Ornstein-Zernike relation from the grand canonical distribution function

Defining the local activity by

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

where $\beta =1/k_{B}T$, 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 k_B} is the Boltzmann constant. Using those definitions the grand canonical partition function can be written 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 \Xi=\sum_N^\infty{1\over N!}\int\dots\int \prod_i^Nz( 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,

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

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)}}} ,

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')={\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,

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

Now, we define the direct correlation function by an inverse relation of 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(r)\over{\delta\rho(r')}}={\delta(r-r')\over{\rho(r')}} -c(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(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.