Chord distribution functions of three-dimensional random media: Approximate first-passage times of Gaussian processes

TL;DR

This paper introduces a semi-analytic approximation for the chord distribution functions in 3D Gaussian random media, linking microstructure analysis to first-passage time problems and validating with experimental data.

Contribution

It presents a novel approximation method for chord distribution functions in 3D Gaussian models, extending to complex structures and validated against experimental images.

Findings

Approximate chord functions match experimental data

Method generalizes to complex random models

Provides insights into microstructure characterization

Abstract

The main result of this paper is a semi-analytic approximation for the chord distribution functions of three-dimensional models of microstructure derived from Gaussian random fields. In the simplest case the chord functions are equivalent to a standard first-passage time problem, i.e., the probability density governing the time taken by a Gaussian random process to first exceed a threshold. We obtain an approximation based on the assumption that successive chords are independent. The result is a generalization of the independent interval approximation recently used to determine the exponent of persistence time decay in coarsening. The approximation is easily extended to more general models based on the intersection and union sets of models generated from the iso-surfaces of random fields. The chord distribution functions play an important role in the characterization of random composite and porous materials. Our results are compared with experimental data obtained from a three-dimensional image of a porous Fontainebleau sandstone and a two-dimensional image of a tungsten-silver composite alloy.