Foundations and applications of a mesoscopic thermodynamic theory of fast phenomena

Abstract

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 frequencywave-number space the corresponding transport coeKcients 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 cruxes, 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 inhuence 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 [

Uniqueness of Landau-Lifshitz energy frame in relativistic dissipative hydrodynamics

“…In fact, we can show that the energy frame is the most natural frame for multi-component systems [27] and that the RG derivation [28] of the mesoscopic dynamics [30] of the RBE naturally leads to the extended thermodynamics in the energy frame. We can thus assert more firmly that the relativistic hydrodynamic equations for viscous fluids must be defined in the energy frame, if it is consistent with the underlying relativistic kinetic equation at all.…”

Derivation of relativistic hydrodynamic equations consistent with relativistic Boltzmann equation by renormalization-group method

“…For shortage of space, we here give a sketch of some of our results for the extended thermodynamics, leaving the detailed derivation in a separate paper [40]. We emphasize that our theory gives an explicit construction of the invariant manifold corresponding to thirteen moments, which has been long sought for [42][43][44][45][46][47].…”

“…In this subsection, we make a brief report on our attempt [40] to extend the RG method so as to obtain the so called mesoscopic dynamics [42,43] in the relativistic case.…”

“…In this article, we also report on our attempt [40] to examine the causality problem (3) by deriving the so called extended thermodynamics [41][42][43][44][45][46][47]: Namely, we derive mesoscopic dynamics of the RBE by constructing the invariant/attractive manifold incorporating some fast modes as well as the zero modes of the linearized collision operator. It turns out that our theory leads to the same expressions for the transport coefficients as given by the Chapman-Enskog method [48] and also novel formulas of the relaxation times in terms of relaxation functions, which allow a natural physical interpretation of the relaxation times.…”

Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus

“…The MC equation, considers fast phenomena in which the relaxation effect takes into account the time derivative of the flux in the constitutive equation. This generalization leads to a hyperbolic transport equation instead of the parabolic one [2]. As an example, the non-local time constitutive equations for the heat flux and phonon hydrodynamics in nanosytems have been presented by Cimmelli, Sellitto and Jou [3].…”