Ornstein-Zernike relation

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

\noindent The {\it Ornstein-Zernike relation} (OZ) integral equation is \begin{equation} h=h[c] \end{equation} where $h[c]$ denotes a functional of $c$. This relation is exact.\\ \end{equation} where $h[c]$ denotes a functional of $c$. This relation is exact.\\ This is complemented by the closure relation \begin{equation} c=c[h] \end{equation} Note that $h$ depends on $c$, and $c$ depends on $h$. Because of this $h$ must be determined {\it 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)\\ \begin{equation} h(1,2) = c(1,2) + \int \rho^{(1)}(3) c(1,3)h(3,2) {\rm d}3 \end{equation} If the system is both homogeneous and isotropic, the OZ relation becomes (\cite{KNAW_1914_17_0793} Eq. 6) ~\\ \fbox{\parbox{\columnwidth}{ \begin{equation} \gamma ({\bf r}) \equiv h({\bf r}) - c({\bf r}) = \rho \int h({\bf r'})~c(|{\bf r} - {\bf r'}|) {\rm d}{\bf r'} \end{equation} }} ~\\ In words, this equation (Hansen \& McDonald \S 5.2 p. 107) ``...describes the fact that the {\it total} correlation between particles 1 and 2, represented by $h(1,2)$, is due in part to the {\it direct} correlation between 1 and 2, represented by $c(1,2)$, but also to the {\it indirect} correlation, $\gamma (r)$, propagated via increasingly large numbers of intermediate particles."\\ Notice that this equation is basically a convolution, {\it i.e.} \begin{equation} h \equiv c + \rho h\otimes c \end{equation} (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).