On Entropic Framework Based on Standard and Fractional Phonon Boltzmann Transport Equations.

Shu-Nan Li,Bing-Yang Cao

doi:10.3390/e21020204

Abstract

Generalized expressions of the entropy and related concepts in non-Fourier heat conduction have attracted increasing attention in recent years. Based on standard and fractional phonon Boltzmann transport equations (BTEs), we study entropic functionals including entropy density, entropy flux and entropy production rate. Using the relaxation time approximation and power series expansion, macroscopic approximations are derived for these entropic concepts. For the standard BTE, our results can recover the entropic frameworks of classical irreversible thermodynamics (CIT) and extended irreversible thermodynamics (EIT) as if there exists a well-defined effective thermal conductivity. For the fractional BTEs corresponding to the generalized Cattaneo equation (GCE) class, the entropy flux and entropy production rate will deviate from the forms in CIT and EIT. In these cases, the entropy flux and entropy production rate will contain fractional-order operators, which reflect memory effects.

Highlights

Entropic and thermodynamic frameworks in heat transport have attracted increasing attention in recent years [1,2,3]
The Clausius statement restricts the direction of heat transfer, which guarantees the non-negativity of entropy generation
These entropic functionals were well-discussed for macroscopic heat conduction models and proved by Grad’s method in the kinetic theory of gases, the entropic framework was not studied as much for phonon heat transport based on the phonon Boltzmann transport equation (BTE)

Summary

Introduction

Entropic and thermodynamic frameworks in heat transport have attracted increasing attention in recent years [1,2,3]. The Clausius statement restricts the direction of heat transfer, which guarantees the non-negativity of entropy generation. Further descriptions such as the entropy production rate require irreversible and non-equilibrium thermodynamics. The nonlocal entropy flux J commonly takes a form as J = JC + K [1], wherein K is the so-called entropy-density extra flux These entropic functionals were well-discussed for macroscopic heat conduction models and proved by Grad’s method in the kinetic theory of gases, the entropic framework was not studied as much for phonon heat transport based on the phonon Boltzmann transport equation (BTE). Besides Equation (2), several fractional BTEs are discussed, which educes a class of fractional-order heat conduction models termed generalized Cattaneo equation (GCE) [23]

Standard Boltzmann Transport Equation

Fractional Models

Conclusions