Mixing rates of particle systems with energy exchange

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.