Editing RSOZ

Given and Stell \cite{JCP_1992_97_04573,PA_1994_209_0495} provided {\bf exact} OZ equations for two-phase random media
based on the original work of Madden and Glandt \cite{JSP_1988_51_0537_nolotengoSpringer,JCP_1992_96_05422}.\\
For a two-species system, for the $(s+1)$ replicated system one has (see Eq.s 2.7 --2.11 \cite{PA_1994_209_0495}):\\
\begin{equation}
h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf}   \otimes h_{mf}
\end{equation}
\begin{equation}
h_{mf} = c_{mf} + \rho_m c_{mm} \otimes h_{mf} + \rho_f  c_{mf}  \otimes h_{ff} + (s-1) \rho_f c_{mf}   \otimes h_{12}
\end{equation}
\begin{equation}
h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f  c_{ff}  \otimes h_{fm} + (s-1) \rho_f c_{12}   \otimes h_{fm}
\end{equation}
\begin{equation}
h_{ff} = c_{ff} + \rho_m c_{fm} \otimes h_{mf} + \rho_f  c_{ff}  \otimes h_{ff} + (s-1) \rho_f c_{12}   \otimes h_{12}
\end{equation}
\begin{equation}
h_{12} = c_{12} + \rho_m c_{fm} \otimes h_{mf} + \rho_f  c_{ff}  \otimes h_{12} + 
 \rho_f  c_{12}  \otimes h_{ff} + (s-2) \rho_f c_{12}   \otimes h_{12}
\end{equation}
In the limit of $s \rightarrow 0$ these equations from the ROZ equations (see Eq.s 2.12 --2.16 \cite{PA_1994_209_0495}):\\
\begin{equation}
h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}
\end{equation}
\begin{equation}
h_{mf} = c_{mf} + \rho_m c_{mm} \otimes h_{mf} + \rho_f  c_{mf}  \otimes h_{ff} - \rho_f c_{mf}   \otimes h_{12}
\end{equation}
\begin{equation}
h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f  c_{ff}  \otimes h_{fm} - \rho_f c_{12}   \otimes h_{fm}
\end{equation} 
\begin{equation}
h_{ff} = c_{ff} + \rho_m c_{fm} \otimes h_{mf} + \rho_f  c_{ff}  \otimes h_{ff} - \rho_f c_{12}   \otimes h_{12}
\end{equation}
\begin{equation}
h_{12} = c_{12} + \rho_m c_{fm} \otimes h_{mf} + \rho_f  c_{ff}  \otimes h_{12} + 
 \rho_f  c_{12}  \otimes h_{ff} -2 \rho_f c_{12}   \otimes h_{12}
\end{equation}
When written in the `percolation terminology'
where $c$ terms `{\it connected}' and $b$  `{\it blocking}' are adapted from the
language of percolation theory.
\begin{equation}
h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}
\end{equation}
\begin{equation}
h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}
\end{equation}
\begin{equation}
h_{ff} = c_{ff} + \rho_m c_{fm} \otimes h_{mf} +  \rho_f c_c \otimes h_{ff} + \rho_f c_b \otimes h_c
\end{equation}
\begin{equation}
h_c = c_c + \rho_f c_c \otimes h_c
\end{equation}
where the direct correlation function is split into
\begin{equation}
c_{ff}(12) = c_c (12) + c_b (12)
\end{equation}
and the total correlation function is also split into
\begin{equation}
h_{ff}(12)= h_c (12) + h_b(12)
\end{equation}
where $m$ denotes the matrix
and $f$ denotes the fluid.
The blocking function $h_b(x)$ accounts for correlations between a pair of 
fluid particles ``blocked" or separated from each other by matrix particles.\\
IMPORTANT NOTE: Unlike an equilibrium mixture, there is only one convolution
integral for $h_{mm}$ because the structure of the medium is
unaffected by the presence of fluid particles.\\ 
Note: $C_{ff}$ (Madden and Glandt) $=h_c$ (Given and Stell)\\
Note: fluid: $f$  (Madden and Glandt), `1'  (Given and Stell)\\
Note: matrix: $m$  (Madden and Glandt), `0'  (Given and Stell)\\
At very low matrix porosities, i.e. very high densities of matrix particles,
the volume accessible to fluid particles is divided into small cavities, each
totally surrounded by a matrix. In this limit, the function $h_c (x)$ 
describes correlations between fluid particles in the same cavity and the 
function $h_b(x)$ describes correlations between particles in different cavities.