Foundations and applications of a mesoscopic thermodynamic theory of fast phenomena

A mesoscopic thermodynamic theory of fast phenomena is proposed. In principle, such a description should include an infinite number of moments of the velocity distribution; in a frequency-wave-number space the corresponding transport coefficients would be expressed in terms of a continued fraction expansion. Our objective is to eliminate from the description a maximum number of fast variables. This is achieved by deriving an asymptotic expression of the continued fraction. This procedure allows for a considerable reduction of the number of relevant variables, which are generally identified as the classical variables, such as energy and velocity, complemented by their corresponding dissipative fluxes, namely, the heat and the momentum fluxes. Two applications are investigated: ultrasound propagation in dilute gases and heat transport in dielectric crystals at very low temperature, where the phenomena of second sound is observed. It is shown that for ultrasound propagation, the influence of the fast variables can be described by introducing so-called effective relaxation times. This results in better agreement with experiments than earlier theoretical models and casts a light on the foundation of mesoscopic formalisms, such as extended irreversible thermodynamics. Concerning heat conduction in dielectric crystals, it is seen that the present description includes the three different modes of transport observed experimentally, namely, ballistic phonons, second sound waves, and diffusion. Our approach is a generalization of the models proposed by Cattaneo [Atti Sem. Univ. Modena 3, 33 (1948)] and by Guyer and Krumhansl [Phys. Rev. 148, 766 (1966); 148, 778 (1966)].