Thermal conductivity in harmonic lattices with random collisions

TL;DR

This paper reviews mathematical results on how harmonic lattices with random velocity exchanges exhibit superdiffusive or diffusive energy transport, depending on chain type, with implications for understanding nonlinear chain behaviors.

Contribution

It provides a rigorous overview of macroscopic energy transport in harmonic chains with random exchanges, linking microscopic dynamics to fractional heat equations.

Findings

Superdiffusion of energy in unpinned acoustic chains

Normal diffusivity in non-acoustic chains

Temperature evolution governed by fractional heat equations

Abstract

We review recent rigorous mathematical results about the macroscopic behaviour of harmonic chains with the dynamics perturbed by a random exchange of velocities between nearest neighbor particles. The random exchange models the effects of nonlinearities of anharmonic chains and the resulting dynamics have similar macroscopic behaviour. In particular there is a superdiffusion of energy for unpinned acoustic chains. The corresponding evolution of the temperature profile is governed by a fractional heat equation. In non-acoustic chains we have normal diffusivity, even if momentum is conserved.