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# Ornstein-Zernike relation

Notation used:

- is the pair distribution function.
- is the pair potential acting between pairs.
- is the total correlation function.
- is the direct correlation function.
- is the indirect (or
*series*or*chain*) correlation function. - is the cavity correlation function.
- is the bridge function.
- is the thermal potential.
- is the Mayer f-function.

The **Ornstein-Zernike relation** integral equation ^{[1]} is given by:

where denotes a functional of . This relation is exact. This is complemented by the closure relation

Note that depends on , and depends on . Because of this must be determined self-consistently. This need for self-consistency is characteristic of all many-body problems. (Hansen and McDonald, section 5.2 p. 106) For a system in an external field, the Ornstein-Zernike relation has the form (5.2.7)

If the system is both homogeneous and isotropic, the Ornstein-Zernike relation becomes (Eq. 6 of Ref. 1)

In words, this equation (Hansen and McDonald, section 5.2 p. 107)

- "...describes the fact that the
*total*correlation between particles 1 and 2, represented by , is due in part to the*direct*correlation between 1 and 2, represented by , but also to the*indirect*correlation, , propagated via increasingly large numbers of intermediate particles."

Notice that this equation is basically a convolution, *i.e.*

(Note: the convolution operation written here as is more frequently written as ) This can be seen by expanding the integral in terms of (here truncated at the fourth iteration):

*etc.*

Diagrammatically this expression can be written as ^{[2]}:

where the bold lines connecting root points denote functions, the blobs denote functions. An arrow pointing from left to right indicates an uphill path from one root point to another. An `uphill path' is a sequence of Mayer bonds passing through increasing particle labels. The Ornstein-Zernike relation can be derived by performing a functional differentiation of the grand canonical distribution function.

## Ornstein-Zernike relation in Fourier space[edit]

The Ornstein-Zernike equation may be written in Fourier space as (^{[3]} Eq. 5):

The carets denote the three-dimensional Fourier transformed quantities which reduce explicitly to:

Note:

## References[edit]

- ↑ L. S. Ornstein and F. Zernike "Accidental deviations of density and opalescence at the critical point of a single substance", Koninklijke Nederlandse Akademie van Wetenschappen Amsterdam Proc. Sec. Sci.
**17**pp. 793- (1914) - ↑ James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A
**45**pp. 816-824 (1992) - ↑ Der-Ming Duh and A. D. J. Haymet "Integral equation theory for uncharged liquids: The Lennard-Jones fluid and the bridge function", Journal of Chemical Physics
**103**pp. 2625-2633 (1995)

**Related reading**

- Jean-Pierre Hansen and I.R. McDonald "Theory of Simple Liquids", Academic Press (2006) (Third Edition) ISBN 0-12-370535-5 § 3.5
- Yan He, Stuart A. Rice, and Xinliang Xu "Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids", Journal of Chemical Physics
**145**234508 (2016)