Vlasov equation for phonons and its macroscopic consequences

Amendments are obtained to the harmonic approximation for the first order of the hyperelasticity theory in the relaxation approximation for a cubic crystal. The Vlasov equation is constructed for a collisionless phonon gas in a self-consistent deformation field. Collisions are considered in the approximation of relaxation to the equilibrium distribution. It is shown that the equations of thermo-elasticity are valid in the thermodynamic limit for the hydrodynamics of a phonon gas. The connection of the kinetic model with the equations of Cattaneo, Guyer-Crumhansl and thermoelasticity in the Bio form is considered.