Hyperbolic heat conduction, effective temperature and third law in the presence of heat flux

Abstract

Some analogies between different nonequilibrium heat conduction models, particularly, random walk, discrete variable model, and Boltzmann transport equation with the single relaxation time approximation, have been discussed. We show that under an assumption of a finite value of the heat carriers velocity, these models lead to the hyperbolic heat conduction equation and the modified Fourier law with the relaxation term. Corresponding effective temperature and entropy have been introduced and analyzed. It has been demonstrated that the effective temperature, defined as a geometric mean of the kinetic temperatures of the heat carriers moving in opposite directions, is governed by a non-linear relation and acts as a criterion for thermalization. It is shown that when the heat flux tends to its maximum possible value, the effective temperature, heat capacity and local entropy go to zero even at a nonzero equilibrium temperature. This provides a possible generalization of the third law to nonequilibrium situations. Analogies between the effective temperature and some other definitions of temperature in nonequilibrium state, particularly, for active systems, disordered semiconductors under electric field, and adiabatic gas flow, have been shown and discussed. Illustrative examples of the behavior of the effective temperature and entropy during nonequilibrium heat conduction in a monatomic gas, a nano film, and a strong shockwave have been analyzed.