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 (\cite{KNAW_1914_17_0793} Eq. 6)

$\gamma (\mathbf{r})\equiv h(\mathbf{r})-c(\mathbf{r})=\rho \int h({\mathbf{r}}^{\prime })c(|\mathbf{r}-{\mathbf{r}}^{\prime }|)\mathrm{d}{\mathbf{r}}^{\prime }$ 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 $h({\bf r})$ (here truncated at the fourth iteration): \begin{eqnarray*} h({\bf r}) = c({\bf r}) &+& \rho \int c(|{\bf r} - {\bf r'}|) c({\bf r'}) {\rm d}{\bf r'} \\ &+& \rho^2 \int \int c(|{\bf r} - {\bf r'}|) c(|{\bf r'} - {\bf r}|) c({\bf r}) {\rm d}{\bf r}{\rm d}{\bf r'} \\ &+& \rho^3 \int\int\int c(|{\bf r} - {\bf r'}|) c(|{\bf r'} - {\bf r}|) c(|{\bf r} - {\bf r'}|) c({\bf r}) {\rm d}{\bf r}{\rm d}{\bf r}{\rm d}{\bf r'}\\ &+& \rho^4 \int \int\int\int c(|{\bf r} - {\bf r'}|) c(|{\bf r'} - {\bf r}|) c(|{\bf r} - {\bf r}|) c(|{\bf r} - {\bf r'}|) h({\bf r'}) {\rm d}{\bf r'} {\rm d}{\bf r}{\rm d}{\bf r}{\rm d}{\bf r'} \end{eqnarray*} {\it etc.}\\ Diagrammatically this expression can be written as \cite{PRA_1992_45_000816}: \begin{figure}[H] \begin{center} \includegraphics[clip,height=30pt,width=350pt]{oz_diag.eps} \end{center} \end{figure} \noindent 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).