Generalized linear Boltzmann equations for particle transport in polycrystals

TL;DR

This paper introduces a new generalized linear Boltzmann equation that accounts for long-range memory effects in particle transport within polycrystals, where traditional models fail due to non-exponential free path length distributions.

Contribution

The paper develops a novel generalized linear Boltzmann equation that accurately models particle transport in polycrystals by incorporating non-exponential free path length distributions.

Findings

The new model captures long-range memory effects in polycrystals.

Free path lengths in polycrystals decay exponentially, unlike in single crystals.

The generalized equation improves predictions over classical models.

Abstract

The linear Boltzmann equation describes the macroscopic transport of a gas of non-interacting point particles in low-density matter. It has wide-ranging applications, including neutron transport, radiative transfer, semiconductors and ocean wave scattering. Recent research shows that the equation fails in highly-correlated media, where the distribution of free path lengths is non-exponential. We investigate this phenomenon in the case of polycrystals whose typical grain size is comparable to the mean free path length. Our principal result is a new generalized linear Boltzmann equation that captures the long-range memory effects in this setting. A key feature is that the distribution of free path lengths has an exponential decay rate, as opposed to a power-law distribution observed in a single crystal.