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

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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,





Revision as of 17:17, 21 May 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 the previous equation,



Inserting these two reults 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.