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## General formulation

Recall the expression for the configurational partition function:

Suppose that the potential can be written as a sum of two contributions

where is, in some sense, small compared to . An extra bonus can be had if the partition function for can be evaluated analytically.

Let

Then, we may express as

where means average with respect to only. If is small, then the average can be expanded in powers of :

The free energy is given by

Separating into two contributions, we have

where is independent of and is given by

and

We wish to develop an expansion for of the general form

where are a set of expansion coefficients that are determined by the condition that such an expansion be consistent with .

Using the fact that

we have that

Equating this expansion to the proposed expansion for , we obtain

This must be solved for each of the undetermined parameters , which can be done by equating like powers of on both sides of the equation. Thus, from the term, we find, from the right side:

and from the left side, the and term contributes:

from which it can be easily seen that

Likewise, from the term,

and from the left side, we see that the and terms contribute:

Thus,

For , the right sides gives:

the left side contributes the , and terms:

Thus,

Now, the free energy, up to the third order term is given by

In order to evaluate , suppose that is given by a pair potential

Then,

The free energy is therefore given by

Next: Derivation of the Van Up: Distribution functions and perturbation Previous: Distribution functions and perturbation
Mark E. Tuckerman 2008-02-24