Polynomial convergence to equilibrium for a system of interacting particles

Abstract

We consider a stochastic particle system in which a finite number of particles interact with one another via a common energy tank. Interaction rate for each particle is proportional to the square root of its kinetic energy, as is consistent with analogous mechanical models. Our main result is that the rate of convergence to equilibrium for such a system is ∼ t −2 , more precisely it is faster than a constant times t −2+ε for any ε > 0. A discussion of exponential vs polynomial convergence for similar particle systems is included.

From billiards to thermodynamic laws: Stochastic energy exchange model

“…However, these known results can not be applied to the stochastic energy exchange model directly. Even proving the simplest case (M = 1) requires advanced techniques and very tedious calculations [39]. It is very difficult to show the speed of convergence through a direct Monte Carlo simulation either.…”

Numerical Simulation of Polynomial-Speed Convergence Phenomenon

“…It may also be nontrivial to prove that a given function is actually a Lyapunov function. We refer [32] for the examples of estimating first passage time by the Lyapunov function method.…”

On the relaxation rate of short chains of rotors interacting with Langevin thermostats

“…In fact, the convergence rate can be very sensitive to the parameters of the model, as nicely illustrated in [8] for a system of two oscillators with two heat baths, one of which has "infinite temperature". Other energyexchange models where subgeometric convergence rates have been observed include [10,11,13,14].…”