Editing Ornstein-Zernike relation from the grand canonical distribution function
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:<math>z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]</math> | :<math>z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]</math> | ||
where <math>\beta | 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 | Using those definitions the [[Grand canonical ensemble | grand canonical partition function]] can be written as | ||
:<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>. | :<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, | By functionally-differentiating <math>\Xi</math> with respect to <math>z({\mathbf r})</math>, and utilizing the mathematical theorem concerning the functional derivative, | ||
:<math>{\delta z({\mathbf r})\over{\delta z({\mathbf r'})}}=\delta({\mathbf r}-{\mathbf r'})</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]]: | we obtain the following equations with respect to the [[density pair correlation functions]]: | ||
:<math>\rho({\mathbf r})={\delta\ln\Xi\over{\delta \ln z({\mathbf r})}}</math>, | :<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>. | :<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, | 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>. | :<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, | 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>{\delta \ln z({\mathbf r})\over{\delta\rho({\mathbf r'})}}={\delta({\mathbf r}-{\mathbf r'})\over{\rho({\mathbf r'})}}</math>. | ||
Inserting these two results into the chain-rule theorem of functional derivatives, | 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>, | :<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, in a sense, a differential form of the | Thus the Ornstein-Zernike relation is, in a sense, a differential form of the partition function. | ||
==See also== | ==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)] | *[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== | ==References== | ||
[[Category:Integral equations]] | [[Category:Integral equations]] |