Toward the Fourier law for a weakly interacting anharmonic crystal

TL;DR

This paper derives a macroscopic stochastic evolution equation for the energies in a weakly interacting anharmonic crystal system, demonstrating a connection to the Ginzburg-Landau system through advanced mathematical properties.

Contribution

It establishes an autonomous stochastic energy evolution for weakly interacting anharmonic oscillators, linking microscopic dynamics to a macroscopic Ginzburg-Landau system.

Findings

Derivation of the stochastic energy evolution equation

Identification of the Ginzburg-Landau system as the macroscopic limit

Use of hypocoercivity and hypoellipticity in the proof

Abstract

For a system of weakly interacting anharmonic oscillators, perturbed by an energy preserving stochastic dynamics, we prove an autonomous (stochastic) evolution for the energies at large time scale (with respect to the coupling parameter). It turn out that this macroscopic evolution is given by the so called conservative (non-gradient) Ginzburg-Landau system of stochastic differential equations. The proof exploits hypocoercivity and hypoellipticity properties of the uncoupled dynamics.