Microstructure of two-phase random media. II. The Mayer-Montroll and Kirkwood-Salsburg hierarchies

It is shown that the Mayer-Montroll (MM) and Kirkwood-Salsburg (KS) hierarchies of equilibrium statistical mechanics for a binary mixture under certain limits become equations for the n-point matrix probability functions S_n associated with two-phase random media. The MM representation proves to be identical to the S_n expression derived by us in a previous paper, whereas the KS representation is different and new. These results are shown to illuminate our understanding of the S_n from both a physical and quantitative point of view. In particular rigorous upper and lower bounds on the S_n are obtained for a two-phase medium formed so as to be in a state of thermal equilibrium. For such a medium consisting of impenetrable-sphere inclusions in a matrix, a new exact expression is also given for S_n in terms of a two-body probability distribution function ρ₂ as well as new expressions for S₃ in terms of ρ₂ and ρ₃, a three-body distribution function. Physical insight into the nature of these results is given by extending some geometrical arguments originally put forth by Boltzmann.