Temperature in and out of equilibrium: a review of concepts, tools and attempts

TL;DR

This review explores the concept of temperature across equilibrium and non-equilibrium systems, addressing recent debates, experimental observations like negative temperatures, and the challenges of defining temperature in small and complex systems.

Contribution

It provides a comprehensive overview of various definitions, recent developments, and open questions regarding temperature in diverse physical contexts, including non-Hamiltonian systems.

Findings

Negative temperatures observed in experiments

Fluctuation-response relations suggest extended temperature concepts

Challenges in defining temperature for small and non-Hamiltonian systems

Abstract

We review the general aspects of the concept of temperature in equilibrium and non-equilibrium statistical mechanics. Although temperature is an old and well-established notion, it still presents controversial facets. After a short historical survey of the key role of temperature in thermodynamics and statistical mechanics, we tackle a series of issues which have been recently reconsidered. In particular, we discuss different definitions and their relevance for energy fluctuations. The interest in such a topic has been triggered by the recent observation of negative temperatures in condensed matter experiments. Moreover, the ability to manipulate systems at the micro and nano-scale urges to understand and clarify some aspects related to the statistical properties of small systems (as the issue of temperature's "fluctuations"). We also discuss the notion of temperature in a dynamical context, within the theory of linear response for Hamiltonian systems at equilibrium and stochastic models with detailed balance, and the generalised fluctuation-response relations, which provide a hint for an extension of the definition of temperature in far-from-equilibrium systems. To conclude we consider non-Hamiltonian systems, such as granular materials, turbulence and active matter, where a general theoretical framework is still lacking.