  We consider the lattice dynamics in the half-space. The initial data are
random according to a probability measure which enforces slow spatial variation
on the linear scale $\varepsilon^{-1}$. We establish two time regimes. For
times of order $\varepsilon^{-\gamma}$, $0<\gamma<1$, 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 of
order $\varepsilon^{-1}$, this covariance changes in time and is governed by a
semiclassical transport equation.
