Ornstein-Zernike relation

Notation:

• $g(r)$ is the pair distribution function.
• $\mathrm{\Phi }(r)$ 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$y(r)\equiv g(r)/e^{-\beta \mathrm{\Phi }(r)}$
• $B(r)$ 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 $f(r)\equiv e^{-\beta \mathrm{\Phi }(r)}-1$.

The Ornstein-Zernike relation (OZ) integral equation is

$h=h[c]$

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

$c=c[h]$

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)

$\gamma (r)\equiv h(r)-c(r)=\rho \int h(r^{\prime })c(|r-r^{\prime }|)d{r}^{\prime }$ 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):

$h(r)=c(r)+\rho \int c(|r-r^{\prime }|)c(r^{\prime })d{r}^{\prime }+{\rho }^2\int \int c(|r-r^{\prime }|)c(|r^{\prime }-r^{″}|)c(r^{″})d{r}^{″}d{r}^{\prime }+{\rho }^3\int \int \int c(|r-r^{\prime }|)c(|r^{\prime }-r^{″}|)c(|r^{″}-r^{‴}|)c(r^{‴})d{r}^{‴}d{r}^{″}d{r}^{\prime }+{\rho }^4\int \int \int \int c(|r-r^{\prime }|)c(|r^{\prime }-r^{″}|)c(|r^{″}-r^{‴}|)c(|r^{‴}-r^{⁗}|)h(r^{⁗})d{r}^{⁗}d{r}^{‴}d{r}^{″}d{r}^{\prime }$

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).

OZ equation in Fourier space

The OZ equation may be written in Fourier space as (Eq. 5 in Ref. 3):

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

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

$\hat{\gamma }(k)=\frac{4\pi }{k}\underset{0}{\overset{\mathrm{\infty }}{\int }}r\sin (kr)\gamma (r)dr$

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

Note:

$\hat{h}(0)=\int h(r)dr$

$\hat{c}(0)=\int c(r)dr$

References

1. [KNAW_1914_17_0793]
2. [PRA_1992_45_000816]
3. [JCP_1995_103_02625]