Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions
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Defining the local activity by | Defining the local activity by | ||
:<math> | :<math>z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]</math> | ||
where <math>\beta:=1/k_BT</math>, and <math>k_B</math> is the [[Boltzmann constant]]. | |||
Using those definitions the [[Grand canonical ensemble | grand canonical partition function]] can be written as | |||
:<math>{\ | :<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 <math>\Xi</math> with respect to <math>z({\mathbf r})</math>, and utilizing the mathematical theorem concerning the functional derivative, | |||
:<math>\ | :<math>{\delta z({\mathbf r})\over{\delta z({\mathbf r'})}}=\delta({\mathbf r}-{\mathbf r'})</math>, | ||
we obtain the following equations with respect to the [[density pair correlation functions]]: | |||
:<math>\rho | :<math>\rho({\mathbf r})={\delta\ln\Xi\over{\delta \ln z({\mathbf r})}}</math>, | ||
:<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 <math>\rho({\mathbf r})</math> and <math>\rho^{(2)}({\mathbf r},{\mathbf r}')</math> can be obtained after some manipulation as, | |||
:<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 the previous equation, | |||
:<math>{\delta \ln z({\mathbf r})\over{\delta\rho({\mathbf r'})}}={\delta({\mathbf r}-{\mathbf r'})\over{\rho({\mathbf r'})}}</math>. | |||
:<math>\int{\delta\rho({\ | Inserting these two results into the chain-rule theorem of functional derivatives, | ||
:<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>, | |||
one obtains the [[Ornstein-Zernike relation]]. | one obtains the [[Ornstein-Zernike relation]]. | ||
Thus the Ornstein-Zernike relation is, | Thus the Ornstein-Zernike relation is, in a sense, a differential form of the [[partition function]]. | ||
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]] | [[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).