Non-equilibrium steady states for chains of four rotors

TL;DR

This paper analyzes a chain of four rotors connected to heat baths, demonstrating convergence to a non-equilibrium steady state at a stretched exponential rate, and addressing resonance challenges with rapid external rotor thermalization.

Contribution

It extends previous work on three rotors by analyzing four rotors, introducing methods to handle resonances and constructing Lyapunov functions for the system.

Findings

System relaxes to a non-equilibrium steady state at a stretched exponential rate.

High-energy rotors decouple, affecting energy dissipation times.

Resonances are managed via rapid thermalization of external rotors.

Abstract

We study a chain of four interacting rotors (rotators) connected at both ends to stochastic heat baths at different temperatures. We show that for non-degenerate interaction potentials the system relaxes, at a stretched exponential rate, to a non-equilibrium steady state (NESS). Rotors with high energy tend to decouple from their neighbors due to fast oscillation of the forces. Because of this, the energy of the central two rotors, which interact with the heat baths only through the external rotors, can take a very long time to dissipate. By appropriately averaging the oscillatory forces, we estimate the dissipation rate and construct a Lyapunov function. Compared to the chain of length three (considered previously by C. Poquet and the current authors), the new difficulty with four rotors is the appearance of resonances when both central rotors are fast. We deal with these resonances using the rapid thermalization of the two external rotors.