Phase Transitions and Change of Type in Low-Temperature Heat Propagation

Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, where dissipative effects are insignificant. 1. Introduction. The analysis in this paper is based on a low-temperature heat propagation model described in (9) and (10). The model is based on experimental results of (3), (2), (5), and (6), which provide evidence of second sound, i.e., hyperbolic, or wavelike, thermal effects where Fourier's law fails, in very pure crystals of sodium fluoride and bismuth. Significantly, these features appear only at certain temperatures below which the materials reach their peak thermal conductivities (at approximately 18.5 K and 4.5 K for NaF and Bi, respectively). No wavelike behavior is found in NaF and Bi at higher temperatures, where only diffusive heat propagation is observed. Further, the speed, UE, at which small amplitude thermal waves propagate is a decreasing function of temperature in the region where the waves can be detected, after which the diffusion process dominates. This hyperbolic region appears separated from the diffusive region by a "critical" temperature, ϑλ, at which UE = 0 (1). The aim of this paper is to understand the dynamics of regular solutions having temperatures close to that of the phase transition. We begin, in section 2, by describing a phenomenological one- dimensional model which uses an internal variable behaving as an order parameter. In section 3, we will examine properties of the phase transition, and in section 4, we obtain conditions under which this class of solutions remain smooth. Some explicit cases are, finally, examined in section 5. 2. Preliminaries. We briefly describe our model and refer to (10) (see also (9)) for further details concerning the thermodynamics of materials with internal parame- ters. In the present context, two forms of heat transmission—diffusive propagation at high temperatures and wavelike propagation at low temperatures—are separated by a phase transition at a critical temperature, ϑλ > 0. At temperatures above ϑλ ,w e