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

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.