1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space and partition functions 225

function depends on extensive variables such asNandV, it can be reexpressed as a
sum of these variables multiplied by their thermodynamic conjugates. SinceG(N,P,T)
depends only one extensive variableN, andμis conjugate toN,G(N,P,T) is a simple
productμN.
The partition function of the isothermal-isobaric ensemble is essentially a canonical
partition function in which the HamiltonianH(x) is replaced by the “instantaneous
enthalpy”H(x) +PVand an additional volume integration is included. SinceIN =
CN/V 0 , it is readily seen that eqn. (5.3.19) is

∆(N,P,T) =

1

V 0

∫∞

0

dVe−βPVQ(N,V,T). (5.3.22)

According to eqn. (5.3.22), the isothermal-isobaric partition function is the Laplace
transform (see Appendix D) of the canonical partition function with respect to volume,
just as the canonical partition function is the Laplace transform of the microcanonical
partition function with respect to energy. In both cases, the variable used to form the
Laplace transform between partition functions is the same variableused to form the
Legendre transform between thermodynamic functions.
We now show that the Gibbs free energy is given by the relation

G(N,P,T) =−

1

β

ln ∆(N,P,T). (5.3.23)

Recall thatG=A+P〈V〉=E+P〈V〉−TS, which can be expressed as

G=〈H(x) +PV〉+T

∂G

∂T

(5.3.24)

with the help of eqn. (5.2.10). Note that the average of the instantaneous enthalpy is

H=〈H(x) +PV〉=

IN

∫∞

0 dV

∫

dx (H(x) +PV)e−β(H(x)+PV)

IN

∫∞

0 dV

∫

dx e−β(H(x)+PV)

=−

1

∆(N,P,T)

∂

∂β

∆(N,P,T)

=−

∂

∂β

ln ∆(N,P,T). (5.3.25)

Therefore, eqn. (5.3.24) becomes

G+

∂

∂β

ln ∆(N,P,T) +β

∂G

∂β

= 0, (5.3.26)

which is analogous to eqn. (4.3.20). Thus, following the procedure in Section 4.3 used
to prove thatA=−(1/β) lnQ, we can easily show thatG=−(1/β) ln ∆ is a solution