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.
All Science Journal Classification (ASJC) codes
- Physics and Astronomy(all)
- Physical and Theoretical Chemistry