Energy Transport in Harmonic Lattices

Abstract

We study the large scale evolution induced by a discrete scalar wave-equation in three dimensions ü_t(γ) = Σ_γ′ α(γ −γ′)u_t(γ′), where γ, γ′ ∈ ℤ³ are lattice sites, and α(γ−γ) is the coupling constant between site γ and γ′. The evolution preserves a Hamiltonian which is the sum of kinetic energy and potential energy. 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 ε. In the limit, where the parameter ε is taken 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 or our result is the complete characterization of energy transport in harmonic lattices with acoustic dispersion relation.

[DOI: 10.1685 / CSC06170] About DOI