Microstructure of two-phase random media. I. The n-point probability functions

The microstructure of a two-phase random medium can be characterized by a set of general n-point probability functions, which give the probability of finding a certain subset of n-points in the matrix phase and the remainder in the particle phase. A new expression for these n-point functions is derived in terms of the n-point matrix probability functions which give the probability of finding all n points in the matrix phase. Certain bounds and limiting values of the S_n follow: the geometrical interpretation of the S_n and their relationship with n-point correlation functions associated with fluctuating bulk properties is also noted. For a bed or suspension of spheres in a uniform matrix we derive a new hierarchy of equations, giving the S _n in terms of the s-body distribution functions ρ_s associated with a statistically inhomogeneous distribution P_N of spheres in the matrix, generalizing expressions of Weissberg and Prager for S₂ and S₃. It is noted that canonical ensemble of mutually impenetrable spheres and the associated set of ρ_s define, in the limit of an unbounded system, a statistically homogeneous and isotropic medium, as does (trivially) a canonical ensemble of mutually penetrable spheres.