The conceptual problems arising in the definition and measurement of temperature in non-equilibrium states are discussed in this paper in situations where the local-equilibrium hypothesis is no longer satisfactory. This is a necessary and urgent discussion because of the increasing interest in thermodynamic theories beyond local equilibrium, in computer simulations, in non-linear statistical mechanics, in new experiments, and in technological applications of nanoscale systems and material sciences. First, we briefly review the concept of temperature from the perspectives of equilibrium thermodynamics and statistical mechanics. Afterwards, we explore which of the equilibrium concepts may be extrapolated beyond local equilibrium and which of them should be modified, then we review several attempts to define temperature in non-equilibrium situations from macroscopic and microscopic bases. A wide review of proposals is offered on effective non-equilibrium temperatures and their application to ideal and real gases, electromagnetic radiation, nuclear collisions, granular systems, glasses, sheared fluids, amorphous semiconductors and turbulent fluids. The consistency between the different relativistic transformation laws for temperature is discussed in the new light gained from this perspective. A wide bibliography is provided in order to foster further research in this field.

/pdf/temperature-in-non-equilibrium-states-a-review-of-open-2vvmx2wgla.pdf

Temperature in non-equilibrium states: a review of open problems and current proposals

This is a review of heat propagation — theory and experiment — in dielectric solids at low temperatures where the phenomenon of second sound occurs. The review does not merely present a list of the various explanations of the observed phenomena. Rather it views them as special cases of a unified theory which is formulated within the framework of extended thermodynamics of phonos. Field equations are derived by averaging over the phonon-Boltzmann equation and initial and boundary value problems are solved. Thus it became possible to achieve a full explanation of the observations of the heat-pulse experiments in which ballistic phonons, second sound and ordinary heat conduction compete.

https://www.engr.uvic.ca/~struchtr/1993CMTphon.pdf

Heat pulse experiments revisited

Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- Newton Method with Incomplete Linearization.- Quasi-chemical Representation.- Hydrodynamics from Grad's Equations: Exact Solutions.- Relaxation Methods.- Method of Invariant Grids.- Method of Natural Projector.- Geometry of Irreversibility: The Film of Nonequilibrium States.- Slow Invariant Manifolds for Open Systems.- Estimation of Dimension of Attractors.- Accuracy Estimation and Post-Processing.- Conclusion.

http://www.math.le.ac.uk/people/ag153/homepage/Beginning.pdf

Invariant Manifolds for Physical and Chemical Kinetics

We review the progress made in extended irreversible thermodynamics during the ten years that have elapsed since the publication of our first review on the same subject (Rep. Frog. Phys. 1988 51 1105-72). During this decade much effort has been devoted to achieving a better understanding of the fundamentals and a broadening of the domain of applications. The macroscopic formulation of extended irreversible thermodynamics is reviewed and compared with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed on the bases of information theory, kinetic theory, stochastic phenomena and computer simulations. Several significant applications are presented, some of them of considerable practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion, microelectronic devices, dielectric relaxation), and some others of special theoretical appeal (superfluids, nuclear collisions, cosmology). We also outline some basic problems which are not yet completely solved, such as the definitions of entropy and temperature out of equilibrium, the selection of the relevant variables, and the status to be reserved to the H-theorem and its relation to the second law. In writing this review, we had four objectives in mind: to show (i) that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics; (ii) that it opens the way to new and useful applications; (iii) that much progress has been achieved during the last decade, and (iv) that the subject is far from being exhausted.

https://iopscience.iop.org/article/10.1088/0034-4885/62/7/201/pdf

Extended irreversible thermodynamics revisited (1988-98)

Phonon hydrodynamics and its applications in nanoscale heat transport

A set of symmetric hyberbolic field equations, describing heat conduction in dielectric solids at low temperatures, is studied with respect to the propagation of temperature shock waves. The field equations have been derived from the Boltzmann-Peierls equation and include the phenomenon of second sound, a special form of wavelike energy transport occuring in some crystals in a temperature range close to absolute zero.

W. Dreyer

Papers

Heat pulse experiments revisited

Entropy and causality as criteria for the existence of shock waves in low temperature heat conduction