Editing Grand canonical Monte Carlo
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[[Monte Carlo]] in the [[grand canonical ensemble | grand-canonical ensemble]]. | |||
== Introduction == | |||
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 | |||
[[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. | [[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. | ||
== Theoretical basis == | == Theoretical basis == | ||
In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether | In the grand canonical ensemble, one first chooses [[Random numbers |randomly]] whether | ||
a trial particle insertion or deletion is attempted. If insertion is chosen, | a trial particle insertion or deletion is attempted. If insertion is chosen, | ||
a particle is placed with uniform probability density inside the system. | a particle is placed with uniform probability density inside the system. | ||
If deletion is chosen, | If deletion is chosen, the one deletes one out of <math>N</math> particles | ||
randomly. The trial move is then accepted or rejected according to the | randomly. The trial move is then accepted or rejected according to the | ||
usual Monte Carlo lottery. | usual Monte Carlo lottery. | ||
As usual, a trial move from | As usual, a trial move from state <math>o</math> to state <math>n</math> is accepted with probability | ||
:<math> acc(o \rightarrow n) = min \left (1, q \right ) </math> | :<math> acc(o \rightarrow n) = min \left (1, q \right ) </math> | ||
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where <math>q</math> is given by: | where <math>q</math> is given by: | ||
:<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o | :<math> q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} </math> | ||
Here, <math> \alpha(o \rightarrow n) </math> is the probability density of | Here, <math> \alpha(o \rightarrow n) </math> is the probability density of | ||
attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying | attempting trial move from state <math>o</math> to state <math>n</math> (also known as underlying | ||
probability), while <math>f(o)</math> is the probability density of state <math>o</math>. | probability), while <math>f(o)</math> is the probability density of state <math>o</math>. | ||
In the grand canonical ensemble, one usually considers the following probability density distribution: | |||
:<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> | |||
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]]. | |||
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: | |||
:<math> | :<math> f( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N \} ) \propto \sum_P | ||
f({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math> | |||
where | where the sum runs over all posible particle label permutations. | ||
Upon trial insertion of an extra particle, one obtains: | Upon trial insertion of an extra particle, one obtains: | ||
:<math> \frac{ | :<math> \frac{f(N+1)}{f(N)} = \Lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )}</math> | ||
The probability density of attempting an insertion is <math> \alpha( N \rightarrow N+1 ) = \frac{1}{2} \frac{1}{V} </math> | 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: | |||
:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math> | :<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1} </math> | ||
where the <math> 1/ | 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: | ||
:<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} = | :<math> \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n)} = | ||
\frac{\alpha( N | \frac{\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N)} = \frac{V}{N+1} </math> | ||
Substitution of Eq. | Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the | ||
acceptance probability for attempted insertions: | acceptance probability for attempted insertions: | ||
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:<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math> | :<math> acc(N \rightarrow N-1) = \frac{N}{V \Lambda^{-3} } e^{-\beta \mu } e^{-\beta ( U_{N-1} - U_N )} </math> | ||
Alternatively, one could derive the acceptance rules by considering the probability density of labelled states, Eq. | 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). | ||
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. | 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. | ||
== References == | |||
== References == | # G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature '''7''' pp. 216-222 (1969) | ||
#[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)] | |||
''' | |||
[[Category: Monte Carlo]] | [[Category: Monte Carlo]] |