1549380323-Statistical Mechanics Theory and Molecular Simulation

Phase space and partition function 141

The link between the macroscopic thermodynamic properties in eqn.(4.2.6) and
the microscopic states contained inQ(N,V,T) is provided through the relation

A(N,V,T) =−kTlnQ(N,V,T) =−

1

β

lnQ(N,V,β). (4.3.17)

In order to see that eqn. (4.3.17) provides the connection between the thermodynamic
state functionA(N,V,T) and the partition functionQ(N,V,T), we note thatA=
E−TSand thatS=−∂A/∂T, from which we obtain

A=E+T

∂A

∂T

. (4.3.18)

We also recognize thatE =〈H(x)〉, the ensemble average of the Hamiltonian. By
definition, this ensemble average is

〈H〉=

CN

∫

dxH(x)e−βH(x)

CN

∫

dx e−βH(x)

=−

1

Q(N,V,T)

∂

∂β

Q(N,V,T)

=−

∂

∂β

lnQ(N,V,T). (4.3.19)

Thus, eqn. (4.3.18) becomes

A+

∂

∂β

lnQ(N,V,β) +β

∂A

∂β

= 0, (4.3.20)

where the fact that

T

∂A

∂T

=T

∂A

∂β

∂β

∂T

=−T

∂A

∂β

1

kT^2

=−β

∂A

∂β

(4.3.21)

has been used. We just need to show that eqn. (4.3.17) is the solution to eqn. (4.3.20),
which is a first-order differential equation forA. Differentiating eqn. (4.3.17) with
respect toβgives

β

∂A

∂β

=

1

β

lnQ(N,V,β)−

∂

∂β

lnQ(N,V,β). (4.3.22)

Substituting eqns. (4.3.17) and (4.3.22) into eqn. (4.3.20) yields

−

1

β

lnQ(N,V,β) +

∂

∂β

lnQ(N,V,β) +

1

β

lnQ(N,V,β)−

∂

∂β

lnQ(N,V,β) = 0,

which verifies thatA=−kTlnQis the solution. Therefore, from eqn. (4.2.6), it is clear
that the macroscopic thermodynamic observables are given in terms of the partition
function by