Ornstein-Zernike relation from the grand canonical distribution function

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Defining the local activity by

$z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]$

where $β: = 1 / k B T$ , 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 $Ξ$ 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).