Generalized heat conduction laws based on thermomass theory and phonon hydrodynamics

Abstract

The Fourier’s law of heat conduction is invalid in extreme conditions, such as the second sound in solids and anomalous heat conduction in nanosystems. The generalized heat conduction law with nonlinear and nonlocal effects is derived from both macroscopic thermomass theory and microscopic phonon Boltzmann method in this paper. The coincidence between thermomass theory and phonon hydrodynamics is also analyzed through their microscopic basis. The convective term in the momentum equation of the thermomass theory comes from the nonlinear terms of the distribution function, which is often neglected in previous phonon hydrodynamics derivations. The Chapman-Enskog expansion leads to the Laplacian term, which is similar to the derivation of Navier-Stokes equation in hydrodynamics and inspires the introduction of a Brinkman extension in the thermomass equation. This comparison reveals how the nonlinear effects could be described by generalized heat conduction laws.