Editing Ornstein-Zernike relation from the grand canonical distribution function

Jump to navigation Jump to search
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.

The edit can be undone. Please check the comparison below to verify that this is what you want to do, and then publish the changes below to finish undoing the edit.

Latest revision Your text
Line 1: Line 1:
Defining the local activity by
Defining the local activity by


:<math>z({\mathbf r})=z\exp[-\beta\psi({\mathbf r})]</math>
:<math>\left. z(r) \right. =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 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>.


By functionally-differentiating <math>\Xi</math>  with respect to <math>z({\mathbf r})</math>, and utilizing the mathematical theorem concerning the functional derivative,
:<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>{\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]]:
By functionally-differentiating <math>\Xi</math>  with respect to <math>z(r)</math>, and utilizing the mathematical theorem concerning the functional derivative,


:<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>{\delta z(r)\over{\delta z(r')}}=\delta(r-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>.
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^{(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,
 
 
:<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 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>.


Inserting these two results into the chain-rule theorem of functional derivatives,
:<math>{\delta \ln z(r)\over{\delta\rho(r')}}={\delta(r-r')\over{\rho(r')}}  -c(r,r').</math>
 
 
Inserting these two reults 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]].
one obtains the [[Ornstein-Zernike relation]].
Thus the Ornstein-Zernike relation is, in a sense, a differential form of the [[partition function]].
Thus the Ornstein-Zernike relation is,
(Note: the material in this page was adapted from a text whose authorship and copyright status are both unknown).
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]]
[[Category:Integral equations]]
Please note that all contributions to SklogWiki are considered to be released under the Creative Commons Attribution Non-Commercial Share Alike (see SklogWiki:Copyrights for details). If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here.
You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource. Do not submit copyrighted work without permission!

To edit this page, please answer the question that appears below (more info):

Cancel Editing help (opens in new window)
Retrieved from "http://www.sklogwiki.org/SklogWiki/index.php/Ornstein-Zernike_relation_from_the_grand_canonical_distribution_function"