Energy Transport by Acoustic Modes of Harmonic Lattices

Abstract

We study the large scale evolution of a scalar lattice excitation u which satisfies a discrete wave equation in three dimensions, $\ddot u_t(\gamma)=-\sum_{\gamma'}\alpha(\gamma-\gamma')u_t(\gamma')$, where $\gamma,\gamma'\in\mathbb{Z}^3$ are lattice sites. We assume that the dispersion relation $\omega$ associated to the elastic coupling constants $\alpha(\gamma-\gamma')$ is acoustic; i.e., it has a singularity of the type $|k|$ near the vanishing wave vector, $k= 0$. To derive equations describing the macroscopic energy transport, we employ a related multiscale Wigner transform and a scale parameter $\varepsilon>0$. The spatial and temporal scales of the Wigner transform are related to the corresponding lattice parameters via a scaling by $\varepsilon$. In the continuum limit, which is achieved by sending the parameter $\varepsilon$ to 0, the Wigner transform disintegrates into three different limit objects: the Wigner transform of a rescaled weak-$L^2$ limit, an H-measure, and a Wigner measure. The first two provide the finer resolution of the energy concentrating at $k=0$ so that a set of closed evolution equations may arise. We demonstrate that these three limit objects satisfy a set of decoupled transport equations: a wave equation for the weak limit, a geometric optics transport equation for the H-measure limit, and a dispersive transport equation for the standard limiting Wigner measure. This yields a complete characterization of macroscopic energy transport in harmonic lattices with regular acoustic dispersion relations.