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.

References in corpus (2)

• On the convergence to statistical equilibrium for harmonic crystals
• On two-temperature problem for harmonic crystals

Cited by in corpus (8)

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• On the Boltzmann Equation for Weakly Nonlinear Wave Equations
• Derivation of the Boltzmann equation and entropy production in functional mechanics
• Hierarchy of space-time structures, Boltzmann equation, and functional mechanics