Ornstein-Zernike relation from the grand canonical distribution function

Defining the local activity by

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

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

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

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

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

we obtain the following equations with respect to the density pair correlation functions:

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

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

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

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

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

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

Inserting these two results into the chain-rule theorem of functional derivatives,

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

one obtains the Ornstein-Zernike relation. Thus the Ornstein-Zernike relation is, in a sense, a differential form of the partition function. (Note: the material in this page was adapted from a text whose authorship and copyright status are both unknown).

See also

• J. A. White and S. Velasco "The Ornstein-Zernike equation in the canonical ensemble", Europhysics Letters 54 pp. 475-481 (2001)