br />
<br />
Several other choices are possible and have been tested since then.<br />
<br />
<br />
==References==<br />
#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8148Wandering interface method2009-05-10T21:39:21Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by considering a canonical ensemble, with the usual constant N, V, T variables, where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval. In the original implementation of WIM, the following biasing function was chosen:<br />
:<math><br />
\exp(W(A)) = <br />
\left \{<br />
\begin{array}{cc}<br />
0 & A < A_{min} \\<br />
1 & A_{min} < A < A_{max} \\<br />
0 & A > A_{max}<br />
\end{array}<br />
\right .<br />
</math><br />
<br />
Several other choices are possible and have been tested since then.<br />
<br />
<br />
==References==<br />
#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8147Wandering interface method2009-05-10T21:37:52Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by considering a canonical ensemble, with the usual constant N, V, T variables, where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval.<br />
<br />
:<math><br />
exp(W(A)) = <br />
\left \{<br />
\begin{array}{cc}<br />
0 & A < A_{min} \\<br />
1 & A_{min} < A < A_{max} \\<br />
0 & A > A_{max}<br />
\end{array}<br />
\right .<br />
</math><br />
<br />
<br />
==References==<br />
#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8146Wandering interface method2009-05-10T21:22:41Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by considering a canonical ensemble, with the usual constant N, V, T variables, where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval.<br />
<br />
==References==<br />
#[http://link.aps.org/doi/10.1103/PhysRevE.75.061609 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8145Wandering interface method2009-05-10T21:19:19Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by considering a canonical ensemble, with the usual constant N, V, T variables, where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval.<br />
<br />
==References==<br />
#[http://dx.doi.org/10.1063/1.2799990 L.G. MacDowell and P. Bryk "Direct Calculation of Interfacial Tensions from Computer Simulation: Results for Freely Jointed Tangent Hard Sphere Chains", Physical Review E '''75''' 061609 (2007)]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8144Wandering interface method2009-05-10T21:07:58Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
This goal is achieved in practice by considering a canonical ensemble, with the usual constant N, V, T variables, where the box shape as described by the surface area, <br />
<math>A</math> is considered as an extra ensemble variable.<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can show that:<br />
:<math> k_BT\ln P(A) \propto -\gamma A </math><br />
Hence, the surface tension, <math>\gamma</math> may be readily evaluated as the slope of <math>P(A)</math> in a logarithmic plot.<br />
<br />
Since the system will tend to stretch continuously so as to achieve a vanishing surface area, in practice one must constraint the extent of deformations in order to bracket the surface area fluctuations within a convenient interval. This is easily achieved in practice by considering a modified canonical ensemble where the microstate probability distribution is given by:<br />
:<math> P(U,A) \propto \exp(-\beta U + W(A) )</math>.<br />
where <math>W</math> is a biasing function chosen so as to constraint the fluctuations within some convenient interval.</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8143Wandering interface method2009-05-10T20:53:22Z<p>Lgmac: </p>
<hr />
<div>The Wandering Interface Method (WIM) is a general, versatile and robust procedure for the calculation of surface free energies. It can be employed to calculate liquid-vapor surface tensions and solid-fluid interfacial tensions alike.<br />
<br />
The method is based on a clear and appealing physical basis: since an interface has usually an extra energy cost, a system will try to stretch in order to minimize the interfacial area. The original feature of WIM is to allow the system's interfacial area to fluctuate freely and extract the surface free energy from the related surface area probability distribution <math> P(A)</math>. <br />
<br />
During the course of the simulation, trial deformations of the simulation box at constant volume are attempted and either accepted or rejected according to the usual metropolis scheme. This provides a markovian random walk of the interfacial area. Collecting the surface area probability distribution, one can readily calculate the surface free energy, since, according to statistical mechanics, one can showv that:<br />
:<math> k_BT\ln P(A) \propto \gamma A </math><br />
<br />
:<math> \ln P(A) = \gamma /k_BT </math></div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Wandering_interface_method&diff=8142Wandering interface method2009-05-10T20:01:42Z<p>Lgmac: New page: :<math> \ln P(A) = \gamma /k_BT </math></p>
<hr />
<div>:<math> \ln P(A) = \gamma /k_BT </math></div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Computer_simulation_techniques&diff=8141Computer simulation techniques2009-05-10T19:38:14Z<p>Lgmac: </p>
<hr />
<div>*[[Molecular dynamics]]<br />
*[[Monte Carlo]]<br />
==Material in common==<br />
*[[Boundary conditions]]<br />
*[[Coarse graining]]<br />
*[[Computation of phase equilibria]]<br />
*[[Configuration analysis]]<br />
*[[Dissipative particle dynamics]]<br />
*[[Electrostatics]]<br />
*[[Ergodic hypothesis]]<br />
*[[Expanded ensemble method]]<br />
*[[Finite size effects]]<br />
*[[Force fields]]<br />
*[[Gibbs-Duhem integration]]<br />
*[[Materials modelling and computer simulation codes]]<br />
*[[Models]]<br />
*[[Self-referential method]]<br />
*[[Smooth Particle methods]]<br />
*[[Tempering methods]]<br />
*[[Test area method]]<br />
*[[Test volume method]]<br />
*[[Verlet neighbour list]]<br />
*[[Wandering Interface Method]]<br />
*[[Widom test-particle method]]<br />
<br />
==Interesting reading==<br />
* W. W. Wood "Early history of computer simulations in statistical mechanics" in "Molecular-dynamics simulation of statistical-mechanical systems" (Eds. Giovanni Ciccotti and William G. Hoover) pp. 3-14 Società Italiana di Fisica (1986)<br />
*[http://physicsworldarchive.iop.org/full/pwa-pdf/9/4/phwv9i4a24.pdf Daan Frenkel and Jean-Pierre Hansen "Understanding liquids: a computer game?", Physics World '''9''' pp. 35–42 (April 1996)]<br />
*[http://arxiv.org/abs/0812.2086 Wm. G. Hoover "50 Years of Computer Simulation -- A Personal View", arXiv:0812.2086v2 (2008)]<br />
[[category: Computer simulation techniques]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Landau_theory_of_phase_transitions&diff=8043Landau theory of phase transitions2009-04-01T13:23:19Z<p>Lgmac: </p>
<hr />
<div>In the '''Landau theory of phase transitions''' of [[Lev Davidovich Landau]], the [[thermodynamic potential]] takes the form of a [[power series]] in the [[order parameters |order parameter]] (<math>\eta</math>) (Ref. 3 Eq. 143.1)<br />
:<math>\left.\Phi(p, T, \eta)\right. = \Phi_0 + \alpha(p,T)\eta + A(p,T)\eta^2 + C(p,T)\eta^3 + B(p,T)\eta^4 + ...,</math><br />
where ''p'' is the [[pressure]] and ''T'' is the [[temperature]].<br />
The the equilibrium state consistent with the external parameters <math>p</math> and <math>T</math> is a minimum of the Landau potential (often also called Landau free energy). This yields a prescription for determining the equilibrium value adopted by <math>\eta</math><br />
<br />
:<math>\left ( \frac{\partial\Phi}{\partial \eta}\right )_{\eta=\eta_{eq}} = 0.</math><br />
<br />
Taking into account very general symmetry properties of the underlying [[Hamiltonian]] one can set a priori the qualitative behaviour of the coefficients in the series. The value of those coefficients then very generally define the nature of the phase transition, which may be found to be either first or second order.<br />
<br />
The Landau theory was the first successful attempt to highlight the universal features of phase transitions. However, the theory is known to fail for [[critical transitions]] of several types because of the neglect of fluctuations (i.e., the fact that the order parameter may fluctuate about the minima of <math>\eta</math> and hence tunnel across free energy barriers separating two such minima.<br />
<br />
==Illustration for the [[Ising model]]==<br />
<br />
For the [[Ising model]] one chooses the magnetization as order parameter. In the absence of external field, the underlying Hamiltonian is invariant to spin exchange. It must then follow that <math>\Phi</math> is also invariant to the sign of the magnetization. Hence, all odd coefficients of <math>\eta</math> must vanish. For qualitative purposes, the free energy may be truncated to fourth order, thus:<br />
<br />
:<math> \Phi(T;\eta) = \Phi_0 + A\eta^2 + C\eta^4 </math><br />
<br />
Equilibrium values for <math>\eta</math> must then obey:<br />
<br />
:<math> \eta \left (2A + 4C\eta^2 \right ) = 0 </math><br />
<br />
This provides three possible candidates for <math>\eta_{\rm eq}</math>, mainly <math>\eta_{\rm eq}=0</math> and <math>\eta_{\rm eq} = \pm (-A/2C)^{1/2} </math>. In order to rule out some candidates, we note that there must be only one possible solution above the [[Curie temperature]], <math> T_c </math> (the system is then not magnetized), while two such solutions are expected below the Curie temperature. These conditions are fulfilled only if <math>A</math> is an odd function of temperature, taking negative values below <math>T_c</math> and positive values at high temperature. The simplest qualitative choice satisfying such conditions is <math>A\propto T-T_c</math>. The full behaviour then follows as:<br />
=== Low temperature solution ===<br />
At low temperature, there are two possible solutions (<math>\eta_{\rm eq}=0</math> is here not a minimum but a maximum) corresponding to finite magnetization of opposite sign:<br />
:<math> \eta_{\rm eq} = \pm \sqrt{-\frac{A}{2C}} </math><br />
The spontaneous emergence of finite magnetization from a Hamiltonian that is invariant to spin exchange is known as [[symmetry breaking]].<br />
<br />
As the critical point is approached, <math>\Delta \eta = \eta_+ - \eta_-</math> may be approximated as a power of the distance away from the critical point, <math>\Delta \eta \propto (T-T_c)^{\beta}</math>. From the solution of the Landau functional for the spontaneous magnetization, we find:<br />
<br />
:<math> \Delta \eta \propto t^{1/2} </math><br />
<br />
Thus, according to the Landau's theory, the approach to criticality is governed by a power law with universal exponent <math>\beta=1/2</math>, regardless of any specific detail of the ferromagnetic fluid. To point out the universal behavior of the exponent <math>\beta</math> is an achievments of Landau's theory. However, experimental studies have shown that the exponent is not quite 1/2, but rather, close to 1/3.<br />
<br />
The explanation of this discrepancy is the goal of renormalization group theory of second order phase transitions. <br />
=== High temperature solution ===<br />
Above the Curie temperature there is only one possible solution corresponding to the disordered phase of zero magnetization,<math>\eta_{\rm eq}=0</math> .<br />
<br />
The qualitative nature of liquid--vapour phase transitions follow the same reasoning, with the order parameter now defined as <math> \eta = (\rho - \rho_c )/\rho_c</math>. The Ising model then usually takes the name of [[Lattice gas]], with spins down standing for empty space, and spins up for molecules (or droplets). Such Lattice gas is a caricature of a compressible fluid along the coexistence line.<br />
== Criticism ==<br />
The Landau theory is phenomenological in nature. Nothing is known a priori about the coefficients of the series. Whereas this is a strong point in order to highlight universal features, it is clear that any useful assumption about the symmetry or vanishing of such coefficients requires a fair understanding of the transition beforehand<br />
==References==<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 26-47 (1937)<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 545-555 (1937)<br />
#L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Volume 5) 3rd Edition Part 1, Chapter XIV, Pergamon Press (1980)<br />
#[http://dx.doi.org/10.1088/0022-3719/9/9/015 A P Cracknell, J Lorenc and J A Przystawa "Landau's theory of second-order phase transitions and its application to ferromagnetism", Journal of Physics C: Solid State Physics '''9''' pp. 1731-1758 (1976)]<br />
[[category: classical thermodynamics]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Landau_theory_of_phase_transitions&diff=7836Landau theory of phase transitions2009-02-19T15:41:39Z<p>Lgmac: Significance + application to Ising model</p>
<hr />
<div>{{Stub-general}}<br />
In the '''Landau theory of phase transitions''' of [[Lev Davidovich Landau]], the [[thermodynamic potential]] takes the form of a [[power series]] in the [[order parameters |order parameter]] (<math>\eta</math>) (Ref. 3 Eq. 143.1)<br />
:<math>\left.\Phi(p, T, \eta)\right. = \Phi_0 + \alpha(p,T)\eta + A(p,T)\eta^2 + C(p,T)\eta^3 + B(p,T)\eta^4 + ...,</math><br />
where ''p'' is the [[pressure]] and ''T'' is the [[temperature]].<br />
The the equilibrium state consistent with the external parameters <math>p</math> and <math>T</math> is a minimum of the Landau potential (often also called Landau free energy). This yields a prescription for determining the equilibrium value adopted by <math>\eta</math><br />
<br />
:<math>\left ( \frac{\partial\Phi}{\partial \eta}\right )_{\eta=\eta_{eq}} = 0.</math><br />
<br />
Taking into account very general symmetry properties of the underlying [[Hamiltonian]] one can set a priori the qualitative behavior of the coefficients in the series. The value of those coefficients then very generally define the nature of the phase transition, which may be found to be either first or second order.<br />
<br />
The Landau theory was the first succesful attempt to highlight the universal features of phase transitions. However, the theory is known to fail for [[critical transitions]] of several types because of the neglect of fluctuations (i.e., the fact that the order parameter may fluctuate about the minima of <math>\eta</math> and hence tunnel across free energy barriers separating two such minima.<br />
<br />
==Illustration for the [[Ising model]]==<br />
<br />
For the [[Ising model]] one chooses the magnetization as order parameter. In the absence of external field, the underlying Hamiltonian is invariant to spin exchange. It must then follow that <math>\Phi</math> is also invariant to the sign of the magnetization. Hence, all odd coefficients of <math>\eta</math> must vanish. For qualitative purposes, the free energy may be truncated to fourth order, thus:<br />
<br />
:<math> \Phi(T;\eta) = \Phi_0 + A\eta^2 + C\eta^4 </math><br />
<br />
Equilibrium values for <math>\eta</math> must then obey:<br />
<br />
:<math> \eta \left (2A + 4C\eta^2 \right ) = 0 </math><br />
<br />
This provides three possible candidates for <math>\eta_{\rm eq}</math>, mainly <math>\eta_{\rm eq}=0</math> and <math>\eta_{\rm eq} = \pm (-A/2C)^{1/2} </math>. In order to rule out some condidates, we note that there must be only one possible solution above the [[Curie temperature]], <math> T_c </math> (the system is then not magnetized), while two such solutions are expected below the Curie temperature. These conditions are fulfilled only if <math>A</math> is an odd function of temperature, taking negative values below <math>T_c</math> and positive values at high temperature. The simplest qualitative choice satisfying such conditions is <math>A\propto T-T_c</math>. The full behavior then follows as:<br />
<br />
=== Low temperature solution ===<br />
<br />
At low temperature, there are two possible solutions (<math>\eta_{\rm eq}=0</math> is here not a minimum but a maximum) corresponding to finite magnetization of opposite sign:<br />
<math> \eta_{\rm eq} = \pm \sqrt{-\frac{A}{2C}} </math><br />
<br />
The spontaneous emergence of finite magnetization from a Hamiltonian that is invariant to spin exchange is known as [[symmetry breaking]].<br />
<br />
=== High temperature solution ===<br />
<br />
Above the Curie temperature there is only one possible solution corresponding to the disordered phase of zero magnetization,<math>\eta_{\rm eq}=0</math> .<br />
<br />
The qualitative nature of liquid--vapor phase transitions follow the same reasoning, with the order parameter now defined as <math> \eta = (\rho - \rho_c )/\rho_c</math>. The Ising model then usually takes the name of [[Lattice gas]], with spins down standing for empty space, and spins up for molecules (or droplets). Such Lattice gas is a charicature of a compressible fluid along the coexistence line.<br />
== Criticism ==<br />
<br />
The Landau theory is phenomenological in nature. Nothing is known a priori about the coefficients of the series. Whereas this is a strong point in order to highlight universal features, it is clear that any useful assumption about the symmetry or vanishing of such coefficients requires a fair understanding of the transition beforehand<br />
<br />
<br />
<br />
<br />
<br />
==References==<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 26-47 (1937)<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 545-555 (1937)<br />
#L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Volume 5) 3rd Edition Part 1, Chapter XIV, Pergamon Press (1980)<br />
#[http://dx.doi.org/10.1088/0022-3719/9/9/015 A P Cracknell, J Lorenc and J A Przystawa "Landau's theory of second-order phase transitions and its application to ferromagnetism", Journal of Physics C: Solid State Physics '''9''' pp. 1731-1758 (1976)]<br />
[[category: classical thermodynamics]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Landau_theory_of_second-order_phase_transitions&diff=7835Talk:Landau theory of second-order phase transitions2009-02-19T15:38:05Z<p>Lgmac: Talk:Landau theory of second-order phase transitions moved to Talk:Landau theory of phase transitions</p>
<hr />
<div>#REDIRECT [[Talk:Landau theory of phase transitions]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Landau_theory_of_phase_transitions&diff=7834Talk:Landau theory of phase transitions2009-02-19T15:38:05Z<p>Lgmac: Talk:Landau theory of second-order phase transitions moved to Talk:Landau theory of phase transitions</p>
<hr />
<div>Landau Theory describes phase transitions generally, whether first or second order. So title of article should change to "Landau theory of phase transitions".<br />
:Good point. Please feel free to re-name the page by clicking on the 'move' tab at the top of the page. --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 15:35, 19 February 2009 (CET)<br />
<br />
Incredible!!! Ising Model not yet defined. I cannot do this whole Wiki myself!!!</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Landau_theory_of_second-order_phase_transitions&diff=7833Landau theory of second-order phase transitions2009-02-19T15:38:05Z<p>Lgmac: Landau theory of second-order phase transitions moved to Landau theory of phase transitions</p>
<hr />
<div>#REDIRECT [[Landau theory of phase transitions]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Landau_theory_of_phase_transitions&diff=7832Landau theory of phase transitions2009-02-19T15:38:05Z<p>Lgmac: Landau theory of second-order phase transitions moved to Landau theory of phase transitions</p>
<hr />
<div>{{Stub-general}}<br />
In the '''Landau theory of phase transitions''' of [[Lev Davidovich Landau]], the [[thermodynamic potential]] takes the form of a [[power series]] in the [[order parameters |order parameter]] (<math>\eta</math>) (Ref. 3 Eq. 143.1)<br />
:<math>\left.\Phi(p, T, \eta)\right. = \Phi_0 + \alpha(p,T)\eta + A(p,T)\eta^2 + C(p,T)\eta^3 + B(p,T)\eta^4 + ...,</math><br />
where ''p'' is the [[pressure]] and ''T'' is the [[temperature]].<br />
The the equilibrium state consistent with the external parameters <math>p</math> and <math>T</math> is a minimum of the Landau potential (often also called Landau free energy). This yields a prescription for determining the equilibrium value adopted by <math>\eta</math><br />
<br />
:<math>\left ( \frac{\partial\Phi}{\partial \eta}\right )_{\eta=\eta_{eq}} = 0.</math><br />
<br />
Taking into account very general symmetry properties of the underlying [[Hamiltonian]] one can set a priori the qualitative behavior of the coefficients in the series. The value of those coefficients then very generally define the nature of the phase transition, which may be found to be either first or second order.<br />
<br />
The Landau theory was the first succesful attempt to highlight the universal features of phase transitions. However, the theory is known to fail for [[critical transitions]] of several types because of the neglect of fluctuations (i.e., the fact that the order parameter may fluctuate about the minima of <math>\eta</math> and hence tunnel across free energy barriers separating two such minima.<br />
<br />
==Illustration for the [[Ising model]]==<br />
<br />
For the [[Ising model]] one chooses the magnetization as order parameter. In the absence of external field, the underlying Hamiltonian is invariant to spin exchange. It must then follow that <math>\Phi</math> is also invariant to the sign of the magnetization. Hence, all odd coefficients of <math>\eta</math> must vanish. For qualitative purposes, the free energy may be truncated to fourth order, thus:<br />
<br />
:<math> \Phi(T;\eta) = \Phi_0 + A\eta^2 + C\eta^4 </math><br />
<br />
Equilibrium values for <math>\eta</math> must then obey:<br />
<br />
:<math> \eta \left (2A + 4C\eta^2 \right ) = 0 </math><br />
<br />
This provides three possible candidates for <math>\eta_{\rm eq}</math>, mainly <math>\eta_{\rm eq}=0</math> and <math>\eta_{\rm eq} = \pm (-A/2C)^{1/2} </math>. In order to rule out some condidates, we note that there must be only one possible solution above the [[Curie temperature]], <math> T_c </math> (the system is then not magnetized), while two such solutions are expected below the Curie temperature. These conditions are fulfilled only if <math>A</math> is an odd function of temperature, taking negative values below <math>T_c</math> and positive values at high temperature. The simplest qualitative choice satisfying such conditions is <math>A\propto T-T_c</math>. The full behavior then follows as:<br />
<br />
=== Low temperature solution ===<br />
<br />
At low temperature, there are two possible solutions (<math>\eta_{\rm eq}=0</math> is here not a minimum but a maximum) corresponding to finite magnetization of opposite sign:<br />
<math> \eta_{\rm eq} = \pm \sqrt{-\frac{A}{2C}} </math><br />
<br />
The spontaneous emergence of finite magnetization from a Hamiltonian that is invariant to spin exchange is known as [[symmetry breaking]].<br />
<br />
=== High temperature solution ===<br />
<br />
Above the Curie temperature there is only one possible solution corresponding to the disordered phase of zero magnetization,<math>\eta_{\rm eq}=0</math> .<br />
<br />
The qualitative nature of liquid--vapor phase transitions follow the same reasoning, with the order parameter now defined as <math> \eta = (\rho - \rho_c )/\rho_c</math>. The Ising model then usually takes the name of [[Lattice gas]], with spins down standing for empty space, and spins up for molecules (or droplets). Such Lattice gas is a charicature of a compressible fluid along the coexistence line.<br />
== Criticism ==<br />
<br />
The Landau theory is phenomenological in nature. Nothing is known a priori about the coefficients of the series. Whereas this is a strong point in order to highlight universal features, it is clear that any useful assumption about the symmetry or vanishing of such coefficients requires a fair understanding of the transition beforehand.<br />
<br />
<br />
<br />
<br />
==References==<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 26-47 (1937)<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 545-555 (1937)<br />
#L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Volume 5) 3rd Edition Part 1, Chapter XIV, Pergamon Press (1980)<br />
#[http://dx.doi.org/10.1088/0022-3719/9/9/015 A P Cracknell, J Lorenc and J A Przystawa "Landau's theory of second-order phase transitions and its application to ferromagnetism", Journal of Physics C: Solid State Physics '''9''' pp. 1731-1758 (1976)]<br />
[[category: classical thermodynamics]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Landau_theory_of_phase_transitions&diff=7831Talk:Landau theory of phase transitions2009-02-19T15:37:23Z<p>Lgmac: </p>
<hr />
<div>Landau Theory describes phase transitions generally, whether first or second order. So title of article should change to "Landau theory of phase transitions".<br />
:Good point. Please feel free to re-name the page by clicking on the 'move' tab at the top of the page. --[[User:Carl McBride | <b><FONT COLOR="#8B3A3A">Carl McBride</FONT></b>]] ([[User_talk:Carl_McBride |talk]]) 15:35, 19 February 2009 (CET)<br />
<br />
Incredible!!! Ising Model not yet defined. I cannot do this whole Wiki myself!!!</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=File:Simeq0.png&diff=7830File:Simeq0.png2009-02-19T15:33:56Z<p>Lgmac: </p>
<hr />
<div>General stub image<br />
<br />
See discussion</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Landau_theory_of_phase_transitions&diff=7829Landau theory of phase transitions2009-02-19T15:33:04Z<p>Lgmac: </p>
<hr />
<div>{{Stub-general}}<br />
In the '''Landau theory of phase transitions''' of [[Lev Davidovich Landau]], the [[thermodynamic potential]] takes the form of a [[power series]] in the [[order parameters |order parameter]] (<math>\eta</math>) (Ref. 3 Eq. 143.1)<br />
:<math>\left.\Phi(p, T, \eta)\right. = \Phi_0 + \alpha(p,T)\eta + A(p,T)\eta^2 + C(p,T)\eta^3 + B(p,T)\eta^4 + ...,</math><br />
where ''p'' is the [[pressure]] and ''T'' is the [[temperature]].<br />
The the equilibrium state consistent with the external parameters <math>p</math> and <math>T</math> is a minimum of the Landau potential (often also called Landau free energy). This yields a prescription for determining the equilibrium value adopted by <math>\eta</math><br />
<br />
:<math>\left ( \frac{\partial\Phi}{\partial \eta}\right )_{\eta=\eta_{eq}} = 0.</math><br />
<br />
Taking into account very general symmetry properties of the underlying [[Hamiltonian]] one can set a priori the qualitative behavior of the coefficients in the series. The value of those coefficients then very generally define the nature of the phase transition, which may be found to be either first or second order.<br />
<br />
The Landau theory was the first succesful attempt to highlight the universal features of phase transitions. However, the theory is known to fail for [[critical transitions]] of several types because of the neglect of fluctuations (i.e., the fact that the order parameter may fluctuate about the minima of <math>\eta</math> and hence tunnel across free energy barriers separating two such minima.<br />
<br />
==Illustration for the [[Ising model]]==<br />
<br />
For the [[Ising model]] one chooses the magnetization as order parameter. In the absence of external field, the underlying Hamiltonian is invariant to spin exchange. It must then follow that <math>\Phi</math> is also invariant to the sign of the magnetization. Hence, all odd coefficients of <math>\eta</math> must vanish. For qualitative purposes, the free energy may be truncated to fourth order, thus:<br />
<br />
:<math> \Phi(T;\eta) = \Phi_0 + A\eta^2 + C\eta^4 </math><br />
<br />
Equilibrium values for <math>\eta</math> must then obey:<br />
<br />
:<math> \eta \left (2A + 4C\eta^2 \right ) = 0 </math><br />
<br />
This provides three possible candidates for <math>\eta_{\rm eq}</math>, mainly <math>\eta_{\rm eq}=0</math> and <math>\eta_{\rm eq} = \pm (-A/2C)^{1/2} </math>. In order to rule out some condidates, we note that there must be only one possible solution above the [[Curie temperature]], <math> T_c </math> (the system is then not magnetized), while two such solutions are expected below the Curie temperature. These conditions are fulfilled only if <math>A</math> is an odd function of temperature, taking negative values below <math>T_c</math> and positive values at high temperature. The simplest qualitative choice satisfying such conditions is <math>A\propto T-T_c</math>. The full behavior then follows as:<br />
<br />
=== Low temperature solution ===<br />
<br />
At low temperature, there are two possible solutions (<math>\eta_{\rm eq}=0</math> is here not a minimum but a maximum) corresponding to finite magnetization of opposite sign:<br />
<math> \eta_{\rm eq} = \pm \sqrt{-\frac{A}{2C}} </math><br />
<br />
The spontaneous emergence of finite magnetization from a Hamiltonian that is invariant to spin exchange is known as [[symmetry breaking]].<br />
<br />
=== High temperature solution ===<br />
<br />
Above the Curie temperature there is only one possible solution corresponding to the disordered phase of zero magnetization,<math>\eta_{\rm eq}=0</math> .<br />
<br />
The qualitative nature of liquid--vapor phase transitions follow the same reasoning, with the order parameter now defined as <math> \eta = (\rho - \rho_c )/\rho_c</math>. The Ising model then usually takes the name of [[Lattice gas]], with spins down standing for empty space, and spins up for molecules (or droplets). Such Lattice gas is a charicature of a compressible fluid along the coexistence line.<br />
== Criticism ==<br />
<br />
The Landau theory is phenomenological in nature. Nothing is known a priori about the coefficients of the series. Whereas this is a strong point in order to highlight universal features, it is clear that any useful assumption about the symmetry or vanishing of such coefficients requires a fair understanding of the transition beforehand.<br />
<br />
<br />
<br />
<br />
==References==<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 26-47 (1937)<br />
#Lev Davidovich Landau "", Physikalische Zeitschrift der Sowjetunion '''11''' pp. 545-555 (1937)<br />
#L. D. Landau and E. M. Lifshitz, "Statistical Physics" (Course of Theoretical Physics, Volume 5) 3rd Edition Part 1, Chapter XIV, Pergamon Press (1980)<br />
#[http://dx.doi.org/10.1088/0022-3719/9/9/015 A P Cracknell, J Lorenc and J A Przystawa "Landau's theory of second-order phase transitions and its application to ferromagnetism", Journal of Physics C: Solid State Physics '''9''' pp. 1731-1758 (1976)]<br />
[[category: classical thermodynamics]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Landau_theory_of_phase_transitions&diff=7826Talk:Landau theory of phase transitions2009-02-19T14:32:43Z<p>Lgmac: </p>
<hr />
<div>Landau Theory describes phase transitions generally, whether first or second order. So title of article should change to "Landau theory of phase transitions".</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Talk:Landau_theory_of_phase_transitions&diff=7825Talk:Landau theory of phase transitions2009-02-19T14:31:37Z<p>Lgmac: New page: Landau Theory describes phase transitions generally, whether first or second order.</p>
<hr />
<div>Landau Theory describes phase transitions generally, whether first or second order.</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Raoult%27s_law&diff=7789Raoult's law2009-02-18T18:49:42Z<p>Lgmac: Extension to other solutions.</p>
<hr />
<div>'''Raoult's law''' states that the [[vapour pressure]] of an [[ideal solution]] of two components is:<br />
:<math>P_v = X_A P^*_{v,A} + X_B P^*_{v,B}</math><br />
where <math>X_i</math> is the [[molar fraction]] of component i, and <math>P^*_{v,i}</math> is the vapour pressure of pure i.<br />
<br />
More generally, '''Raoult's law''' describes the [[partial pressure]] of component A in the vapour coexisting with a liquid mixture as:<br />
<br />
:<math> P_A = X_A P^*_{v,A} </math>.<br />
<br />
This law is obeyed for all components of an ideal solution, and is also obeyed for the solvent of an [[ideal dilute solution]]. The solute's partial pressure of such solutions then obey [[ Henry's law]]. Ideal dilute solutions describe the limiting behavior of a mixture of infinite dilution. Therefore, all solutions in the limit of infinite dilution obey Raoult's law, i.e.:<br />
<br />
:<math> \lim_{X_A \rightarrow 1} P_A = X_A P^*_{v,A} </math>.<br />
<br />
<br />
[[category: classical thermodynamics]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=User:Lgmac&diff=7123User:Lgmac2008-08-28T09:06:20Z<p>Lgmac: New page: {{Author_Lgmac| <!-- Use your Arxiv name here --> first_name = Luis| middle_initial = G.| last_name = MacDowell| image_name = xxx.jpg| image_caption = Me.| position = Profesor Titular...</p>
<hr />
<div>{{Author_Lgmac|<br />
<br />
<!-- Use your Arxiv name here --><br />
<br />
first_name = Luis|<br />
middle_initial = G.|<br />
last_name = MacDowell|<br />
<br />
image_name = xxx.jpg|<br />
image_caption = Me.|<br />
<br />
position = Profesor Titular|<br />
institution = [http://www.ucm.es Universidad Complutense de Madrid] |<br />
research_group = [http://www.ucm.es/info/molecsim/ Grupo de Termodinámica Estadística de Fluidos Moleculares]| <br />
<br />
homepage_link = [http://www.ucm.es/info/molecsim/luis.html Luis G. MacDowell] |<br />
<br />
<br />
groups = |<br />
collaborators = |<br />
<br />
undergrad_institution = [http://www.ucm.es Universidad Complutense de Madrid]|<br />
undergrad_years = 1989-1994|<br />
grad_institution = [http://www.ucm.es Universidad Complutense de Madrid]|<br />
grad_years = 1996-2000|<br />
grad_advisor = [Dr. Carlos Vega] |<br />
thesis = Termodinámica Estadística de Moléculas Flexibles: Teoría y Simulación. |<br />
<br />
blog_link = None. |<br />
email = See my home page. |<br />
grad_students = None. |<br />
postdocs = None. |<br />
undergrad_students = None. |<br />
}}</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Template:Author_Lgmac&diff=7122Template:Author Lgmac2008-08-28T08:56:55Z<p>Lgmac: New page: Current Information - Affiliations - Education - Students - Biography__NOTOC__ <hr> '''[http://www....</p>
<hr />
<div>[[#Current Information|Current Information]] - [[#Affiliations|Affiliations]] - [[#Education|Education]] - [[#Students|Students]] - [[#Biography|Biography]]__NOTOC__<br />
<br />
<hr><br />
<br />
'''[http://www.google.com/search?q={{{first_name}}}+{{{last_name}}} Google me!]''' - '''[http://scholar.google.com/scholar?hl=en&lr=&safe=off&q=author%3A%22{{{first_name}}}+{{{last_name}}}%22&btnG=Search Google Scholar]''' - '''[http://arxiv.org/find/cond-mat/1/au:+{{{last_name}}}_{{{first_name}}}/0/1/0/all/0/1 Arxiv papers]''' - '''[http://scitation.aip.org/vsearch/servlet/VerityServlet?KEY=ALL&smode=strresults&CURRENT=&ONLINE=&SMODE=&possible1zone=article&maxdisp=10&possible1={{{first_name}}}+{{{last_name}}}&submit.x=0&submit.y=0&submit=search&sType=on Journal papers on Scitation]'''<br />
<br />
<hr><br />
<br />
<h2 id="Current Information">Current Information</h2><br />
<br />
[[Image:{{{image_name}}}|thumb|right|200px|{{{image_caption}}}]]<br />
<br />
'''Position:''' {{{position}}}<br><br />
'''Institution:''' {{{institution}}}<br><br />
'''Keywords:''' {{{keywords}}}<br><br />
<br />
'''Homepage:''' {{{homepage_link}}}<br><br />
'''Blog:''' {{{blog_link}}}<br><br />
'''Email:''' {{{email}}}<br />
<br />
<h2 id="Affiliations">Affiliations</h2><br />
<br />
'''Groups:'''<br><br />
{{{groups}}}<br />
<br />
'''Collaborators:'''<br><br />
{{{collaborators}}}<br />
<br />
<h2 id="Education">Education</h2><br />
<br />
<u>'''Undergraduate'''</u><br><br />
'''Institution:''' {{{undergrad_institution}}} {{{undergrad_years}}}<br />
<br />
<u>'''Graduate'''</u><br><br />
'''Institution:''' {{{grad_institution}}} {{{grad_years}}}<br><br />
'''Advisor:''' {{{grad_advisor}}}<br><br />
'''Thesis:''' {{{thesis}}}<br />
<br />
<h2 id="Students">Students</h2><br />
<br />
'''Graduate students:'''<br><br />
{{{grad_students}}}<br />
<br />
'''Postdocs:'''<br><br />
{{{postdocs}}}<br />
<br />
'''Undergraduate students:'''<br><br />
{{{undergrad_students}}}<br />
<br />
<h2 id="Biography">Biography</h2><br />
<br />
'''Born:''' {{{birth_date}}} | {{{birth_place}}} <br><br />
<br />
'''Biography/Timeline:''' {{{biography}}}</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Keesom_potential&diff=6825Keesom potential2008-07-15T14:04:29Z<p>Lgmac: </p>
<hr />
<div>{{Stub-general}}<br />
The '''Keesom potential''' is a [[Boltzmann average]] over the dipolar section of the [[Stockmayer potential]], resulting in<br />
<br />
:<math> \Phi_{12}(r) = 4 \epsilon \left[ \left(\frac{\sigma}{r} \right)^{12}- \left( \frac{\sigma}{r}\right)^6 \right] - \frac{1}{3}\frac{\mu^2_1 \mu^2_2}{(4\pi\epsilon_0)^2 k_BT r_{12}^6}</math><br />
<br />
where:<br />
* <math> \Phi_{12}(r) </math> is the [[intermolecular pair potential]] between two particles at a distance r; <br />
* <math> \sigma </math> is the diameter (length), i.e. the value of <math>r</math> at <math> \Phi(r)=0</math> ;<br />
* <math> \epsilon </math> : well depth (energy)<br />
* <math>\mu</math> is the [[dipole moment]]<br />
* <math>T</math> is the [[temperature]]<br />
* <math>k_B</math> is the [[Boltzmann constant]]<br />
* <math>\epsilon_0</math> is the permitiviy of free space.<br />
<br />
<br />
For dipoles disolved in a dielectric medium, this equation may be generalized by including the dielectric constant of the medium within the <math>4\pi\epsilon_0</math> term.<br />
==References==<br />
#[http://dx.doi.org/10.1080/00268979600100661 Richard J. Sadus "Molecular simulation of the vapour-liquid equilibria of pure fluids and binary mixtures containing dipolar components: the effect of Keesom interactions", Molecular Physics '''97''' pp. 979-990 (1996)]<br />
[[category:models]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5471Grand canonical Monte Carlo2008-01-28T13:13:31Z<p>Lgmac: </p>
<hr />
<div>'''Grand-canonical ensemble Monte Carlo''' (GCEMC) is a very versatile and powerful [[Monte Carlo]] technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, then one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} \qquad\qquad\text{(1)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
<br />
As noted by Filinov, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) grand canonical probability density: <br />
<br />
:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}\qquad\qquad\text{(2)} </math> <br />
<br />
where <math>N</math> is the total number of particles, <math>\mu</math> is the [[chemical potential]], <math>\beta := 1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
The sub-index ''L'' makes emphasis on a particular definition of microstate in which particles are assumed to be distinguishable. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles has no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the <math>U</math> subindex stands for ''unlabelled'', the coordinates in parentheis indicate that the positions are not atributted to any particular choice of labelling and the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f_U(N+1)}{f_U(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}\qquad\qquad\text{(3)} </math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math>. The <math> 1/2 </math> factor accounts for the probability of attempting an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/(N+1) </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} \qquad\qquad\text{(4)}</math><br />
<br />
Substitution of Eq.(3) and Eq.(4) into Eq.(1) yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq.(2) but taking into account that there are then <math>N+1</math> labelled microstates leading to the original <math>N</math> particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "Investigations of phase transitions by a Monte-Carlo method", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5460Grand canonical Monte Carlo2008-01-25T15:07:45Z<p>Lgmac: </p>
<hr />
<div>== Introduction ==<br />
Grand-Canonical [[Monte Carlo]] is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, then one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
<br />
As noted by Filinov, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) grand canonical probability density: <br />
<br />
:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math> <br />
<br />
where ''N'' is the total number of particles, <math>\mu</math> is the chemical potential, <math>\beta=1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
The subindex ''L'' makes emphasis on a particular definition of microstate in which particles are assumed distinguishible. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the ''U'' subindex stands for ''unlabelled'', the coordinates in parentheis indicate that the positions are not atributted to any particular choice of labelling and the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f_U(N+1)}{f_U(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math>. The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/(N+1) </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq., but taking into account that there are then <math>N+1</math> labelled microstates leading to the original <math>N</math> particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5459Grand canonical Monte Carlo2008-01-25T14:57:37Z<p>Lgmac: </p>
<hr />
<div>== Introduction ==<br />
Grand-Canonical [[Monte Carlo]] is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, then one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
<br />
As noted by Filinov, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) grand canonical probability density: <br />
<br />
:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math> <br />
<br />
where ''N'' is the total number of particles, <math>\mu</math> is the chemical potential, <math>\beta=1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
The subindex ''L'' denotes a particular definition of microstate in which particules are assumed distinguishible. Thus, <math> f_L </math> should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate of undistinguishable particles, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the ''U'' subindex stands for ''unlabelled'', the coordinates in parentheis indicate that the positions are not atributted to any particular choice of labelling and the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f_U(N+1)}{f_U(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math>. The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/(N+1) </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq., but taking into account that there are then <math>N+1</math> labelled microstates leading to the original <math>N</math> particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5457Grand canonical Monte Carlo2008-01-25T14:40:15Z<p>Lgmac: /* Theoretical basis */</p>
<hr />
<div>[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
== Introduction ==<br />
Grand-Canonical Monte Carlo is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, then one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
<br />
In the grand canonical ensemble, one usually considers the following probability density distribution:<br />
<br />
:<math> f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math> <br />
<br />
where ''N'' is the total number of particles, <math>\mu</math> is the chemical potential, <math>\beta=1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
<br />
This should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the ''U'' subindex stands for unlabelled, the coordinates in parentheis indicate that the positions are not atributted to any particular choice of labelling and the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f_U(N+1)}{f_U(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math>. The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/(N+1) </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq., but taking into account that there are then N+1 labelled microstates leading to the original N particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5456Grand canonical Monte Carlo2008-01-25T14:33:51Z<p>Lgmac: /* Theoretical basis */</p>
<hr />
<div>[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
== Introduction ==<br />
Grand-Canonical Monte Carlo is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, then one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
<br />
In the grand canonical ensemble, one usually considers the following probability density distribution:<br />
<br />
:<math> f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math> <br />
<br />
where ''N'' is the total number of particles, <math>\mu</math> is the chemical potential, <math>\beta=1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
<br />
This should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position <math>{\mathbf r}_1</math>, labelled particle 2 in position <math>{\mathbf r}_2</math> and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f(N+1)}{f(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math> The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/N+1 </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq., but taking into account that there are then N+1 labelled microstates leading to the original N particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5455Grand canonical Monte Carlo2008-01-25T14:26:06Z<p>Lgmac: </p>
<hr />
<div>[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
== Introduction ==<br />
Grand-Canonical Monte Carlo is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, the one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
In the grand canonical ensemble, one usually considers the following probability density distribution:<br />
<br />
:<math> f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where ''N'' is the total number of particles, <math>\mu</math> is the chemical potential, <math>\beta=1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
<br />
This should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position {\mathbf r}_1</math>, labelled particle 2 in position {\mathbf r}_2</math> and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P<br />
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f(N+1)}{f(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math> The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/N+1 </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq., but taking into account that there are then N+1 labelled microstates leading to the original N particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal.<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5454Grand canonical Monte Carlo2008-01-25T14:16:43Z<p>Lgmac: /* Theoretical basis */</p>
<hr />
<div>[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
== Introduction ==<br />
Grand-Canonical Monte Carlo is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, the one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
In the grand canonical ensemble, one usually considers the following probability density distribution:<br />
<br />
:<math> f('''r'''_1,'''r'''_2, ..., '''r'''_N) \propto \frac{\Lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where ''N'' is the total number of particles, <math>\mu</math> is the chemical potential, <math>\beta=1/k_B T</math> and <math>\Lambda</math> is the de [[De Broglie thermal wavelength]].<br />
<br />
<br />
This should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position '''r'''_1, labelled particle 2 in position '''r'''_2 and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f( \{ '''r'''_1,'''r'''_2, ..., '''r'''\} ) \propto \sum_P<br />
f('''r'''_1,'''r'''_2, ..., '''r'''_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f(N+1)}{f(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math> The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/N+1 </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal. Alternatively, one could derive the acceptance rules<br />
by considering the probability density of labelled states, Eq.\ref{eq:f}, but taking into account that there are then N+1 labelled microstates leading to the original N particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5453Grand canonical Monte Carlo2008-01-25T14:12:24Z<p>Lgmac: /* Theoretical basis */</p>
<hr />
<div>[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
== Introduction ==<br />
Grand-Canonical Monte Carlo is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
== Theoretical basis ==<br />
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, the one deletes one out of <math>N</math> particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual Monte Carlo lottery.<br />
As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability<br />
<br />
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where <math>q</math> is given by:<br />
<br />
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of<br />
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying<br />
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.<br />
In the grand canonical ensemble, one usually considers the following probability density distribution:<br />
<br />
:<math> f('''r'''_1,'''r'''_2, ..., '''r'''_N) \propto \frac{\lambda^{-3N}}{N!} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
This should be interpreted as the probability density of a classical state with labelled particles, having labelled particle 1 in position '''r'''_1, labelled particle 2 in position '''r'''_2 and so on. Since labelling of the particles is of no physical significance whatsoever, there are <math>N!</math> identical states which result from permutation of the labels (this explains the <math>N!</math> term in the denominator). Hence, the probability of the significant microstate, i.e., one with <math>N</math> particles at positions <math> {\mathbf r}_1</math>, <math> {\mathbf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
:<math> f( \{ '''r'''_1,'''r'''_2, ..., '''r'''\} ) \propto \sum_P<br />
f('''r'''_1,'''r'''_2, ..., '''r'''_N) = \lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math><br />
<br />
where the sum runs over all posible particle label permutations.<br />
<br />
Upon trial insertion of an extra particle, one obtains:<br />
<br />
:<math> \frac{f(N+1)}{f(N)} = \lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math><br />
<br />
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math> The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V</math> factor results from placing the particle with uniform probability anywhere inside the simulation box. The reverse attempt (moving from state of <math>N+1</math> particles to the original <math>N</math> particle state) is chosen with probability: <br />
<br />
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
<br />
where the <math> 1/N+1 </math> factor results from random removal of one among <math>N+1</math> particles. Therefore, the ratio of underlying probabilities is:<br />
<br />
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} =<br />
\frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
:<math> acc(N \rightarrow N+1) = \frac{V \lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
:<math> acc(N \rightarrow N-1) = \frac{N}{V \lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math><br />
<br />
The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvented by ignoring the labelling problem and assuming that the underlying probabilities for insertion and removal are equal. Alternatively, one could derive the acceptance rules<br />
by considering the probability density of labelled states, Eq.\ref{eq:f}, but taking into account that there are then N+1 labelled microstates leading to the original N particle labelled state upon deletion (one for each possible label permutation of the deleted particle).<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5450Grand canonical Monte Carlo2008-01-25T12:43:30Z<p>Lgmac: /* Introduction */</p>
<hr />
<div>{{Stub-general}}<br />
[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
== Introduction ==<br />
Grand-Canonical Monte Carlo is a very versatile and powerful technique that explicitly accounts for density fluctuations at fixed volume and [[temperature]]. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the preferred choice for the study of [[Interface |interfacial]] phenomena, in the last decade <br />
[[grand canonical ensemble | grand-canonical ensemble]] simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the [[Configurational bias Monte Carlo |configurational bias]] grand canonical technique has very much improved the situation.<br />
<br />
== Theoretical basis ==<br />
<br />
In the grand canonical ensemble, one first chooses randomly whether<br />
a trial particle insertion or deletion is attempted. If insertion is chosen,<br />
a particle is placed with uniform probability density inside the system.<br />
If deletion is chosen, the one deletes one out of N particles<br />
randomly. The trial move is then accepted or rejected according to the<br />
usual MC lotery.<br />
<br />
As usual, a trial move from state o to state n is accepted with probability<br />
<br />
<math> acc(o \rightarrow n) = min \left (1, q \right ) </math><br />
<br />
where q is given by:<br />
<br />
<math> \label{eq:q} q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow<br />
n)} \times \frac{f(n)}{f(o)} } </math><br />
<br />
Here, <math> \alpha(o \rightarrow n) </math> is the<br />
probability density of<br />
attempting trial move from state o to state n (also known as underlying<br />
probability), while f(o) is the<br />
probability density of state o.<br />
<br />
In the grand canonical ensemble, one usually considers the following<br />
probability density distribution:<br />
<math> \label{eq:f} f({\bf r}_1,{\bf r}_2, ..., {\bf r}_N) \propto<br />
\frac{\lambda^{-3N}{N!} e^{\beta \mu N} e^{-\beta U_N}<br />
</math><br />
This should be interpreted as the probability density of a clasical<br />
state with labeled particles, having labeled particle 1 in position<br />
<math> {\bf r}_1</math>, labeled particle 2 in position <math> {\bf<br />
r}_2</math> and so on. Since labeling of the particles is of no<br />
physical significance whatsoever, there are N! identical states wich<br />
result from permutation of the labels (this explains the N! term in the<br />
denominator). Hence, the probability of the significant microstate,<br />
i.e., one with N particles at positions <math> {\bf r}_1</math>, <math><br />
{\bf r}_2</math>, etc., irrespective of the labels, will be given by:<br />
<br />
<math> f( \{ {\bf r}_1,{\bf r}_2, ..., {\bf r}_N} ) \propto \sum_P<br />
f({\bf r}_1,{\bf r}_2, ..., {\bf r}_N) =<br />
\lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}<br />
</math><br />
<br />
Upon trial insertion of an extra particle, one obtains:<math> \label{eq:fratio) \frac{f(N+1)/f(N)} = \lambda^{-3} e^{\beta \mu }<br />
e^{-\beta ( U_{N+1} - U_N )}<br />
</math><br />
<br />
The probability density of attempting an insertion is<br />
<math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math><br />
The <math> 1/2 </math> factor accounts for the probability of attempting<br />
an insertion (from the choice of insertion or deletion). The <math> 1/V<br />
</math> factor results from placing the particle with uniform<br />
probability anywhere inside the simulation box.<br />
<br />
The reverse attempt (moving from state of N+1 particles to the original<br />
N particle state) is chosen with probability:<br />
<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math><br />
where the <math> 1/N+1 </math> factor results from random removal of one among<br />
N+1 particles.<br />
<br />
Therefore, the ratio of underlying probabilities is:<br />
<br />
<math> \label{eq:alpharatio}<br />
\frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n) =<br />
{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N} = \frac{V}{N+1}<br />
</math><br />
<br />
Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into<br />
Eq.\ref{eq:q} yields the<br />
acceptance probability for attempted insertions:<br />
<br />
<math> acc(N \rightarrow N+1) = \frac{V \lambda^{-3} }{N+1}<br />
e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}<br />
</math><br />
<br />
For the inverse deletion process, similar arguments yield:<br />
<br />
<math> acc(N \rightarrow N-1) = \frac{N}{V \lambda^{-3} }<br />
e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )}<br />
</math><br />
<br />
The same acceptance rules are obtained in reference books. Usually the<br />
problem<br />
of proper counting of states is circumvected by ignoring the labeling<br />
problem and assuming that the underlying probabilites for insertion and<br />
removal are equal. Alternatively, one could derive the acceptance rules<br />
by considering the probability density of labeled states, Eq.\ref{eq:f}, but<br />
taking into account that there are then N+1 labeled microstates leading to the original N particle labeled state upon deletion<br />
(one for each posible label permutation of the deleted particle).<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmachttp://www.sklogwiki.org/SklogWiki/index.php?title=Grand_canonical_Monte_Carlo&diff=5448Grand canonical Monte Carlo2008-01-25T11:29:38Z<p>Lgmac: /* References */</p>
<hr />
<div>{{Stub-general}}<br />
Monte Carlo in the [[grand canonical ensemble | grand-canonical ensemble]].<br />
<br />
[[Category: Monte Carlo]]<br />
<br />
== Introduction ==<br />
<br />
Grand-Canonical Monte Carlo is a very versatile and powerful techinque that explicitely accounts for density fluctuations at fixed volume and temperature. This is achieved by means of trial insertion and deletion of molecules. Although this feature has made it the prefered choice for the study of interfacial phenomena, in the last decade grand-canonical simulations have also found widespread applications in the study of bulk properties. Such applications had been hitherto limited by the very low particle insertion and deletion probabilities, but the development of the configurational bias grand canonical technique has very much improved the situation. <br />
<br />
<br />
== References == <br />
# G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969)<br />
#[http://dx.doi.org/10.1080/00268977400102551 D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics '''28''' pp. 1241-1252 (1974)]<br />
<br />
[[Category: Monte Carlo]]</div>Lgmac