On second sound at the critical temperature

Abstract

Based on low-temperature experimental data in solid dielectric crystals, we derive a model of heat conduction for rigid materials using the theory of thermo-dynamic internal state variables. The model is intended to admit wavelike propagation of heat below—and diffusive conduction above—a particular temperature value ϑ λ {\vartheta {_\lambda }} . A rapid decay of the speed of thermal waves occurs just below this temperature, coincident with the conductivity of the material reaching a peak. An analysis of weak and strong discontinuity waves is given in order to exhibit several main features of the proposed model.

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

“…The functions , 1 , and 2 present in (6) and (8) are material functions. The second law of thermodynamics imposes the restrictions that ( ) = 20 2 1 ( ) and 2 ( ) < 0, where the constant 20 comes from the Helmholtz free energy which has the form = 1 ( ) + (1/2) 20 P 2 .…”

“…Finally, by employing the substitution ( ) = V with = G 1 ∘ −1 , = 2 ∘ −1 , and ℎ = ∘ −1 , system (9) is exactly system (1).…”

“…The above system is derived in [1,2] and describes the propagation of heat wave for rigid solids at very low temperatures, below about 20 K. The first equation in (1) comes from the balance of energy, which takes the form…”

“…[1][2][3][4] and references therein). To improve the model, one may generalize the history dependence of q by modifying (4) or, as was done in [2], by introducing a suitable nonlinear dependence in (7),…”

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

“…This then requires the use of an extended form of thermodynamics, in which we employ an internal parameter, p , to appropriately model observations. The internal parameter satisfies a particular form of evolution equation (see [10], [9]), and heat flow is described below ϑ λ by the equations ε(ϑ) t + q x = 0, (2.3) (2.5) where g 1 (ϑ) ≥ 0 ≥ g 2 (ϑ) and α(ϑ) ≥ 0 are material functions. The second law of thermodynamics imposes the restriction that α(ϑ) = ψ 20 ϑ 2 g 1 (ϑ), where the constant ψ 20 > 0 comes from the Helmholtz free energy, ψ, which has the form ψ = ψ 1 (ϑ) + 1 2 ψ 20 ϑp 2 .…”