Adiabatic switching and thermodynamic integration 319In Section 4.2, we showed that derivatives ofAwith respect toN,V, orTgive the
chemical potential, pressure, or entropy, respectively. ForA(N,V,T,λ), what does the
derivative with respect toλrepresent? According to eqn. (8.2.3),
∂A
∂λ=−
kT
Q∂Q
∂λ=−
kT
Z∂Z
∂λ. (8.2.4)
Computing the derivative ofZwith respect toλ, we find
kT
Z∂Z
∂λ=
kT
Z∂
∂λ∫
dNre−βU(r^1 ,...,rN,λ)=
kT
Z∫
dNr(
−β∂U
∂λ)
e−βU(r^1 ,...,rN,λ)=−
〈
∂U
∂λ〉
. (8.2.5)
Note that the free energy difference ∆AABcan be obtained trivially from the relation
∆AAB=
∫ 1
0∂A
∂λ
dλ. (8.2.6)Substituting eqns. (8.2.4) and (8.2.5) into eqn. (8.2.6) yields the freeenergy difference
as
∆AAB=∫ 1
0〈
∂U
∂λ〉
λdλ, (8.2.7)where〈···〉λdenotes an average over the canonical ensemble described by thedistri-
bution exp[−βU(r 1 ,...,rN,λ)] withλfixed at a particular value. The special choice
off(λ) = 1−λandg(λ) =λhas a simple interpretation: With these functions, eqn.
(8.2.7) becomes
∆AAB=∫ 1
0〈UB−UA〉λdλ. (8.2.8)Eqn. (8.2.8) recalls the relationship between work and free energy from the second law
of thermodynamics. If, in transforming the system from stateAto stateB, an amount
of workWis performed on the system, then
W≥∆AAB, (8.2.9)where equality holds only if the transformation is carried out along a reversible path.
We will refer to this inequality as the “work–free–energy inequality.”Since reversible
work is related to a change in potential energy (see Section 1.6), eqn. (8.2.8) is actually
a statistical version of eqn. (8.2.9) for the special case of equality. Eqn. (8.2.8) tells us
that the free energy difference is the ensemble average of the microscopic reversible
work needed to change the potential energy of each configuration fromUA toUB
along the chosenλ-path. Note, however, that eqn. (8.2.7), which is known as the
thermodynamic integrationformula (Kirkwood, 1935), is independent of the choice of