Mixing rates of particle systems with energy exchange

TL;DR

This paper investigates the convergence rates to equilibrium in energy exchange particle systems on finite lattices, establishing spectral gap scaling and classifying stationary distributions, advancing understanding of macroscopic heat equations from microscopic models.

Contribution

It proves spectral gap scaling as N^{-2} for a class of energy exchange models and classifies their reversible stationary distributions, linking microscopic dynamics to macroscopic heat equations.

Findings

Spectral gap scales as order N^{-2}.

Complete classification of reversible stationary distributions.

Results apply to models similar to billiard lattice models.

Abstract

A fundamental problem of non-equilibrium statistical mechanics is the derivation of macroscopic transport equations in the hydrodynamic limit. The rigorous study of such limits requires detailed information about rates of convergence to equilibrium for finite sized systems. In this paper we consider the finite lattice $\{1, 2,..., N\}$, with an energy $\EnergyStateI{i} \in (0,\infty)$ associated to each site. The energies evolve according to a Markov jump process with nearest neighbor interaction such that the total energy is preserved. We prove that for an entire class of such models the spectral gap of the generator of the Markov process scales as $\Order(N^{-2})$. Furthermore, we provide a complete classification of reversible stationary distributions of product type. We demonstrate that our results apply to models similar to the billiard lattice model considered in \cite{10297039,10863485}, and hence provide a first step in the derivation of a macroscopic heat equation for a microscopic stochastic evolution of mechanical origin.