1549380323-Statistical Mechanics Theory and Molecular Simulation

296 Monte Carlo

∆(N,P,T) =

1

V 0 N!λ^3 N

∫∞

0

dVe−βPVVN

×

∫

ds 1 ···dsNe−βU(V

1 / (^3) s 1 ,...,V 1 / (^3) sN)

. (7.3.43)

From eqn. (7.3.43), we see that the acceptance probability for volume moves is, there-
fore, given by

A(V′|V) = min

[

1 ,e−βP(V

′−V)

e−β[U(r

′)−U(r)]

eNln(V

′/V)]

. (7.3.44)

Since a volume move leads to a change in potential energy that increases withN,
volume moves have a low probability of acceptance unlessδis small. Moreover, the fact
that all terms in the potential energy must be updated when the volume changes means
that volume moves are computationally demanding. For this reason,volume moves are
usually made less frequently and have a slightly higher target average acceptance rate
than particle moves.

Sampling the grand canonical distribution

The grand canonical partition function for a system of particles maintained at constant
chemical potentialμin a volumeV at temperatureTis

Z(μ,V,T) =

∑∞

N=0

eβμNQ(N,V,T)

=

∑∞

N=0

eβμN

1

N!λ^3 N

∫

dr 1 ···drNe−βU(r^1 ,...,rN). (7.3.45)

The relative ease with which this ensemble can be sampled in Monte Carlois an
interesting advantage over molecular dynamics.
Sampling the grand canonical ensemble requires sampling the particlecoordinates
and the particle numberN. Particle moves can, once again, be generated using the
scheme of eqn. (7.3.37). Sampling the particle numberNis achieved via attempted
particle insertionandparticle deletionmoves. As these names imply, periodic attempts
are made to insert a particle at a randomly chosen spatial location orto delete a
randomly chosen particle from the system as a means of generatingparticle-number
fluctuations. From eqn. (7.3.45), the acceptance probability of a trial insertion move
can be seen to be

A(N+ 1|N) = min

[

1 ,

V

λ^3 (N+ 1)

eβμe−β[U(r

′)−U(r)]

]

, (7.3.46)

wherer′is the configuration of an (N+ 1)-particle system generated by the insertion
andris the originalN-particle configuration. The volume factor in eqn. (7.3.46)