Effective temperatures in a simple model of non-equilibrium, non-Markovian dynamics

TL;DR

This paper investigates the concept of effective temperatures in a non-Markovian, non-equilibrium diffusion model with two heat baths at different temperatures, comparing various definitions and their validity.

Contribution

It provides an exact analysis of effective temperature definitions in a non-Markovian model, highlighting their relation to different bath temperatures and time scale separations.

Findings

Fluctuation-dissipation based temperature reflects the slow bath temperature.

Quasi-static and thermodynamic temperatures align with the fast bath temperature.

Validity of temperature definitions depends on system parameters and time scales.

Abstract

The concept of effective temperatures in nonequilibrium systems is studied within an exactly solvable model of non-Markovian diffusion. The system is coupled to two heat baths which are kept at different temperatures: one ('fast') bath associated with an uncorrelated Gaussian noise and a second ('slow') bath with an exponential memory kernel. Various definitions of effective temperatures proposed in the literature are evaluated and compared. The range of validity of these definitions is discussed. It is shown in particular, that the effective temperature defined from the fluctuation-dissipation relation mirrors the temperature of the slow bath in parameter regions corresponding to a separation of time scales. On the contrary, quasi-static and thermodynamic definitions of an effective temperature are found to display the temperature of the fast bath in most parameter regions.