Energy transport by acoustic modes of harmonic lattices
  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. 
Lisa Harris, Jani Lukkarinen, Stefan Teufel, and Florian Theil