Lattice Dynamics in the Half-Space, II. Energy Transport Equation

TL;DR

This paper studies lattice dynamics in a half-space with random initial data, showing convergence to Gaussian measures and deriving a semiclassical transport equation for energy distribution over different time scales.

Contribution

It introduces a detailed analysis of energy transport in lattice systems with random initial conditions, deriving a new semiclassical transport equation for the covariance evolution.

Findings

Gaussian measure convergence at intermediate times

Covariance evolution governed by a transport equation

Different regimes of energy transport depending on time scale

Abstract

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.