Ornstein-Zernike relation from the grand canonical distribution function

Defining the local activity by $z(r)=z\exp[-\beta \psi (r)]$ where $\beta =1/k_{B}T$, and $k_{B}$ is the Boltzmann constant. Using those definitions the grand canonical partition function can be written as

$\Xi =\sum _{N}^{\infty }{1 \over N!}\int \dots \int \prod _{i}^{N}z(r_{i})\exp(-\beta U_{N})dr_{1}\dots dr_{N}$.

By functionally-differentiating $\Xi$ with respect to $z(r)$, and utilizing the mathematical theorem concerning the functional derivative,

${\delta z(r) \over {\delta z(r')}}=\delta (r-r')$,

we get the following equations with respect to the density pair correlation functions.

$\rho (r)={\delta \ln \Xi \over {\delta \ln z(r)}}$,

$\rho ^{(2)}(r,r')={\delta ^{2}\ln \Xi \over {\delta \ln z(r)\delta \ln z(r')}}$.

A relation between $\rho (r)$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho^{(2)}(r,r')} can be obtained after some manipulation as,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\delta\rho({\bf r})\over{\delta \ln z({\bf r'})}}=\rho^{(2)}({\bf r,r'})-\rho({\bf r})\rho({\bf r'})+\delta({\bf r}-{\bf r'})\rho({\bf r}).}

Now, we define the direct correlation function by an inverse relation of Eq. (\ref{deltarho}),

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\delta \ln z({\bf r})\over{\delta\rho({\bf r'})}}={\delta({\bf r}-{\bf r'})\over{\rho({\bf r'})}} \label{deltalnz}-c({\bf r,r'}).}

Inserting Eqs. (\ref{deltarho}) and (\ref{deltalnz}) into the chain-rule theorem of functional derivatives,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int{\delta\rho({\bf r})\over{\delta \ln z({\bf r}^{\prime\prime})}}{\delta \ln z({\bf r}^{\prime\prime})\over{\delta\rho({\bf r'})}}{\rm d}{\bf r}^{\prime\prime}=\delta({\bf r}-{\bf r'}),}

one obtains the Ornstein-Zernike relation. Thus the Ornstein-Zernike relation is, in a sense, a differential form of the partition function.