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

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.

Published at http://dx.doi.org/10.1214/14-AOP916 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (6)

• Spectral gap for the zero range process with constant rate
• On the derivation of Fourier's law in stochastic energy exchange systems
• Macroscopic energy diffusion for a chain of anharmonic oscillators
• Mixing rates of particle systems with energy exchange
• Heat transport in stochastic energy exchange models of locally confined hard spheres
• Analysis of top-swap shuffling for genome rearrangements

Cited by in corpus (5)

• Integrable heat conduction model
• On the limiting Markov process of energy exchanges in a rarely interacting ball-piston gas
• Role of conserved quantities in Fourier's law for diffusive mechanical systems
• Dynamical contribution to the heat conductivity in stochastic energy exchanges of locally confined gases
• Thermal conductivity and local thermodynamic equilibrium of stochastic energy exchange models