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

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.

Local temperatures out of equilibrium

“…They have concluded that under the local equilibrium condition, different definitions give the same temperature, while in a non-equilibrium state beyond local equilibrium, the values of temperature defined by different approaches may deviate from one another. Puglisi et al have reviewed different definitions of temperature in classical systems and discussed their relevance to energy fluctuations [139]. Particularly, a generalized fluctuation-dissipation relation (FDR) was introduced, which provides a statistical basis for extending the definition of temperature to far-from-equilibrium classical systems.…”

“…The direct application of Eq. (15) to open quantum systems is difficult [139,146], because different definitions of entropy do not lead to a consistent value even at the same temperature [147,148]. For example, the von Neumann entropy of the reduced system, S VN ≡ −tr S (ρ S ln ρ S ), has been used to define the local temperature via Eq.…”

Initial, effective, and kinetic freeze-out temperatures from transverse momentum spectra in high-energy proton(deuteron)–nucleus and nucleus–nucleus collisions

“…The concept of temperature used in thermodynamics and statistical mechanics [1] is also used in subatomic physics due to similar thermal and statistical property. At least the initial, chemical freeze-out, thermal or kinetic freeze-out, and effective temperatures are used in high energy collisions.…”

On the fluctuation-dissipation relation in non-equilibrium and non-Hamiltonian systems

“…Our work represents an update of the previous review 13 . The interested reader can find more exhaustive reviews of the general subject in 12,[14][15][16][17][18] . For a more rigorous discussion see also 19 .…”