SPECTRAL GAP FOR STOCHASTIC ENERGY EXCHANGE MODEL WITH NONUNIFORMLY POSITIVE RATE FUNCTION

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...

Full description

Saved in:
Bibliographic Details
Published in:The Annals of probability 2015-07, Vol.43 (4), p.1663-1711
Main Author: Sasada, Makiko
Format: Article
Language:English
Subjects:
60K35
82C31
energy exchange
Kinetics
Lattice theory
locally confined hard balls
Mathematical functions
nonuniformly positive rate function
Spectral gap
Stochastic models
Studies
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 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(ℰ)N−2 where C(ℰ) is a positive constant depending on 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.
ISSN:0091-1798
2168-894X
DOI:10.1214/14-AOP916