Grand canonical Monte Carlo

Monte Carlo in the 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 interfacial phenomena, in the last decade 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 grand canonical technique has very much improved the situation.

Theoretical basis

In the grand canonical ensemble, one first chooses randomly whether a trial particle insertion or deletion is attempted. If insertion is chosen, a particle is placed with uniform probability density inside the system. If deletion is chosen, the one deletes one out of N particles randomly. The trial move is then accepted or rejected according to the usual MC lotery.

As usual, a trial move from state o to state n is accepted with probability

$acc(o\rightarrow n)=min\left(1,q\right)$

where q is given by:

Failed to parse (unknown function "\label"): {\displaystyle  \label{eq:q}     q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow  n)} \times \frac{f(n)}{f(o)} } }

Here, $\alpha (o\rightarrow n)$ is the
 probability density of
attempting trial move from state o to state n (also known as underlying
probability), while f(o) is the
probability density of state o.

In the grand canonical ensemble, one usually considers the following
probability density distribution:
Failed to parse (unknown function "\label"): {\displaystyle  \label{eq:f} f({\bf r}_1,{\bf r}_2, ..., {\bf r}_N) \propto   \frac{\lambda^{-3N}{N!} e^{\beta \mu N} e^{-\beta U_N}   }

This should be interpreted as the probability density of a clasical state with labeled particles, having labeled particle 1 in position ${\bf {r}}_{1}$, labeled particle 2 in position ${\bf {r}}_{2}$ and so on. Since labeling of the particles is of no physical significance whatsoever, there are N! identical states wich result from permutation of the labels (this explains the N! term in the denominator). Hence, the probability of the significant microstate, i.e., one with N particles at positions ${\bf {r}}_{1}$, ${\bf {r}}_{2}$, etc., irrespective of the labels, will be given by:

Failed to parse (syntax error): {\displaystyle f( \{ {\bf r}_1,{\bf r}_2, ..., {\bf r}_N} ) \propto \sum_P f({\bf r}_1,{\bf r}_2, ..., {\bf r}_N) = \lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N} }

Upon trial insertion of an extra particle, one obtains:Failed to parse (unknown function "\label"): {\displaystyle \label{eq:fratio) \frac{f(N+1)/f(N)} = \lambda^{-3} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} }

The probability density of attempting an insertion is $\alpha (N\rightarrow N+1)={\frac {1}{2}}{\frac {1}{V}}$ The $1/2$ factor accounts for the probability of attempting an insertion (from the choice of insertion or deletion). The $1/V$ factor results from placing the particle with uniform probability anywhere inside the simulation box.

The reverse attempt (moving from state of N+1 particles to the original N particle state) is chosen with probability: $\alpha (N+1\rightarrow N)={\frac {1}{2}}{\frac {1}{N+1}}$ where the $1/N+1$ factor results from random removal of one among N+1 particles.

Therefore, the ratio of underlying probabilities is:

Failed to parse (unknown function "\label"): {\displaystyle \label{eq:alpharatio} \frac{\alpha(n\rightarrow o)}{\alpha(o\rightarrow n) = {\alpha( N \rightarrow N+1 )}{\alpha( N+1 \rightarrow N} = \frac{V}{N+1} }

Substitution of Eq.\ref{eq:alpharatio} and Eq.\ref{eq:fratio} into Eq.\ref{eq:q} yields the acceptance probability for attempted insertions:

$acc(N\rightarrow N+1)={\frac {V\lambda ^{-3}}{N+1}}e^{\beta \mu }e^{-\beta (U_{N+1}-U_{N})}$

For the inverse deletion process, similar arguments yield:

$acc(N\rightarrow N-1)={\frac {N}{V\lambda ^{-3}}}e^{-\beta \mu }e^{-\beta (U_{N-1}-U_{N})}$

The same acceptance rules are obtained in reference books. Usually the problem of proper counting of states is circumvected by ignoring the labeling problem and assuming that the underlying probabilites for insertion and removal are equal. Alternatively, one could derive the acceptance rules by considering the probability density of labeled states, Eq.\ref{eq:f}, but taking into account that there are then N+1 labeled microstates leading to the original N particle labeled state upon deletion (one for each posible label permutation of the deleted particle).

References

1. G. E. Norman and V. S. Filinov "INVESTIGATIONS OF PHASE TRANSITIONS BY A MONTE-CARLO METHOD", High Temperature 7 pp. 216-222 (1969)
2. D. J. Adams "Chemical potential of hard-sphere fluids by Monte Carlo methods", Molecular Physics 28 pp. 1241-1252 (1974)