Ornstein-Zernike relation from the grand canonical distribution function

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Defining the local activity by

where , and is the Boltzmann constant. Using those definitions the grand canonical partition function can be written as

.

By functionally-differentiating with respect to , and utilizing the mathematical theorem concerning the functional derivative,

,

we obtain the following equations with respect to the density pair correlation functions:

,
.

A relation between and can be obtained after some manipulation as,

.

Now, we define the direct correlation function by an inverse relation of the previous equation,

.

Inserting these two results into the chain-rule theorem of functional derivatives,

,

one obtains the Ornstein-Zernike relation. Thus the Ornstein-Zernike relation is, in a sense, a differential form of the partition function. (Note: the material in this page was adapted from a text whose authorship and copyright status are both unknown).

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