Equation of motion of a phonon gas and non-Fourier heat conduction

Abstract

Heat conduction in solids is due to the motion of the phonon gas. A more general description of the heat transport in solids includes consideration of the mass, pressure, and inertial force of the phonon gas. The mass of the phonon gas refers to the equivalent mass of its energy based on Einstein’s mass-energy relation. The thermal vibration of the lattice creates the phonon gas pressure and the momentum change of the phonon gas results in an inertial force. The phonon gas velocity is directly proportional to the heat flux. These concepts are used to establish an equation of motion for the phonon gas including the driving, inertial, and resistant forces using Newtonian dynamics. This equation reduces to Fourier’s law of heat conduction when the inertial force can be neglected relative to the other terms so that heat conduction becomes pure diffusion. However, Fourier’s law of heat conduction no longer holds if the heat flux is very high, such that the inertial force of the phonon gas is not negligible. In such cases, the heat conduction behavior deviates from Fourier’s law even for steady-state conditions so that the heat conduction is characterized by a nonlinear relationship between the heat flux and the temperature gradient.