Chord length distribution in d-dimensional anisotropic Markov media

Abstract

Markov media are often used as a prototype model in the analysis of linear particle transport in disordered materials. For this class of stochastic geometries, it is assumed that the chord lengths must follow an exponential distribution, with a direction-dependent average if anisotropy effects are to be taken into account. The practical realizability of Markov media in arbitrary dimension has been a long-standing open question. In this work we show that Poisson hyperplane tessellations provide an explicit construction for random media satisfying the Markov property and easily including anisotropy. The average chord length can be computed explicitly and is be shown to be intimately related to the statistical properties of the tessellation cells and in particular to their surface-to-volume ratio. A computer code that is able to generate anisotropic Poisson tessellations in arbitrary dimension restricted to a given finite domain is developed, and the convergence to exact asymptotic formulas for the chord length distribution and the polyhedral features of the tessellation cells is established by extensive Monte Carlo simulations in the limit of domains having an infinite size.

Cauchy formulas for linear transport in random media

“…Quenched disorder models satisfying the Markov property for the chord lengths 6 were introduced by Pomraning and co-workers for one-dimensional geometries, based on a Poisson point process on the line [1]. We have recently shown that Markov media for any dimension d can be generated by using homogeneous Poisson tessellations, i.e., stochastic partitions of a domain into convex polyhedra obtained by homogeneously sampling hyperplanes, and then assigning each polyhedron of the tessellation a label α with probability p α [16,34]. For this purpose, we have developed a computer code that can construct such geometries by Monte Carlo methods, with arbitrary anisotropy laws [16,34]; examples of realizations for isotropic Poisson tessellations are provided in fig.…”