How to define temperature in active systems?

TL;DR

This paper compares various definitions of temperature in active matter systems, revealing that different measures often coincide far from equilibrium, and discusses their practical relevance and limitations.

Contribution

It systematically analyzes and compares multiple temperature definitions in active systems, highlighting conditions under which they agree or differ, and evaluates their measurement implications.

Findings

Different temperature measures often coincide far from equilibrium.

Kinetic, configurational, and higher-moment temperatures form a class with similar values.

Effective temperature and virial-based temperatures form a separate class with similar values.

Abstract

We are used to measure temperature with a thermometer and we know from everyday life that different types of thermometers measure the same temperature. This experience can be based on equilibrium thermodynamics, which explains the equivalence of different possibilities to define temperature. In contrast, for systems out of equilibrium such as active matter, measurements performed with different thermometers can generally lead to different temperature values. In the present work, we systematically compare different possibilities to define temperature for active systems. Based on simulations and theory for inertial active Brownian particles, we find that different temperatures generally lead to different temperature values, as expected. Remarkably, however, we find that different temperatures not only lead to the same values near equilibrium (low P\'eclet number or high particle mass), but even far from equilibrium, several different temperatures approximately coincide. In particular, we find that the kinetic temperature, the configurational temperature, and temperatures based on higher moments of the velocity distribution constitute a class of temperatures that all assume very similar values over a wide parameter range. Notably, the effective temperature and temperatures exploiting the virial theorem, the Stokes-Einstein relation, or a harmonic confinement form a second class of temperatures whose values approximately coincide with each other but which strongly differ from those of the first class. Finally, we identify advantages and disadvantages of the different possibilities to define temperature and discuss their relevance for measuring the temperature of active systems.