Microstructure of two-phase random media. III. The n-point matrix probability functions for fully penetrable spheres

We examine the n-point matrix probability functions S_n (which give the probability of finding n points in the matrix phase of a two-phase random medium), for a model in which the included material consists of fully penetrable spheres of equal diameter (i.e., a system of identical spheres such that their centers are randomly distributed in the matrix). Exploiting the special simplicity of the model we give an explicit closed-form expression for S₃ as well as sharp bounds on S₃ and S₄. Our best lower bound on S₃ and our corresponding upper bound on S ₄ satisfy certain asymptotic forms (for both small and large separation of points) that are satisfied by the exact S₃ and S ₄ for impenetrable as well as penetrable spheres, even though the bounding properties of our expressions can only be guaranteed for penetrable spheres. These expressions (and the resulting approximation for S₄ in terms of S₁ and S₂ obtained from them) are thus highly appropriate approximants for both systems to be used in composite-media transport-coefficient expressions that involve integrals over the S_n. The S₃ expression has in fact been suggested some time ago by Weissberg and Prager; our methods here provide further justification for this expression as well as one means of systematically generalizing it to S _n for higher n.