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.

Constitutive equations for heat conduction in nanosystems and nonequilibrium processes: an overview

“…Three such approaches are the phonon hydrodynamics [5][6][7], the thermomass theory [12][13][14], and the dualphase-lag model [15,16]. All these models consider the heat carriers as a fluid, whose hydrodynamic-like equations of motion describe the heat transport.…”

“…(1) to the nonlinear regime may be obtained in the framework of Extended Irreversible Thermodynamics (EIT) [6,7,21,27,28], a theory which upgrades the dissipative fluxes to the rank of independent thermodynamic variables [10,22,[29][30][31], while nonlinear generalizations of Eq. (2) have been obtained either in EIT [5], or in the framework of the thermomass theory [12][13][14], which also pays a special attention to nonlinear terms in the heat transport equation.…”

“…This collection is made by quasiparticles of heat carriers, called thermons, which are representative of the vibrations of the molecules generated by heating the conductor and whose mass may be calculated from the Einstein's mass-energy duality. For crystals, the thermomass gas is just the phonon gas, for pure metals it will be attached on the electron gas, and for semi-metals it will be constituted by both these different gases [13,[49][50][51].…”

“…The Thermomass (TM) theory [13,[49][50][51] provides a further example of nonlinear heat-transport equation with thermal lag. In TM theory the heat transport is due to the motion of a gas-like collection of heat carriers, characterized by an effective mass density and flowing through the medium due to a thermomass-pressure gradient.…”

Non-Fourier heat conduction study for steady states in metallic nanofilms

“…In another words, it is known so far that non-Fourier heat conduction occurs only in non-steady states. Recently, Guo and co-workers [6][7][8] proposed the notion of thermomass based on Einstein's mass-energy relation; thermomass is defined as the equivalent mass of thermal energy. Heat conduction is understood as the flow of thermomass and Newtonian mechanics can be used to analyze its motion:…”