10.4 Other ensembles 311
The(NPT)ensemble average of a physical quantityAdepending on the positions
R=r 1 ,...,rNis given as
〈A〉(NPT)=
∫∞
0 dVe−βPV∫dRA(R)e−βU(R)
Q(N,P,T). (10.29)
Qis the partition function which is related to the Gibbs free energy:G =
−kBTlnQ(N,P,T); see Section 7.1. As the volume is allowed to vary, we extend
the notion of a configuration to include the volume in addition to the set of particle
positions,R=(r 1 ,r 2 ,...rN), with the restriction that the latter should all lie
within that volume. A Markov chain must be constructed in which particle moves
and volume changes are allowed. It is, however, impossible to change the volume
independently from the particle positions, as a decrease of the volume might cause
particles close to the wall to fall outside the volume. Therefore, a change in volume
must be accompanied by an appropriate rescaling of the particle positions.
To be more specific, let us consider a cubic volumeL×L×Lwith the edges along
the positive Cartesian axes. We scale the particle positions according tori=siL
so that the positionssilie within the unit cube. The average(10.29)can now be
written as
〈A〉(NPT)=∫∞
0 dVe−βPVVN∫dSA(LS)e−βU(LS)
Q(N,P,T)(10.30)
whereSdenotes the combined positionss 1 ,...,sN. When changing the volume,S
remains the same; the change in the real positionsriis accounted for by a change
in the JacobianVNand the various factorsLin (10.30). The Boltzmann weight of
the(NVT)ensemble method is replaced by
ρ(V,S)=e−βPVVNe−βU(LS). (10.31)A step in the Metropolis Markov chain consists of either a particle move, which
is performed exactly like the particle moves in the (NVT) ensemble method, or a
volume change. The calculation of the ratio of the weights before and after the
volume change consists of calculating the change due to the potential energy,
exp{−β[U(LnewS)−U(LoldS)]} (10.32)and multiplying this by the ratio of the terms involving the volume coordinate:
exp[−βP(Vnew−Vold)](
Vnew
Vold)N
. (10.33)
The product of(10.32)and(10.33)defines the acceptance ratio of the new config-
uration according to the Metropolis recipe. Eppenga and Frenkel[22]have applied
the method with the logarithm of the volume as the extra coordinate rather than the
volume itself.