Editing RSOZ

Given and Stell (Refs 1 and 2) provided '''exact''' [[Ornstein-Zernike relation]]s  for two-phase random media
based on the original work of Madden and Glandt (Refs 3 and 4).
For a two-species system, for the <math>(s+1)</math> replicated system one has (see Eq.s 2.7 --2.11 Ref. 2):

:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm} + s\rho_f c_{mf}   \otimes h_{mf}</math>


:<math>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}</math>


:<math>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}</math>


:<math>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}</math>

:<math>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} </math> 


In the limit of <math>s \rightarrow 0</math> these equations from the [[ROZ]] equations (see Eq.s 2.12 --2.16 Ref. 2):

:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>


:<math>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}</math>


:<math>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}</math>


:<math>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}</math>


:<math>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}</math>

When written in the `percolation terminology'
where <math>c</math> terms ''connected'' and <math>b</math>  ''blocking'' are adapted from the
language of percolation theory.

:<math>h_{mm} = c_{mm} + \rho_m c_{mm} \otimes h_{mm}</math>

:<math>h_{fm} = c_{fm} + \rho_m c_{fm} \otimes h_{mm} + \rho_f c_c \otimes h_{fm}</math>

:<math>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</math>

:<math>h_c = c_c + \rho_f c_c \otimes h_c</math>

where the direct correlation function is split into

:<math>\left.c_{ff}(12)\right. = c_c (12) + c_b (12)</math>
and the total correlation function is also split into
:<math>\left.h_{ff}(12)\right.= h_c (12) + h_b(12)</math>

where <math>m</math> denotes the matrix
and <math>f</math> denotes the fluid.
The blocking function <math>h_b(x)</math> 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 <math>h_{mm}</math> because the structure of the medium is
unaffected by the presence of fluid particles.

*Note: <math>C_{ff}</math> (Madden and Glandt) <math>=h_c</math> (Given and Stell)
*Note: fluid: <math>f</math>  (Madden and Glandt), `1'  (Given and Stell)
*Note: matrix: <math>m</math>  (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 <math>h_c (x)</math> 
describes correlations between fluid particles in the same cavity and the 
function <math>h_b(x)</math> describes correlations between particles in different cavities.
==Polydisperse systems==
For a polydisperse fluid, composed of <math>n_f</math> components, in a polydisperse matrix,
composed of <math>n_m</math> components, written in matrix form in [[Fourier space]] (see Eq. 18 of Ref. 5):

:<math>\tilde{\mathbf H}_{mm} = \tilde{\mathbf C}_{mm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf H}_{mm}
</math>

:<math>\tilde{\mathbf H}_{fm} = \tilde{\mathbf C}_{fm} + \rho_m \tilde{\mathbf C}_{mm} \tilde{\mathbf H}_{fm} + \rho_f  \tilde{\mathbf C}_{fm}   \tilde{\mathbf H}_{ff} - \rho_f \tilde{\mathbf C}_{12}    \tilde{\mathbf H}_{fm}
</math>

:<math>\tilde{\mathbf H}_{ff} = \tilde{\mathbf C}_{ff} + \rho_m \tilde{\mathbf C}_{fm}^T \tilde{\mathbf H}_{fm} + \rho_f  \tilde{\mathbf C}_{ff}   \tilde{\mathbf H}{ff} - \rho_f \tilde{\mathbf C}_{12}   \tilde{\mathbf H}_{12}</math>

:<math>\tilde{\mathbf H}_{12} = \tilde{\mathbf C}_{12} + \rho_m \tilde{\mathbf C}_{fm}^T \tilde{\mathbf H}_{fm} + \rho_f  \tilde{\mathbf C}_{ff}  \tilde{\mathbf H}_{12} + 
 \rho_f  \tilde{\mathbf C}_{12}  \tilde{\mathbf H}_{ff} -2 \rho_f \tilde{\mathbf C}_{12}   \tilde{\mathbf H}_{12}</math>

Note: <math>{\mathbf c}_{fm} = {\mathbf c}_{mf}^T</math> and <math>{\mathbf h}_{fm} = {\mathbf h}_{mf}^T</math>.

==References==
#[http://dx.doi.org/10.1063/1.463883  James A. Given and George Stell "Comment on: Fluid distributions in two-phase random media: Arbitrary matrices", Journal of Chemical Physics '''97''' pp. 4573 (1992)]
#[http://dx.doi.org/10.1016/0378-4371(94)90200-3  James A. Given and George R. Stell "The replica Ornstein-Zernike equations and the structure of partly quenched media",Physica A '''209''' pp. 495-510 (1994)]
#[http://dx.doi.org/10.1007/BF01028471 W. G. Madden and E. D. Glandt "Distribution functions for fluids in random media", J. Stat. Phys. '''51''' pp. 537- (1988)]
#[http://dx.doi.org/10.1063/1.462726  William G. Madden, "Fluid distributions in random media: Arbitrary matrices",  Journal of Chemical Physics '''96''' pp. 5422 (1992)]
#[http://dx.doi.org/10.1080/0026897031000085128 S. Jorge;  Elisabeth Schöll-Paschinger;  Gerhard Kahl; María-José Fernaud "Structure and thermodynamic properties of a polydisperse fluid in contact with a polydisperse matrix", Molecular Physics '''101''' pp. 1733-1740 (2003)]
[[Category: Integral equations]]