Chord-length and free-path distribution functions for many-body systems

We study fundamental morphological descriptors of disordered media (e.g., heterogeneous materials, liquids, and amorphous solids): the chord-length distribution function p(z) and the free-path distribution function p(z,a). For concreteness, we will speak in the language of heterogeneous materials composed of two different materials or “phases.” The probability density function p(z) describes the distribution of chord lengths in the sample and is of great interest in stereology. For example, the first moment ofp(z) is the “mean intercept length” or “mean chord length.” The chord-length distribution function is of importance in transport phenomena and problems involving “discrete free paths” of point particles (e.g., Knudsen diffusion and radiative transport). The free-path distribution function p(z,a) takes into account the finite size of a simple particle of radius a undergoing discrete free-path motion in the heterogeneous material and we show that it is actually the chord-length distribution function for the system in which the “pore space” is the space available to a finite-sized particle of radius a. Thus it is shown that p(z) =p(z,O). We demonstrate that the functions p(z) and p(z,a) are related to another fundamentally important morphological descriptor of disordered media, namely, the so-called lineal-path function L(z) studied by us in previous work [Phys. Rev. A 45, 922 ( 1992)]. The lineal path function gives the probability of finding a line segment of length z wholly in one of the “phases” when randomly thrown into the sample. We derive exact series representations of the chord-length and free-path distribution functions for systems of spheres with a polydispersivity in size in arbitrary dimension D. For the special case of spatially uncorrelated spheres (i.e., fully penetrable spheres) we evaluate exactly the aforementioned functions, the mean chord length, and the mean free path. We also obtain corresponding analytical formulas for the case of mutually impenetrable (i.e., spatially correlated) polydispersed spheres.