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

$\Xi =\sum _{N}^{\infty }{1 \over N!}\int \dots \int \prod _{i}^{N}z({\mathbf {r} }_{i})\exp(-\beta U_{N}){\rm {d}}{\mathbf {r} }_{1}\dots {\rm {d}}{\mathbf {r} }_{N}$.

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

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

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

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

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

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

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

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

${\delta \ln z({\mathbf {r} }) \over {\delta \rho ({\mathbf {r} '})}}={\delta ({\mathbf {r} }-{\mathbf {r} '}) \over {\rho ({\mathbf {r} '})}}$.

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

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

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

See also

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