Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities

TL;DR

This paper studies how energy and elongation evolve in a harmonic chain with random velocity flips, showing they follow diffusive equations in a large-scale limit, with energy dynamics influenced by elongation gradients.

Contribution

It proves the macroscopic diffusive behavior of energy and elongation in a harmonic chain with velocity flips, highlighting the coupled evolution of these quantities.

Findings

Elongation follows a linear diffusive equation.

Energy evolution depends on elongation gradient.

Profiles converge to solutions of diffusive PDEs.

Abstract

We consider an unpinned chain of harmonic oscillators with periodic boundary conditions, whose dynamics is perturbed by a random flip of the sign of the velocities. The dynamics conserves the total volume (or elongation) and the total energy of the system. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation.