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# Ornstein-Zernike relation

Notation used:

The Ornstein-Zernike relation integral equation [1] is given by:

$h=h\left[c\right]$

where $h[c]$ denotes a functional of $c$. This relation is exact. This is complemented by the closure relation

$c=c\left[h\right]$

Note that $h$ depends on $c$, and $c$ depends on $h$. Because of this $h$ 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)

$h(1,2)=c(1,2)+\int \rho ^{{(1)}}(3)c(1,3)h(3,2)d3$

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

$\gamma ({{\mathbf r}})\equiv h({{\mathbf r}})-c({{\mathbf r}})=\rho \int h({{\mathbf r}'})~c(|{{\mathbf r}}-{{\mathbf r}'}|){{\rm {d}}}{{\mathbf r}'}$

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 $h(1,2)$, is due in part to the direct correlation between 1 and 2, represented by $c(1,2)$, but also to the indirect correlation, $\gamma (r)$, propagated via increasingly large numbers of intermediate particles."

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

$h\equiv c+\rho h\otimes c$

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

$h({{\mathbf r}})=c({{\mathbf r}})+\rho \int c(|{{\mathbf r}}-{{\mathbf r}'}|)c({{\mathbf r}'}){{\rm {d}}}{{\mathbf r}'}$
$+\rho ^{2}\iint c(|{{\mathbf r}}-{{\mathbf r}'}|)c(|{{\mathbf r}'}-{{\mathbf r}''}|)c({{\mathbf r}''}){{\rm {d}}}{{\mathbf r}''}{{\rm {d}}}{{\mathbf r}'}$
$+\rho ^{3}\iiint c(|{{\mathbf r}}-{{\mathbf r}'}|)c(|{{\mathbf r}'}-{{\mathbf r}''}|)c(|{{\mathbf r}''}-{{\mathbf r}'''}|)c({{\mathbf r}'''}){{\rm {d}}}{{\mathbf r}'''}{{\rm {d}}}{{\mathbf r}''}{{\rm {d}}}{{\mathbf r}'}$
$+\rho ^{4}\iiiint c(|{{\mathbf r}}-{{\mathbf r}'}|)c(|{{\mathbf r}'}-{{\mathbf r}''}|)c(|{{\mathbf r}''}-{{\mathbf r}'''}|)c(|{{\mathbf r}'''}-{{\mathbf r}''''}|)h({{\mathbf r}''''}){{\rm {d}}}{{\mathbf r}''''}{{\rm {d}}}{{\mathbf r}'''}{{\rm {d}}}{{\mathbf r}''}{{\rm {d}}}{{\mathbf r}'}$
etc.

Diagrammatically this expression can be written as [2]:

where the bold lines connecting root points denote $c$ functions, the blobs denote $h$ 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

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

${\hat {\gamma }}=({\mathbf {I}}-\rho {\mathbf {{\hat {c}}}})^{{-1}}{\mathbf {{\hat {c}}}}\rho {\mathbf {{\hat {c}}}}$

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

${\hat {\gamma }}(k)={\frac {4\pi }{k}}\int _{0}^{\infty }r~\sin(kr)\gamma (r)dr$

$\gamma (r)={\frac {1}{2\pi ^{2}r}}\int _{0}^{\infty }k~\sin(kr){\hat {\gamma }}(k)dk$

Note:

${\hat {h}}(0)=\int h(r){{\rm {d}}}{{\mathbf r}}$

${\hat {c}}(0)=\int c(r){{\rm {d}}}{{\mathbf r}}$

## References

1. 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)
2. James A. Given "Liquid-state methods for random media: Random sequential adsorption", Physical Review A 45 pp. 816-824 (1992)
3. 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)