Question

Consider two systems, A and B, with the same number of
particles N, the first described by the potential energy UA(r1, r2, ...rN ), and
second by UB(r1, r2, ...rN ) = UA(r1, r2, ...rN ) + ∆U(r1, r2, ...rN ), where ∆U
is the perturbation of the potential energy of interaction relative to system
A. Find, in the thermodynamic perturbation approach, an expression for the
Helmoltz free energy difference ∆F = FB − FA between the two systems at
temperature T that involves averages in the canonical ensemble corresponding to system A.

Answer 1

The general expression for partition function is

N (H 1 П «Рр,т, Z ь3N U({ri}) В exp ААК 2m

and the Helmoltz free energy is

F =--In Z

So the difference is

AF -In(ZB/ZA 3

The momentum integrals for both systems is same so that

f...Je-BUltT r ZA

By Taylor's series expansion we can write this as

epUA(1 BAU)&r 1 _ BAU ZB ZA

different expression may be obtained on using different level of approximation.

AF In(1 BAU)