Hydrodynamic approximation to the phonon Boltzmann equation

The eigenvectors of the normal-process collision matrix are used to derive equations for energy and crystal-momentum balance from the linear phonon Boltzmann equation. The balance equations obtained are valid when ωτN1 and τNτR, where ω characterizes the rate of change of the energy and crystal-momentum densities, and τN and τR are the relaxation times for normal-process and resistive-process scattering, respectively. Different branches of the phonon frequency spectrum are allowed to possess different group velocities. No restrictions to an isotropic and dispersionless spectrum are required. Terms through order ωτN and τNτR are kept. The results obtained are compared with those of Sussmann and Thellung, Guyer and Krumhansl, and Kwok.