Editing Grand canonical Monte Carlo

'''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 
[[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 ==
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 particle is placed with uniform probability density inside the system.
If   deletion is chosen, then one deletes one out of <math>N</math> particles
randomly. The trial move is then accepted or rejected according to the
usual Monte Carlo  lottery.
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>

where <math>q</math> is given by:

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

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
probability), while <math>f(o)</math> is the probability density of state <math>o</math>.

As noted by Filinov, evaluation of the proper acceptance rules requires very carefull interpretation of the (clasical) 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 [[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({\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 parentheis indicate that the positions are not atributted to any particular choice of labelling and the sum runs over all posible particle label permutations.

Upon trial insertion of an extra particle, one obtains:

:<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: 

:<math> \alpha( N+1 \rightarrow N ) = \frac{1}{2} \frac{1}{N+1}  </math>

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

Substitution of Eq.(3) and  Eq.(4) into Eq.(1) yields the
acceptance probability for attempted insertions:

:<math> acc(N \rightarrow N+1) = \frac{V \Lambda^{-3} }{N+1} e^{\beta \mu } e^{-\beta ( U_{N+1} - U_N )} </math>

For the inverse deletion process, similar arguments yield:

:<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).

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 == 
# 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)]
#[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)]
[[Category: Monte Carlo]]