Lattice dynamics in the half-space. Energy transport equation

Abstract

We consider the lattice dynamics in the half-space with zero boundary condition. The initial data are random according to a probability measure which enforces slow spatial variation on the linear scale ε−1. We establish two time regimes. For times on the order of ε−κ, 0<κ<1, locally the measure converges to a Gaussian measure, which is time stationary with a covariance inherited from the initial measure (non-Gaussian, in general). For times on the order of ε−1, this covariance changes in time and is governed by a semiclassical transport equation.