Superdiffusive Heat Transport in a class of Deterministic One-Dimensional Many-Particle Lorentz gases

TL;DR

This paper demonstrates that a deterministic one-dimensional Lorentz gas model exhibits superdiffusive heat transport, with numerical evidence supporting the Boltzmann equation as a valid limit and revealing anomalous transport behavior.

Contribution

It provides numerical validation of the Boltzmann limit for a deterministic Lorentz gas and uncovers superdiffusive heat transport in the absence of conserved quantities.

Findings

Boltzmann description is a valid limit of the particle model.

Deterministic dynamics lead to superdiffusive heat transport.

Model behaves as a persistent random walker with broad waiting times.

Abstract

We study heat transport in a one-dimensional chain of a finite number $N$ of identical cells, coupled at its boundaries to stochastic particle reservoirs. At the center of each cell, tracer particles collide with fixed scatterers, exchanging momentum. In a recent paper, \cite{CE08}, a spatially continuous version of this model was derived in a scaling regime where the scattering probability of the tracers is $\gamma\sim1/N$, corresponding to the Grad limit. A Boltzmann type equation describing the transport of heat was obtained. In this paper, we show numerically that the Boltzmann description obtained in \cite{CE08} is indeed a bona fide limit of the particle model. Furthermore, we also study the heat transport of the model when the scattering probability is one, corresponding to deterministic dynamics. At a coarse grained level the model behaves as a persistent random walker with a broad waiting time distribution and strong correlations associated to the deterministic scattering. We show, that, in spite of the absence of global conserved quantities, the model leads to a superdiffusive heat transport.