Linear Boltzmann-like equation, describing non-classical particle transport, and related asymptotic solutions for small mean free paths

TL;DR

This paper derives a generalized linear Boltzmann-like equation for non-classical particle transport with arbitrary free path distributions and constructs asymptotic solutions that approximate diffusion behavior for small mean free paths.

Contribution

It introduces a Boltzmann-like equation accommodating general free path distributions and develops asymptotic diffusion approximations for small mean free paths.

Findings

Derived a generalized Boltzmann-like equation for arbitrary free path distributions.

Constructed asymptotic solutions leading to diffusion approximations.

Applicable to models with finite first and second moments of free path distributions.

Abstract

In classical kinetic or kinetic-like models a particle free path distribution is exponensial, but this is more likely to be an exception than a rule. In this paper we derive a linear Boltzmann-like equation for a general free path distribution in the framework of Alt's model J. Math. Biol. 9:147 (1980). In the special case that the free path distribution has at least first and second finite moments we construct an asymptotic solution of the equation for small mean free paths. The asymptotic solution becomes a diffusion approximation to the one-speed Boltzmann-like equation.