  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.
