Ornstein-Zernike relation from the grand canonical distribution function
Defining the local activity by
\[z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]\]
where \(\beta:=1/k_BT\), and \(k_B\) is the Boltzmann constant. Using those definitions the grand canonical partition function can be written as
\[\Xi=\sum_N^\infty{1\over N!}\int\dots\int \prod_i^Nz({\mathbf r}_i)\exp(-\beta U_N){\rm d}{\mathbf r}_1\dots{\rm d}{\mathbf r}_N\].
By functionally-differentiating \(\Xi\) with respect to \(z({\mathbf r})\), and utilizing the mathematical theorem concerning the functional derivative,
\[{\delta z({\mathbf r})\over{\delta z({\mathbf r'})}}=\delta({\mathbf r}-{\mathbf r'})\],
we obtain the following equations with respect to the density pair correlation functions:
\[\rho({\mathbf r})={\delta\ln\Xi\over{\delta \ln z({\mathbf r})}}\],
\[\rho^{(2)}({\mathbf r},{\mathbf r}')={\delta^2\ln\Xi\over{\delta \ln z({\mathbf r})\delta\ln z({\mathbf r'})}}\].
A relation between \(\rho({\mathbf r})\) and \(\rho^{(2)}({\mathbf r},{\mathbf r}')\) can be obtained after some manipulation as,
\[{\delta\rho({\mathbf r})\over{\delta \ln z({\mathbf r'})}}=\rho^{(2)}({\mathbf r,r'})-\rho({\mathbf r})\rho({\mathbf r'})+\delta({\mathbf r}-{\mathbf r'})\rho({\mathbf r})\].
Now, we define the direct correlation function by an inverse relation of the previous equation,
\[{\delta \ln z({\mathbf r})\over{\delta\rho({\mathbf r'})}}={\delta({\mathbf r}-{\mathbf r'})\over{\rho({\mathbf r'})}}\].
Inserting these two results into the chain-rule theorem of functional derivatives,
\[ \int{\delta\rho({\mathbf r})\over{\delta \ln z({\mathbf r}^{\prime\prime})}}{\delta \ln z({\mathbf r}^{\prime\prime})\over{\delta\rho({\mathbf r'})}}{\rm d}{\mathbf r}^{\prime\prime}=\delta({\mathbf r}-{\mathbf r'})\],
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).
