Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions
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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( | :<math>{\delta\rho(r)\over{\delta \ln z(r')}}=\rho^{(2)}(r,r')-\rho(r)\rho(r')+\delta(r-r')\rho(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( | :<math>{\delta \ln z(r)\over{\delta\rho(r')}}={\delta(r-r')\over{\rho(r')}} -c(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, |
Revision as of 14:58, 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}),
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.