Two-point cluster function for continuum percolation

Abstract

We introduce a two-point cluster function C2(r1,r2) which reflects information about clustering in general continuum–percolation models. Specifically, for any two-phase disordered medium, C2(r1,r2) gives the probability of finding both points r1 and r2 in the same cluster of one of the phases. For distributions of identical inclusions whose coordiantes are fully specified by center-of-mass positions (e.g., disks, spheres, oriented squares, cubes, ellipses, or ellipsoids, etc.), we obtain a series representation of C2 which enables one to compute the two-point cluster function. Some general asymptotic properties of C2 for such models are discussed. The two-point cluster function is then computed for the adhesive-sphere model of Baxter. The two-point cluster function for arbitrary media provides a better signature of the microstructure than does a commonly employed two-point correlation function defined in the text.