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
$z({\bf r})=z\exp[-\beta\psi({\bf r})]$
<math>z(r)=z\exp[-\beta\psi(r)]</math>
where $\beta=1/k_BT$, and $k_B$ is the Boltzmann constant.
where <math>\beta=1/k_BT</math>, and <math>k_B</math> is the [[Boltzmann constant]].
Using those definitions the grand canonical partition
Using those definitions the [[grand canonical partition function]] can be written as
function can be written as
 
\begin{eqnarray}
<math>\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.</math>
\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}
By functionally-differentiating <math>\Xi</math> with respect to <math>z(r)</math>, and utilizing the mathematical theorem concerning the functional derivative,
By functionally-differentiating $\Xi$ with respect to $z({\bf r})$, and utilizing the mathematical theorem concerning the functional derivative,
 
\begin{eqnarray}
<math>{\delta z({\bf r})\over{\delta z({\bf r'})}}=\delta({\bf r}-{\bf r'}),</math>
{\delta z({\bf r})\over{\delta z({\bf r'})}}=\delta({\bf r}-{\bf r'}),
 
\end{eqnarray}
we get the following equations with respect to the density pair correlation functions.
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})}},
 
\end{eqnarray}
<math>\rho({\bf r})={\delta\ln\Xi\over{\delta \ln z({\bf r})}},</math>
\begin{eqnarray}
 
\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)}({\bf r,r'})={\delta^2\ln\Xi\over{\delta \ln z({\bf r})\delta\ln z({\bf 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(r)</math> and <math>\rho^{(2)}(r,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({\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}).</math>
 
Now, we define the direct correlation function by an inverse relation of Eq. (\ref{deltarho}),
Now, we define the direct correlation function by an inverse relation of Eq. (\ref{deltarho}),
\begin{eqnarray}
 
{\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'}).
<math>{\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'}).</math>
\end{eqnarray}
 
Inserting Eqs. (\ref{deltarho}) and (\ref{deltalnz}) into the chain-rule theorem of functional derivatives,
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'}),
 
\end{eqnarray}
<math>\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'}),</math>
one get the Ornstein-Zernike Equation.
 
Thus the O-Z equation is,
one obtains the [[Ornstein-Zernike equation]].
Thus the Ornstein-Zernike equation is,
in a sense, a differential form of the partition function.
in a sense, a differential form of the partition function.
[[Category:Integral equations]]

Revision as of 14:51, 27 February 2007

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

Failed to parse (unknown function "\b"): {\displaystyle \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.}

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

we get 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 Eq. (\ref{deltarho}),

Failed to parse (unknown function "\label"): {\displaystyle {\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'}).}

Inserting Eqs. (\ref{deltarho}) and (\ref{deltalnz}) into the chain-rule theorem of functional derivatives,

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

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