Diffusive Propagation of Energy in a Non-Acoustic Chain

TL;DR

This paper rigorously demonstrates that in a non-acoustic chain of harmonic oscillators with momentum-conserving stochastic perturbations, energy diffuses normally while curvature and momentum follow a damped Euler-Bernoulli beam equation, revealing a novel diffusive energy transfer mechanism.

Contribution

It provides the first rigorous proof of normal energy diffusion in a one-dimensional momentum-conserving harmonic oscillator chain with stochastic perturbations.

Findings

Energy density evolves via a nonlinear diffusive equation.

Curvature and momentum follow a linear damped Euler-Bernoulli system.

Energy transfer in this model is diffusive in nature.

Abstract

We consider a non acoustic chain of harmonic oscillators with the dynamics perturbed by a random local exchange of momentum, such that energy and momentum are conserved. The macroscopic limits of the energy density, momentum and the curvature (or bending) of the chain satisfy a system of evolution equations}. We prove that, in a diffusive space-time scaling, the curvature and momentum evolve following a linear system that corresponds to a damped Euler-Bernoulli beam equation. The macroscopic energy density evolves following a non linear diffusive equation. In particular the energy transfer is diffusive in this dynamics. This provides a first rigorous example of a normal diffusion of energy in a one dimensional dynamics that conserves the momentum.