Generalized Boltzmann transport theory for relaxational heat conduction

Abstract

Relaxational heat conduction lacks a modified transport theory at the mesoscopic level. We establish a modified Boltzmann transport theory with the generalized collision term, which can give rise to the convolution relationship between the heat flux and temperature gradient as well as fractional Fourier law. The macroscopic relaxational behaviors are thereafter connected to mesoscopic memory effects in the generalized collision term. The modified Boltzmann transport theory not only provides an underlying explanation for macroscopic relaxational heat conduction but also possesses engineering applications to situations far from equilibrium. The generalized collision term is not unique framework for relaxational heat conduction, and generalizing the drift term in the Boltzmann transport equation (BTE) can also cover macroscopic models. However, this framework will be paired with anomalies in energy continuity and entropy balance, and as a consequence, it is not suggested for relaxational heat conduction.