Editing Grand canonical Monte Carlo

'''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  

If  deletion is chosen, then one deletes one out of <math>N</math> particles

As usual, a trial move from an original state (<math>o</math>) to a new state (<math>n</math>) is accepted with probability

:<math>  q = \frac{ \alpha(n \rightarrow o)}{\alpha(o \rightarrow n)} \times \frac{f(n)}{f(o)} \qquad\qquad\text{(1)}  </math>

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:

 

:<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>     

 

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]].

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:

:<math> f_U( \{{\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N  \} ) \propto \sum_P

f_L({\mathbf r}_1,{\mathbf r}_2 ..., {\mathbf r}_N) = \Lambda^{-3N} e^{\beta \mu N} e^{-\beta U_N}</math>

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.

:<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>

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:  

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:

\frac{\alpha( N+1 \rightarrow N )}{\alpha( N \rightarrow N+1)} =    \frac{V}{N+1} \qquad\qquad\text{(4)}</math>

Substitution of Eq.(3) and  Eq.(4) into Eq.(1) yields the

:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} 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.(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).