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

$\mathrm{\Xi }=\sum _N^{\mathrm{\infty }}\frac{1}{N!}\int \dots \int \prod _i^{N}z(r_i)\exp (-\beta U_N)d{r}_1\dots d{r}_N$.

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

$\frac{\delta z(r)}{\delta z(r^{\prime })}=\delta (r-r^{\prime })$,

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

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

${\rho }^{(2)}(r,r^{\prime })=\frac{{\delta }^2\ln \mathrm{\Xi }}{\delta \ln z(r)\delta \ln z(r^{\prime })}$.

A relation between $\rho (r)$ and ${\rho }^{(2)}(r,r^{\prime })$ can be obtained after some manipulation as,

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

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

$\frac{\delta \ln z(r)}{\delta \rho (r^{\prime })}=\frac{\delta (r-r^{\prime })}{\rho (r^{\prime })}-c(r,r^{\prime }).$

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

$\int \frac{\delta \rho (r)}{\delta \ln z(r^{\prime \prime })}\frac{\delta \ln z(r^{\prime \prime })}{\delta \rho (r^{\prime })}d{r}^{\prime \prime }=\delta (r-r^{\prime })$,

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