# Ornstein-Zernike relation from the grand canonical distribution function

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).