Thermalization in harmonic particle chains with velocity flips

TL;DR

This paper introduces a new mathematical approach using dynamical system replication to analyze thermalization in harmonic chains with velocity flips, demonstrating rapid local Fourier's law compliance before diffusive timescales.

Contribution

The paper develops a novel dynamical replica method to study thermalization in lattice models, providing explicit bounds and extending analysis to systems with anharmonic potentials.

Findings

Thermalization occurs within a time scale of order L^(2/3).

The kinetic temperature profile satisfies Fourier's law locally before diffusive times.

The method applies to systems with velocity flips and potentially to anharmonic onsite potentials.

Abstract

We propose a new mathematical tool for the study of transport properties of models for lattice vibrations in crystalline solids. By replication of dynamical degrees of freedom, we aim at a new dynamical system where the "local" dynamics can be isolated and solved independently from the "global" evolution. The replication procedure is very generic but not unique as it depends on how the original dynamics are split between the local and global dynamics. As an explicit example, we apply the scheme to study thermalization of the pinned harmonic chain with velocity flips. We improve on the previous results about this system by showing that after a relatively short time period the average kinetic temperature profile satisfies the dynamic Fourier's law in a local microscopic sense without assuming that the initial data is close to a local equilibrium state. The bounds derived here prove that the above thermalization period is at most of the order L^(2/3), where L denotes the number of particles in the chain. In particular, even before the diffusive time scale Fourier's law becomes a valid approximation of the evolution of the kinetic temperature profile. As a second application of the dynamic replica method, we also briefly discuss replacing the velocity flips by an anharmonic onsite potential.