Ornstein-Zernike relation from the grand canonical distribution function

Defining the local activity by

$\left.z(r)\right.=z\exp[-\beta \psi (r)]$

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 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 \Xi} with respect to 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 z(r)} , and utilizing the mathematical theorem concerning the functional derivative,

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 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 (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(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 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(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(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,

$\int {\delta \rho (r) \over {\delta \ln z(r^{\prime \prime })}}{\delta \ln z(r^{\prime \prime }) \over {\delta \rho (r')}}dr^{\prime \prime }=\delta (r-r')$,

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