Local stationarity for lattice dynamics in the harmonic approximation

Abstract

We consider the lattice dynamics in the harmonic approximation for We consider the lattice dynamics in the harmonic approximation for a simple hypercubic lattice with arbitrary unit cell. 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 of order ${\epsilon }^{-\gamma }$, $0<\gamma <1$, locally the measure converges to a Gaussian measure which is space-time stationary with a covariance inherited from the initial (in general, non-Gaussian) measure. For times of order ${\epsilon }^{-1}$ this local space covariance changes in time and is governed by a semiclassical transport equation.