170 Canonical ensemble
〈a〉 0 =1
Z(0)(N,V,T)
∫
dr 1 ···drNa(r 1 ,...,rN) e−βU^0 (r^1 ,...,rN). (4.7.5)IfU 1 is a small perturbation toU 0 , then the average〈exp(−βU 1 )〉 0 can be expanded
in powers ofU 1 :
〈e−βU^1 〉 0 = 1−β〈U 1 〉 0 +
β^2
2!〈U 12 〉 0 −
β^3
3!〈U 13 〉 0 +···=
∑∞
l=0(−β)l
l!〈U 1 l〉 0. (4.7.6)Since the total partition function is given by
Q(N,V,T) =
Z(N,V,T)
N!λ^3 N, (4.7.7)
the Helmholtz free energy becomes
A(N,V,T) =−
1
βln(
Z(N,V,T)
N!λ^3 N)
=−
1
β
ln(
Z(0)(N,V,T)
N!λ^3 N)
−
1
β
ln〈e−βU^1 〉 0. (4.7.8)The free energy naturally separates into two contributions
A(N,V,T) =A(0)(N,V,T) +A(1)(N,V,T), (4.7.9)where
A(0)(N,V,T) =−1
βln(
Z(0)(N,V,T)
N!λ^3 N)
(4.7.10)
is independent ofU 1 and
A(1)(N,V,T) =−
1
βln〈e−βU^1 〉 0 =−1
βln∑∞
l=0(−β)l
l!〈U 1 l〉 0 , (4.7.11)where, in the second expression, we have expanded the exponential in a power series.
We easily see thatA(0)is the free energy of the unperturbed system, andA(1)is a
correction to be determined perturbatively. To this end, we propose an expansion for
A(1)of the general form
A(1)=∑∞
k=1(−β)k−^1
k!ωk, (4.7.12)where{ωk}is a set of (as yet) unknown expansion coefficients. These coefficients are
determined by the condition that eqn. (4.7.12) be consistent with eqn. (4.7.11) at each
order in the two expansions.