Ornstein-Zernike relation

Notation:

• $g(r)$ is the pair distribution function.
• $\Phi (r)$ is the pair potential acting between pairs.
• $h(1,2)$ is the total correlation function $h(1,2)\equiv g(r)-1$.
• $c(1,2)$ is the direct correlation function.
• $\gamma (r)$ is the indirect (or series or chain) correlation function $\gamma (r)\equiv h(r)-c(r)$.
• $y(r_{12})$ is the cavity correlation function$y(r)\equiv g(r)/e^{-\beta \Phi (r)}$
• $B(r)$ is the bridge function.
• $\omega (r)$ is the thermal potential, $\omega (r)\equiv \gamma (r)+B(r)$.
• $f(r)$ is the [[Mayer $f$-function]], defined as $f(r)\equiv e^{-\beta \Phi (r)}-1$.

The Ornstein-Zernike relation (OZ) integral equation is

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

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 \& McDonald \S 5.2 p. 106) For a system in an external field, the OZ 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 OZ relation becomes (Ref. 1Eq. 6)

$\gamma (r)\equiv h(r)-c(r)=\rho \int h(r')~c(|r-r'|)dr'$ In words, this equation (Hansen \& McDonald \S 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):

$h(r)=c(r)+\rho \int c(|r-r'|)c(r')dr'+\rho ^{2}\int \int c(|r-r'|)c(|r'-r''|)c(r'')dr''dr'+\rho ^{3}\int \int \int c(|r-r'|)c(|r'-r''|)c(|r''-r'''|)c(r''')dr'''dr''dr'+\rho ^{4}\int \int \int \int c(|r-r'|)c(|r'-r''|)c(|r''-r'''|)c(|r'''-r''''|)h(r'''')dr''''dr'''dr''dr'$

etc. Diagrammatically this expression can be written as (Ref. 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 OZ relation can be derived by performing a functional differentiation of the grand canonical distribution function (HM check this).

References

1. [KNAW_1914_17_0793]
2. [PRA_1992_45_000816]