Kinetic Limit for Wave Propagation in a Random Medium

TL;DR

This paper proves that in a weakly disordered crystal, the averaged wave dynamics at a kinetic scale follow a linear Boltzmann equation, linking microscopic disorder to macroscopic wave behavior.

Contribution

It establishes a rigorous derivation of the linear Boltzmann equation for wave propagation in a weakly disordered harmonic crystal.

Findings

Disorder averaged Wigner function converges to a solution of the linear Boltzmann equation.

The result applies in the limit of small disorder parameter epsilon.

The dispersion relation is a Morse function that suppresses crossed recollisions.

Abstract

We study crystal dynamics in the harmonic approximation. The atomic masses are weakly disordered, in the sense that their deviation from uniformity is of order epsilon^(1/2). The dispersion relation is assumed to be a Morse function and to suppress crossed recollisions. We then prove that in the limit epsilon to 0 the disorder averaged Wigner function on the kinetic scale, time and space of order epsilon^(-1), is governed by a linear Boltzmann equation.