Lattice dynamics in the half-space. Energy transport equation

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 epsilon(-1). We establish two time regimes. For times on the order of epsilon(-kappa), 0 < kappa < 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 epsilon(-1), this covariance changes in time and is governed by a semiclassical transport equation. (C) 2010 American Institute of Physics. [doi:10.1063/1.3451109]