Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions

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(New page: Defining the local activity by $z({\bf r})=z\exp[-\beta\psi({\bf r})]$ where $\beta=1/k_BT$, and $k_B$ is the Boltzmann constant. Using those definitions the grand canonical partition func...)
 
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Defining the local activity by
Defining the local activity by
$z({\bf r})=z\exp[-\beta\psi({\bf r})]$
 
where $\beta=1/k_BT$, and $k_B$ is the Boltzmann constant.
:<math>z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]</math>
Using those definitions the grand canonical partition
 
function can be written as
where <math>\beta:=1/k_BT</math>, and <math>k_B</math> is the [[Boltzmann constant]].
\begin{eqnarray}
Using those definitions the [[Grand canonical ensemble | grand canonical partition function]] can be written as
\Xi=\sum_N^\infty{1\over N!}\int\dots\int \prod_i^Nz({\b f r}_i)\exp(-\beta U_N){\rm d}{\bf r}_1\dots{\rm d}{\bf r}_N.
 
\end{eqnarray}
:<math>\Xi=\sum_N^\infty{1\over N!}\int\dots\int \prod_i^Nz({\mathbf r}_i)\exp(-\beta U_N){\rm d}{\mathbf r}_1\dots{\rm d}{\mathbf r}_N</math>.
By functionally-differentiating $\Xi$ with respect to $z({\bf r})$, and utilizing the mathematical theorem concerning the functional derivative,
 
\begin{eqnarray}
By functionally-differentiating <math>\Xi</math> with respect to <math>z({\mathbf r})</math>, and utilizing the mathematical theorem concerning the functional derivative,
{\delta z({\bf r})\over{\delta z({\bf r'})}}=\delta({\bf r}-{\bf r'}),
 
\end{eqnarray}
:<math>{\delta z({\mathbf r})\over{\delta z({\mathbf r'})}}=\delta({\mathbf r}-{\mathbf r'})</math>,
we get the following equations with respect to the density pair correlation functions.
 
\begin{eqnarray}\rho({\bf r})={\delta\ln\Xi\over{\delta \ln z({\bf r})}},
we obtain the following equations with respect to the [[density pair correlation functions]]:
\end{eqnarray}
 
\begin{eqnarray}
:<math>\rho({\mathbf r})={\delta\ln\Xi\over{\delta \ln z({\mathbf r})}}</math>,
\rho^{(2)}({\bf r,r'})={\delta^2\ln\Xi\over{\delta \ln z({\bf r})\delta\ln z({\bf r'})}}.
 
\end{eqnarray}
:<math>\rho^{(2)}({\mathbf r},{\mathbf r}')={\delta^2\ln\Xi\over{\delta \ln z({\mathbf r})\delta\ln z({\mathbf r'})}}</math>.
A relation between $\rho({\bf r})$ and $\rho^{(2)}({\bf r,r'})$ can be obtained after some manipulation as,
 
\begin{eqnarray}
A relation between <math>\rho({\mathbf r})</math> and <math>\rho^{(2)}({\mathbf r},{\mathbf r}')</math> can be obtained after some manipulation as,
{\delta\rho({\bf r})\over{\delta \ln z({\bf r'})}}=\rho^{(2)}({\bf r,r'})-\rho({\bf r})\rho({\bf r'})+\delta({\bf r}-{\bf r'})\rho({\bf r}).\label{deltarho}
 
\end{eqnarray}
:<math>{\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})</math>.
Now, we define the direct correlation function by an inverse relation of Eq. (\ref{deltarho}),
 
\begin{eqnarray}
Now, we define the [[direct correlation function]] by an inverse relation of the previous equation,
{\delta \ln z({\bf r})\over{\delta\rho({\bf r'})}}={\delta({\bf r}-{\bf r'})\over{\rho({\bf r'})}} \label{deltalnz}-c({\bf r,r'}).
 
\end{eqnarray}
:<math>{\delta \ln z({\mathbf r})\over{\delta\rho({\mathbf r'})}}={\delta({\mathbf r}-{\mathbf r'})\over{\rho({\mathbf r'})}}</math>.
Inserting Eqs. (\ref{deltarho}) and (\ref{deltalnz}) into the chain-rule theorem of functional derivatives,
 
\begin{eqnarray} \int{\delta\rho({\bf r})\over{\delta \ln z({\bf r}^{\prime\prime})}}{\delta \ln z({\bf r}^{\prime\prime})\over{\delta\rho({\bf r'})}}{\rm d}{\bf r}^{\prime\prime}=\delta({\bf r}-{\bf r'}),
Inserting these two results  into the chain-rule theorem of functional derivatives,
\end{eqnarray}
 
one get the Ornstein-Zernike Equation.
:<math> \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'})</math>,
Thus the O-Z equation is,
 
in a sense, a differential form of the partition function.
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==
*[http://dx.doi.org/10.1209/epl/i2001-00270-x  J. A. White and S. Velasco "The Ornstein-Zernike equation in the canonical ensemble", Europhysics Letters '''54''' pp. 475-481 (2001)]
==References==
[[Category:Integral equations]]

Latest revision as of 14:39, 29 April 2008

Defining the local activity by

where , and is the Boltzmann constant. Using those definitions the grand canonical partition function can be written as

.

By functionally-differentiating with respect to , and utilizing the mathematical theorem concerning the functional derivative,

,

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

,
.

A relation between and can be obtained after some manipulation as,

.

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

.

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

,

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[edit]

References[edit]

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