Editing 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).