State equation of phonon gas and conservation equations for phonon gas motion

Based on the mass-energy relation of Einstein's relativity theory, the mass of solid lattices is divided into two parts: the rest mass of solid atoms and the equivalent mass of thermal vibration energy of lattices. The latter is exactly the equivalent mass of the phonon gas in a solid. The vibration energy of the solid lattices includes the thermal energies consisting of the rest mass of the solid lattices and the equivalent mass of the phonon gas. The state equations for the lattice rest mass and the phonon gas are deduced based on the state equation of solids. The heat conduction is just the motion of the phonon gas in a solid. The conservation equations for the phonon gas motion are established. It is found that the conservation equation of phonon gas momentum degenerates to the Fourier's conduction law when the inertial force of the phonon gas can be ignored. The physical nature of the Fourier's law is the balance between the driving force and the resistance for the motion of the phonon gas. Under ultrahigh heat flux conditions where the inertial force is too high to be ignored, the Fourier's law is no longer valid even under the steady condition.