Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions
Carl McBride (talk | contribs) No edit summary |
Carl McBride (talk | contribs) No edit summary |
||
Line 1: | Line 1: | ||
Defining the local activity by | Defining the local activity by | ||
<math>z(r)=z\exp[-\beta\psi(r)]</math> | |||
:<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]]. | 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 | 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( 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( r_i)\exp(-\beta U_N)dr_1\dots dr_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(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(r)\over{\delta z(r')}}=\delta(r-r')</math>, | ||
we get the following equations with respect to the density pair correlation functions. | we get 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(r)={\delta\ln\Xi\over{\delta \ln z(r)}}</math>, | ||
Line 16: | Line 23: | ||
:<math>\rho^{(2)}(r,r')={\delta^2\ln\Xi\over{\delta \ln z(r)\delta\ln z(r')}}</math>. | :<math>\rho^{(2)}(r,r')={\delta^2\ln\Xi\over{\delta \ln z(r)\delta\ln z(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(r)</math> and <math>\rho^{(2)}(r,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(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(r)\over{\delta\rho(r')}}={\delta(r-r')\over{\rho(r')}} -c(r,r').</math> | :<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, | ||
:<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(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>, | ||
one obtains the [[Ornstein-Zernike relation]]. | one obtains the [[Ornstein-Zernike relation]]. |
Revision as of 15:00, 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.