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,

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({\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,

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({\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. (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)