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

Latest revision as of 14:39, 29 April 2008

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. (Note: the material in this page was adapted from a text whose authorship and copyright status are both unknown).

References[edit]

@@ Line 1: / Line 1: @@
 Defining the local activity by
-<math>z(r)=z\exp[-\beta\psi(r)]</math>
-where <math>\beta=1/k_BT</math>, and <math>k_B</math> is the [[Boltzmann constant]].
-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({\b f r}_i)\exp(-\beta U_N){\rm d}{\bf r}_1\dots{\rm d}{\bf r}_N.</math>
+:<math>z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]</math>
-By functionally-differentiating <math>\Xi</math>  with respect to <math>z(r)</math>, and utilizing the mathematical theorem concerning the functional derivative,
+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>{\delta z({\bf r})\over{\delta z({\bf r'})}}=\delta({\bf r}-{\bf r'}),</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>.
-we get the following equations with respect to the density pair correlation functions.
+By functionally-differentiating <math>\Xi</math>  with respect to <math>z({\mathbf r})</math>, and utilizing the mathematical theorem concerning the functional derivative,
-<math>\rho({\bf r})={\delta\ln\Xi\over{\delta \ln z({\bf 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]]:
-<math>\rho^{(2)}({\bf r,r'})={\delta^2\ln\Xi\over{\delta \ln z({\bf r})\delta\ln z({\bf r'})}}.</math>
+:<math>\rho({\mathbf r})={\delta\ln\Xi\over{\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,
+:<math>\rho^{(2)}({\mathbf r},{\mathbf r}')={\delta^2\ln\Xi\over{\delta \ln z({\mathbf r})\delta\ln z({\mathbf 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>
+A relation between <math>\rho({\mathbf r})</math> and <math>\rho^{(2)}({\mathbf r},{\mathbf r}')</math> can be obtained after some manipulation as,
-Now, we define the direct correlation function by an inverse relation of Eq. (\ref{deltarho}),
+:<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 \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>
+Now, we define the [[direct correlation function]] by an inverse relation of the previous equation,
-Inserting Eqs. (\ref{deltarho}) and (\ref{deltalnz}) into the chain-rule theorem of functional derivatives,
+:<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({\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>
+Inserting these two results  into the chain-rule theorem of functional derivatives,
-one obtains the [[Ornstein-Zernike equation]].
+:<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>,
-Thus the Ornstein-Zernike equation is,
-in a sense, a differential form of the partition function.
+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).
+==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

Latest revision as of 14:39, 29 April 2008

See also[edit]

References[edit]

Navigation menu

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

Latest revision as of 14:39, 29 April 2008

See also[edit]

References[edit]

Navigation menu

Search