Ornstein-Zernike relation from the grand canonical distribution function

Defining the local activity by

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.z(r)\right.=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.

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho (r)={\delta \ln \Xi \over {\delta \ln z(r)}}} ,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho ^{(2)}(r,r')={\delta ^{2}\ln \Xi \over {\delta \ln z(r)\delta \ln z(r')}}} .

A relation between Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho (r)} and $\rho ^{(2)}(r,r')$ can be obtained after some manipulation as,

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\delta \rho (r) \over {\delta \ln z(r')}}=\rho ^{(2)}(r,r')-\rho (r)\rho (r')+\delta (r-r')\rho (r).}

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

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(r)\over{\delta\rho(r')}}={\delta(r-r')\over{\rho(r')}} -c(r,r').}

Inserting these two reults 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(r)\over{\delta \ln z(r^{\prime\prime})}}{\delta \ln z(r^{\prime\prime})\over{\delta\rho(r')}}dr^{\prime\prime}=\delta(r-r')} ,

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