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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="theta Subscript lamda"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ϑ<!-- ϑ --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi /> <mml:mi>λ<!-- λ --></mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\vartheta {_\lambda }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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.