Energy Transport in Stochastically Perturbed Lattice Dynamics

Abstract

We consider lattice dynamics with a small stochastic perturbation of order ε and prove that for a space-time scale of order ε −1 the local spectral density (Wigner function) evolves according to a linear transport equation describing inelastic collisions. For an energy and momentum conserving chain the transport equation predicts a slow decay, as 1/ √ t, for the energy current correlation in equilibrium. This is in agreement with previous studies using a different method.Proof. We use the evolution equation (33) where K is the bound on the total energy from condition (b3). Then by Gronwall's inequality J ρ , E ε (t) ≤ 2ρK + J ρ , E ε (0) e c 3 γt ,

Energy Diffusion in Harmonic System with Conservative Noise

“…In [2], [24], [20] [29] a kinetic limit is performed for chains of an-harmonic oscillators, and in [23] a linear Boltzmann equation is rigorously derived for the harmonic chain of oscillators with random masses. In [5] the authors consider a system of harmonic oscillators in ∂ t u α (t, r,k) + v(k) · ∇u α (t, r, k)…”

“…showing that thermal conductivity is infinite in dimension one and two for a system of harmonic oscillators perturbed by a conservative noise ( [5], [4]). …”

“…The key point is the derivation of a Nash type inequality which provides an estimate for convergence rates slower than exponential ( [22], [6], [30]). The diffusion coefficient is given by an infrared regularization of the thermal conductivity obtained in [4], [5], with a proper renormalization (13).…”

“…In particular the rescaled solution of the Boltzmann equation converges to a diffusion equation, with a diffusion coefficient given by the thermal conductivity obtained in [4], [5].…”

Energy Diffusion and Superdiffusion in Oscillators Lattice Networks

“…In [5] we consider the hyperbolic scaling in a weak noise limit. Noise is rescaled by multiplying its strength γ by .…”