How hot can a heat bath get?

TL;DR

This paper analyzes a simple yet rich model of two interacting particles connected to heat reservoirs at different temperatures, revealing diverse long-term behaviors including existence of steady states and tail distributions of energy.

Contribution

It provides a comprehensive analysis of the long-time dynamics, steady state existence, and tail behavior in a non-equilibrium system with extreme temperature differences.

Findings

No steady state exists if the potential is too stiff.

Energy tails can be algebraic, fractional exponential, or exponential.

Convergence speed varies from algebraic to exponential depending on parameters.

Abstract

We study a model of two interacting Hamiltonian particles subject to a common potential in contact with two Langevin heat reservoirs: one at finite and one at infinite temperature. This is a toy model for 'extreme' non-equilibrium statistical mechanics. We provide a full picture of the long-time behaviour of such a system, including the existence / non-existence of a non-equilibrium steady state, the precise tail behaviour of the energy in such a state, as well as the speed of convergence toward the steady state. Despite its apparent simplicity, this model exhibits a surprisingly rich variety of long time behaviours, depending on the parameter regime: if the surrounding potential is 'too stiff', then no stationary state can exist. In the softer regimes, the tails of the energy in the stationary state can be either algebraic, fractional exponential, or exponential. Correspondingly, the speed of convergence to the stationary state can be either algebraic, stretched exponential, or exponential. Regarding both types of claims, we obtain matching upper and lower bounds.