Spectral gap for stochastic energy exchange model with nonuniformly positive rate function

TL;DR

This paper establishes a lower bound on the spectral gap for a class of stochastic energy exchange models with nonuniform rate functions, showing dependence on system size and average energy, extending previous results.

Contribution

It extends spectral gap bounds to models with nonuniform rate functions, incorporating energy dependence, and applies results to billiard lattice and stick process models.

Findings

Spectral gap lower bound of order C(𝓔)N^{-2} for nonuniform rate functions.

Dependence of spectral gap on average energy 𝓔.

Application to billiard lattice and stick process models.

Abstract

We give a lower bound on the spectral gap for a class of stochastic energy exchange models. In 2011, Grigo et al. introduced the model and showed that, for a class of stochastic energy exchange models with a uniformly positive rate function, the spectral gap of an $N$-component system is bounded from below by a function of order $N^{-2}$. In this paper, we consider the case where the rate function is not uniformly positive. For this case, the spectral gap depends not only on $N$ but also on the averaged energy $\mathcal{E}$, which is the conserved quantity under the dynamics. Under some assumption, we obtain a lower bound of the spectral gap which is of order $C(\mathcal{E})N^{-2}$ where $C(\mathcal{E})$ is a positive constant depending on $\mathcal {E}$. As a corollary of the result, a lower bound of the spectral gap for the mesoscopic energy exchange process of billiard lattice studied by Gaspard and Gilbert [J. Stat. Mech. Theory Exp. 2008 (2008) p11021, J. Stat. Mech. Theory Exp. 2009 (2009) p08020] and the stick process studied by Feng et al. [Stochastic Process. Appl. 66 (1997) 147-182] are obtained.