Ornstein-Zernike relation from the grand canonical distribution function: Difference between revisions
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Defining the local activity by | Defining the local activity by | ||
:<math>z(r)=z\exp[-\beta\psi(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]]. | ||
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Now, we define the direct correlation function by an inverse relation of | Now, we define the direct correlation function by an inverse relation of the previous equation, | ||
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Inserting | Inserting these two reults into the chain-rule theorem of functional derivatives, | ||
Revision as of 15:02, 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 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.