Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions
Carl McBride (talk | contribs) mNo edit summary |
Carl McBride (talk | contribs) m (Note concerning authorship and copyright.) |
||
Line 32: | Line 32: | ||
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, 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== | ==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]] |
Latest revision as of 14:39, 29 April 2008
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 obtain 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 the previous equation,
- .
Inserting these two results 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. (Note: the material in this page was adapted from a text whose authorship and copyright status are both unknown).