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

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Abstract
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 Sn associated with two‐phase random media. The MM representation proves to be identical to the Sn 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 Sn from both a physical and quantitative point of view. In particular rigorous upper and lower bounds on the Sn 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 Sn 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.