# Ornstein–Zernike equation

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In statistical mechanics the Ornstein–Zernike equation (named after Leonard Ornstein and Frits Zernike) is an integral equation for defining the direct correlation function. It basically describes how the correlation between two molecules can be calculated. Its applications can mainly be found in fluid theory. In molecular theories of ionic solutions, this type of integral equation can be used as a probabilistic description how molecules (i.e. particles, ions, and colloids), distribute space and time given their interaction energies.[1]

## Derivation

The derivation below is heuristic in nature: rigorous derivations require extensive graph analysis or functional techniques. The interested reader is referred to the text book for the full derivation.[2]

It is convenient to define the total correlation function:

${\displaystyle h(r_{12})=g(r_{12})-1\,}$

which is a measure for the "influence" of molecule 1 on molecule 2 at a distance ${\displaystyle r_{12}}$ away with ${\displaystyle g(r_{12})}$ as the radial distribution function. In 1914 Ornstein and Zernike proposed [3] to split this influence into two contributions, a direct and indirect part. The direct contribution is defined to be given by the direct correlation function, denoted ${\displaystyle c(r_{12})}$. The indirect part is due to the influence of molecule 1 on a third molecule, labeled 3, which in turn affects molecule 2, directly and indirectly. This indirect effect is weighted by the density and averaged over all the possible positions of particle 3. This decomposition can be written down mathematically as

${\displaystyle h(r_{12})=c(r_{12})+\rho \int d\mathbf {r} _{3}c(r_{13})h(r_{32})\,}$

which is called the Ornstein–Zernike equation. Its interest is that, by eliminating the indirect influence, ${\displaystyle c(r)}$ is shorter-ranged than ${\displaystyle h(r)}$ and can be more easily described.

If we define the distance vector between two molecules ${\displaystyle \mathbf {r} _{ij}:=\mathbf {r} _{i}-\mathbf {r} _{j}}$ for ${\displaystyle i,j=1,2,3}$, the OZ equation can be rewritten using a convolution.

${\displaystyle h(\mathbf {r} _{12})=c(\mathbf {r} _{12})+\rho \int d\mathbf {r} _{3}c(\mathbf {r} _{12}-\mathbf {r} _{32})h(\mathbf {r} _{32})=c(\mathbf {r} _{12})+\rho \int d\mathbf {r} _{32}c(\mathbf {r} _{12}-\mathbf {r} _{32})h(\mathbf {r} _{32})=c(\mathbf {r} _{12})+\rho (c\,*\,h)(\mathbf {r} _{12})\,}$.

If we then denote the Fourier transforms of ${\displaystyle h(\mathbf {r} )}$ and ${\displaystyle c(\mathbf {r} )}$ by ${\displaystyle {\hat {H}}(\mathbf {k} )}$ and ${\displaystyle {\hat {C}}(\mathbf {k} )}$, respectively, and use the convolution theorem we obtain

${\displaystyle {\hat {H}}(\mathbf {k} )={\hat {C}}(\mathbf {k} )+\rho {\hat {H}}(\mathbf {k} ){\hat {C}}(\mathbf {k} ),\,}$

which yields

${\displaystyle {\hat {C}}(\mathbf {k} )={\frac {{\hat {H}}(\mathbf {k} )}{1+\rho {\hat {H}}(\mathbf {k} )}}\,\,\,\,{\text{and}}\,\,\,\,\,\,\,{\hat {H}}(\mathbf {k} )={\frac {{\hat {C}}(\mathbf {k} )}{1-\rho {\hat {C}}(\mathbf {k} )}}.\,}$

One needs to solve for both ${\displaystyle h(r)}$ and ${\displaystyle c(r)}$ (or, equivalently, their Fourier transforms). This requires an additional equation, known as a closure relation. The Ornstein–Zernike equation can be formally seen as a definition of the direct correlation function ${\displaystyle c(r)}$ in terms of the total correlation function ${\displaystyle h(r)}$. The details of the system under study (most notably, the shape of the interaction potential ${\displaystyle u(r)}$) are taken into account by the choice of the closure relation. Commonly used closures are the Percus–Yevick approximation, well adapted for particles with an impenetrable core, and the hypernetted-chain equation, widely used for "softer" potentials. More information can be found in.[4]

## Closure relations

Closure relations are independent second equations that connect the total correlation ${\displaystyle h(r)}$ and the direct correlation ${\displaystyle c(r)}$. The Ornstein-Zernike equation and a second equation are needed in order to solve for two unknowns: the total correlation ${\displaystyle h(r)}$ and the direct correlation ${\displaystyle c(r)}$.[1] The word "closure" means that it closes or "completes" the conditions for a unique determination of ${\displaystyle h(r)}$ and ${\displaystyle c(r)}$.[1]

## References

1. ^ a b c Lee, Lloyd (2008). Molecular Thermodynamics of Electrolyte Solutions. World Scientific. pp. 103–107. ISBN 9812814191.
2. ^ Frisch, H.; Lebowitz, J.L. (1964). The Equilibrium Theory of Classical Fluids. New York: Benjamin. ASIN B000PHQPES.
3. ^ Ornstein, L. S.; Zernike, F. (1914). "Accidental deviations of density and opalescence at the critical point of a single substance" (pdf). Royal Netherlands Academy of Arts and Sciences (KNAW). Proceedings. 17: 793–806. Bibcode:1914KNAB...17..793. Archived 24 Sep 2010 at the 'Digital Library' of the Dutch History of Science Web Center
4. ^ McQuarrie, D.A. (May 2000) [1976]. Statistical Mechanics. University Science Books. p. 641. ISBN 9781891389153.