Mixing rates of particle systems with energy exchange

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.

References in corpus (4)

• Heat conduction and Fourier's law in a class of many particle dispersing billiards
• On the derivation of Fourier's law in stochastic energy exchange systems
• Heat transport in stochastic energy exchange models of locally confined hard spheres
• Analysis of top-swap shuffling for genome rearrangements

Cited by in corpus (9)

• Equilibration of energy in slow-fast systems
• Spectral gap for stochastic energy exchange model with nonuniformly positive rate function
• Sub-exponential mixing of random billiards driven by thermostats
• On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas
• A two-stage approach to relaxation in billiard systems of locally confined hard spheres
• Numerical simulation of polynomial-speed convergence phenomenon
• Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases
• Assortative Exchange Processes
• Thermal conductivity and local thermodynamic equilibrium of stochastic energy exchange models