Polynomial convergence to equilibrium for a system of interacting particles

TL;DR

This paper proves that a stochastic particle system with energy-dependent interactions converges to equilibrium at a polynomial rate of approximately t^{-2}, with detailed analysis contrasting exponential and polynomial convergence behaviors.

Contribution

It establishes the polynomial convergence rate of the system to equilibrium and discusses conditions affecting exponential versus polynomial decay.

Findings

Convergence rate is approximately t^{-2}

Convergence is faster than any constant times t^{-2+ε} for ε>0

Includes discussion on exponential vs polynomial convergence

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 $\sim t^{-2}$, more precisely it is faster than a constant times $t^{-2+\varepsilon}$ for any $\varepsilon>0$. A discussion of exponential vs polynomial convergence for similar particle systems is included.