Hyperbolic Heat Transport in Rigid Conductors

Abstract

There exists an impressive amount of literature on heat conduction and hyperbolic equations. It is not our purpose to examine this topic exhaustively, but to stress the more significant and illustrative aspects in connection with extended irreversible thermodynamics (EIT). Two main motivations underlie such a vast literature. One of them, of a theoretical nature, refers to the so-called ‘paradox’ of propagation of thermal signals with infinite speed. The second, more closely related to experimental observations, deals with the propagation of second sound, ballistic phonon propagation, and phonon hydrodynamics in solids at low temperatures, where heat transport departs dramatically from the usual parabolic description.

In the classical theory of heat transport, thermal signals obey a parabolic equation. In the linear approximation, this implies that the influence of such a signal is felt instantaneously throughout the whole system, or, otherwise stated, that the thermal signal propagates at infinite velocity. One of the first motivations for EIT was precisely to remove the paradox of infinite speed of propagation.

A macroscopic theory with finite speed of propagation finds its roots in the kinetic theory and in experiments. From an experimental point of view, the search for generalising the Fourier equation was launched in the 1960s by the discovery of second sound and ballistic phonon propagation in some dielectric crystals at low temperature. This observation stimulated also the development of microscopic models of heat conduction supporting the generalised macroscopic transport equations. Such analyses are of interest in solid-state physics because they provide useful and relevant information on phonon scattering processes. However, most works on heat propagation are concerned with the dynamical consequences of the transport equations, without paying much attention to their thermodynamic implications. In Chap. 2 we have already emphasized the need to deal consistently with both dynamical and thermodynamical aspects. In Chap. 10 we complete the analysis of heat transport by focussing our attention on nanosystems, because of their technological applications and their conceptual challenges.