Ornstein-Zernike relation from the grand canonical distribution function

Defining the local activity by

\left.z(r)\right.=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

\Xi =\sum _{N}^{\infty }{1 \over N!}\int \dots \int \prod _{i}^{N}z(r_{i})\exp(-\beta U_{N})dr_{1}\dots dr_{N}

.

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

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

,

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

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

,

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

.

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

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

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

{\delta \ln z(r) \over {\delta \rho (r')}}={\delta (r-r') \over {\rho (r')}}-c(r,r').

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

\int {\delta \rho (r) \over {\delta \ln z(r^{\prime \prime })}}{\delta \ln z(r^{\prime \prime }) \over {\delta \rho (r')}}dr^{\prime \prime }=\delta (r-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