A Nonlocal Thermodynamic Analysis of 2nd Sound-propagation in Crystalline Dielectrics

A thermodynamic model of second sound propagation in rigid solids like dielectric crystals is proposed: this is achieved within !he framework of extended irreversible thermodynamics. The independent variables are the temperature, the heat flux vector plus a supplementary variable that is identified as the flux of the heat flux; to include non-local effects, the constitutive equations are assumed to depend on the gradients of the temperature and the heat flux vector. After establishing the evolution equations governing the behaviour of the basic variables in the course of space and time, the entropy production is calculated and a generalized Gibbs equation is derived. The present model is shown to be rather general as it encompasses the particular models of Cattaneo and Guyer-Krumhansl. Onsager-like reciprocal relations are also displayed and discussed. Working within the lowest-order approximation, a general wave equation for the temperature is derived. This relation is a third-order hyperbolic differential equation with respect to time, allowing for propagation of waves at finite velocity. A dispersion relation between the wavevector and the frequency is established and the corresponding phase velocity is calculated.