1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space and partition functions 221

the volume of the container adjusts to keep the internal pressure equal to the external
applied pressure such thatH(x) +PVis constant. The termPVin the instantaneous
enthalpy represents the work done by the system against the external pressure.
The fact thatH(x) +PVis conserved implies that the ensemble is the collection
of all microstates on the constant enthalpy hypersurface defined by the condition

H(x) +PV=H, (5.3.1)

analogous to the constant energy hypersurface in the microcanonical ensemble. Since
the ensemble distribution function must satisfy the equilibrium Liouvilleequation and
therefore be a functionF(H(x)) of the Hamiltonian, the appropriate solution for the
isoenthalpic-isobaric ensemble is simply aδ-function expressing the conservation of
the instantaneous enthalpy,

f(x) =F(H(x)) =Mδ(H(x) +PV−H), (5.3.2)

whereMis an overall normalization constant.
As in the microcanonical ensemble, the partition function (the number of accessi-
ble microstates) is obtained by integrating over the constant enthalpy hypersurface.
However, as the volume is not fixed in this ensemble, each volume accessible to the
system has an associated manifold of accessible phase space pointsbecause the size
of the configuration is determined by the volume. The partition function must, there-
fore, contain an integration over both the phase spaceandthe volume. Denoting the
partition function as Γ(N,P,H), we have

Γ(N,P,H) =M

∫∞

0

dV

∫

dp 1 ···

∫

dpN

×

∫

D(V)

dr 1 ···

∫

D(V)

drNδ(H(r,p) +PV−H), (5.3.3)

where the volume can, in principle, be any positive number. It is important to note
that the volume and position integrations cannot be interchanged,since the position
integration is restricted to the domain defined by each volume. For this reason, the
volume integration cannot be used to integrate over theδ-function. The definition of
the normalization constantMis similar to the microcanonical ensemble except that an
additional reference volumeV 0 is needed to make the partition function dimensionless:

M≡MN=

H 0

V 0 N!h^3 N

. (5.3.4)

Although we can write eqn. (5.3.3) more compactly as

Γ(N,P,H) =M

∫∞

0

dV

∫

dxδ(H(x) +PV−H), (5.3.5)

where the volume dependence of the phase space integration is implicit, this volume
dependence must be determined before the integration overVcan be performed.