Thermalization with a multibath: an investigation in simple models

TL;DR

This paper investigates how simple spin models coupled to two thermal baths reach stationary states, analyzing the fluctuation-dissipation relations and thermalization processes in non-equilibrium conditions.

Contribution

It provides analytical and numerical insights into the stationary states and fluctuation-dissipation relations of spin models with two baths, especially in the slow bath limit.

Findings

Systems reach a stationary state in finite time regardless of bath parameters.

The fluctuation-dissipation relation deviates from equilibrium form in non-trivial ways.

Thermalization with both baths depends on the relative timescales and temperatures.

Abstract

We study analytically and numerically a couple of paradigmatic spin models, each described in terms of two sets of variables attached to two different thermal baths with characteristic timescales $T$ and $\tau$ and inverse temperatures $B$ and $\beta$. In the limit in which one bath becomes extremely slow ($\tau \to \infty$), such models amount to a paramagnet and to a one-dimensional ferromagnet, in contact with a single bath. We show that these systems reach a stationary state in a finite time for any choice of $B$ and $\beta$. We determine the non-equilibrium fluctuation-dissipation relation between the autocorrelation and the response function in such state and, from that, we discuss if and how thermalization with the two baths occurs and the emergence of a non-trivial fluctuation-dissipation ratio.