Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors

TL;DR

This paper studies a chain of three coupled rotors connected to heat baths, proving ergodicity with a stretched exponential rate and analyzing the energy dissipation of the middle rotor using averaging techniques.

Contribution

It establishes the existence and uniqueness of a non-equilibrium steady state and provides a novel analysis of energy dissipation for the middle rotor via effective dynamics.

Findings

Unique invariant measure exists under non-degeneracy conditions.

The process exhibits ergodicity with a stretched exponential convergence rate.

Numerical illustrations of the steady state properties are provided.

Abstract

We consider a chain of three rotors (rotators) whose ends are coupled to stochastic heat baths. The temperatures of the two baths can be different, and we allow some constant torque to be applied at each end of the chain. Under some non-degeneracy condition on the interaction potentials, we show that the process admits a unique invariant probability measure, and that it is ergodic with a stretched exponential rate. The interesting issue is to estimate the rate at which the energy of the middle rotor decreases. As it is not directly connected to the heat baths, its energy can only be dissipated through the two outer rotors. But when the middle rotor spins very rapidly, it fails to interact effectively with its neighbors due to the rapid oscillations of the forces. By averaging techniques, we obtain an effective dynamics for the middle rotor, which then enables us to find a Lyapunov function. This and an irreducibility argument give the desired result. We finally illustrate numerically some properties of the non-equilibrium steady state.