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

TL;DR

This paper investigates the relaxation dynamics of a two-rotor system with Langevin interaction, demonstrating that the convergence to equilibrium occurs at a stretched exponential rate with an optimal exponent of 1/2.

Contribution

The authors establish the optimality of the stretched exponential relaxation rate for short chains of rotors interacting with Langevin thermostats.

Findings

Relaxation to equilibrium is at most a stretched exponential with exponent 1/2.

The exponent 1/2 for the relaxation rate is proven to be optimal.

The result applies specifically to a two-rotor system with Langevin heat bath.

Abstract

In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.