Energy transport by acoustic modes of harmonic lattices

Lisa Harris, Jani Lukkarinen, Stefan Teufel, Florian Theil

SIAM Journal on Mathematical Analysis Vol 40, 4 (2008).
[Lattice (487.8 kB)]

Zusammenfassung

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 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 Wignertransform and change variables so that the spatial and temporal scales are of the order of ε. In the continuumlimit, which is achieved by sending the parameter ε to 0, the Wignertransform disintegrate sinto three different limit objects: the transform of the weak limit, theH-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 Wignermeasure, 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.