Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions
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Using those definitions the [[grand canonical partition function]] can be written as | Using those definitions the [[grand canonical partition function]] can be written as | ||
<math>\Xi=\sum_N^\infty{1\over N!}\int\dots\int \prod_i^Nz( r_i)\exp(-\beta U_N)dr_1\dots dr_N | :<math>\Xi=\sum_N^\infty{1\over N!}\int\dots\int \prod_i^Nz( r_i)\exp(-\beta U_N)dr_1\dots dr_N</math>. | ||
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 <math>\Xi</math> with respect to <math>z(r)</math>, and utilizing the mathematical theorem concerning the functional derivative, | ||
<math>{\delta z(r)\over{\delta z(r')}}=\delta(r-r') | :<math>{\delta z(r)\over{\delta z(r')}}=\delta(r-r')</math>, | ||
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. | ||
<math>\rho( | :<math>\rho(r)={\delta\ln\Xi\over{\delta \ln z(r)}}</math>, | ||
<math>\rho^{(2)}( | :<math>\rho^{(2)}(r,r')={\delta^2\ln\Xi\over{\delta \ln z(r)\delta\ln z(r')}}</math>. | ||
A relation between <math>\rho(r)</math> and <math>\rho^{(2)}(r,r')</math> can be obtained after some manipulation as, | A relation between <math>\rho(r)</math> and <math>\rho^{(2)}(r,r')</math> can be obtained after some manipulation as, | ||
<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> | :<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}), | ||
<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> | :<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> | ||
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, | ||
<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> | :<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 obtains the [[Ornstein-Zernike relation]]. | one obtains the [[Ornstein-Zernike relation]]. |
Revision as of 14:56, 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
- .
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 relation. Thus the Ornstein-Zernike relation is, in a sense, a differential form of the partition function.