Energy transport through rare collisions

TL;DR

This paper investigates energy transport in a one-dimensional Hamiltonian chain with rare collisions and velocity-flipping noise, deriving a stochastic energy evolution equation in the limit of vanishing cell overlap.

Contribution

It introduces a model combining Hamiltonian dynamics with rare collisions and noise, deriving the limiting stochastic energy evolution as cell overlap tends to zero.

Findings

Energy evolves according to a stochastic equation in the limit

Limiting process matches the symmetric exclusion process with two energies

Hydrodynamic limit aligns with known stochastic models

Abstract

We study a one-dimensional hamiltonian chain of masses perturbed by an energy conserving noise. The dynamics is such that, according to its hamiltonian part, particles move freely in cells and interact with their neighbors through collisions, made possible by a small overlap of size $\epsilon > 0$ between near cells. The noise only randomly flips the velocity of the particles. If $\epsilon \rightarrow 0$, and if time is rescaled by a factor $1/{\epsilon}$, we show that energy evolves autonomously according to a stochastic equation, which hydrodynamic limit is known in some cases. In particular, if only two different energies are present, the limiting process coincides with the simple symmetric exclusion process.