Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions

Revision as of 17:06, 10 July 2007

Defining the local activity by

z({\mathbf {r} })=z\exp[-\beta \psi ({\mathbf {r} })]

where $\beta =1/k_{B}T$ , and $k_{B}$ is the Boltzmann constant. Using those definitions the grand canonical partition function can be written as

\Xi =\sum _{N}^{\infty }{1 \over N!}\int \dots \int \prod _{i}^{N}z({\mathbf {r} }_{i})\exp(-\beta U_{N}){\rm {d}}{\mathbf {r} }_{1}\dots {\rm {d}}{\mathbf {r} }_{N}

.

By functionally-differentiating $\Xi$ with respect to $z({\mathbf {r} })$ , and utilizing the mathematical theorem concerning the functional derivative,

{\delta z({\mathbf {r} }) \over {\delta z({\mathbf {r} '})}}=\delta ({\mathbf {r} }-{\mathbf {r} '})

,

we obtain the following equations with respect to the density pair correlation functions:

\rho ({\mathbf {r} })={\delta \ln \Xi  \over {\delta \ln z({\mathbf {r} })}}

,

\rho ^{(2)}({\mathbf {r} },{\mathbf {r} }')={\delta ^{2}\ln \Xi  \over {\delta \ln z({\mathbf {r} })\delta \ln z({\mathbf {r} '})}}

.

A relation between $\rho ({\mathbf {r} })$ and $\rho ^{(2)}({\mathbf {r} },{\mathbf {r} }')$ can be obtained after some manipulation as,

{\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} })

.

Now, we define the direct correlation function by an inverse relation of the previous equation,

{\delta \ln z({\mathbf {r} }) \over {\delta \rho ({\mathbf {r} '})}}={\delta ({\mathbf {r} }-{\mathbf {r} '}) \over {\rho ({\mathbf {r} '})}}

.

Inserting these two results into the chain-rule theorem of functional derivatives,

\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} '})

,

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

References

@@ Line 1: / Line 1: @@
 Defining the local activity by
-:<math>\left. z(r) \right. =z\exp[-\beta\psi(r)]</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]].
@@ Line 7: / Line 7: @@
-:<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>.
+:<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(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(r)\over{\delta z(r')}}=\delta(r-r')</math>,
+:<math>{\delta z({\mathbf r})\over{\delta z({\mathbf r'})}}=\delta({\mathbf r}-{\mathbf r'})</math>,
-we get 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(r)={\delta\ln\Xi\over{\delta \ln z(r)}}</math>,
+:<math>\rho({\mathbf r})={\delta\ln\Xi\over{\delta \ln z({\mathbf r})}}</math>,
-:<math>\rho^{(2)}(r,r')={\delta^2\ln\Xi\over{\delta \ln z(r)\delta\ln z(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(r)</math> and <math>\rho^{(2)}(r,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(r)\over{\delta \ln z(r')}}=\rho^{(2)}(r,r')-\rho(r)\rho(r')+\delta(r-r')\rho(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>.
@@ Line 34: / Line 34: @@
-:<math>{\delta \ln z(r)\over{\delta\rho(r')}}={\delta(r-r')\over{\rho(r')}}  -c(r,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 reults  into the chain-rule theorem of functional derivatives,
+Inserting these two results  into the chain-rule theorem of functional derivatives,
-:<math>\int{\delta\rho(r)\over{\delta \ln z(r^{\prime\prime})}}{\delta \ln z(r^{\prime\prime})\over{\delta\rho(r')}}dr^{\prime\prime}=\delta(r-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]].
-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.
 ==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]]

Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions

Revision as of 17:06, 10 July 2007

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