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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \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:

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

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho ^{(2)}({\mathbf {r} },{\mathbf {r} }')={\delta ^{2}\ln \Xi \over {\delta \ln z({\mathbf {r} })\delta \ln z({\mathbf {r} '})}}} .

A relation between Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \rho ({\mathbf {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)}({\mathbf r},{\mathbf r}')} can be obtained after some manipulation 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 {\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.

See also

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