Nonequilibrium thermodynamics of phonon hydrodynamic model for nanoscale heat transport

A phonon hydrodynamic equation has been recently derived from the kinetic theory of phonons for nanoscale heat transport at ordinary temperatures. The classical irreversible thermodynamics is no longer valid due to the failure of the local equilibrium hypothesis from temporal and spatial strong nonequilibrium effects. In the present paper, we investigate the thermodynamic consistency of the phonon hydrodynamic equation for heat transport based on the phonon kinetic theory. The macroscopic expressions of entropy density and entropy flux in the bulk region as well as interfacial entropy generation in the boundary region are derived from their mesoscopic definitions in terms of the nonequilibrium phonon distribution solution. The phonon hydrodynamic equation is demonstrated to be consistent with the second law in the frame of extended irreversible thermodynamics. This paper provides a solid mesoscopic theoretical foundation to the previous nonequilibrium thermodynamics of the phonon hydrodynamic model for heat transport in nanosystems on a macroscopic basis.