Energy transport by acoustic modes of harmonic lattices

TL;DR

This paper analyzes how energy propagates in three-dimensional harmonic lattices with acoustic dispersion, deriving macroscopic transport equations from microscopic wave dynamics using Wigner transforms and limit measures.

Contribution

It introduces a detailed derivation of macroscopic energy transport equations in harmonic lattices with acoustic dispersion relations using advanced measure-theoretic methods.

Findings

Decoupling of energy transport into wave, dispersive, and geometric optics equations.

Complete characterization of energy transport in acoustic harmonic lattices.

Use of Wigner, H-measure, and Wigner-measure to analyze large-scale limits.

Abstract

We study the large scale evolution of a scalar lattice excitation which satisfies a discrete wave-equation in three dimensions. We assume that the dispersion relation associated to the elastic coupling constants of the wave-equation is acoustic, i.e., it has a singularity of the type |k| near the vanishing wave vector, k=0. To derive equations that describe the macroscopic energy transport we introduce the Wigner transform and change variables so that the spatial and temporal scales are of the order of epsilon. In the continuum limit, which is achieved by sending the parameter epsilon to 0, the Wigner transform disintegrates into three different limit objects: the transform of the weak limit, the H-measure and the Wigner-measure. We demonstrate that these three limit objects satisfy a set of decoupled transport equations: a wave-equation for the weak limit of the rescaled initial data, a dispersive transport equation for the regular limiting Wigner measure, and a geometric optics transport equation for the H-measure limit of the initial data concentrating to k=0. A simple consequence of our result is the complete characterization of energy transport in harmonic lattices with acoustic dispersion relations.