The Diffusion Approximation for the Linear Boltzmann Equation with Vanishing Scattering Coefficient

TL;DR

This paper studies the diffusion approximation of the linear Boltzmann equation in media with non-uniform scattering, including optically thin inclusions, extending the understanding of particle transport in complex media.

Contribution

It introduces a diffusion approximation framework for the linear Boltzmann equation with vanishing scattering in parts of the domain, applicable to composite media with thin inclusions.

Findings

Diffusion approximation holds even with non-uniform scattering.

The limit equation involves an infinite diffusion coefficient in optically thin regions.

Applicable to radiative transfer in composite media.

Abstract

The present paper discusses the diffusion approximation of the linear Boltzmann equation in cases where the collision frequency is not uniformly large in the spatial domain. Our results apply for instance to the case of radiative transfer in a composite medium with optically thin inclusions in an optically thick background medium. The equation governing the evolution of the approximate particle density coincides with the limit of the diffusion equation with infinite diffusion coefficient in the optically thin inclusions.