Relaxation Spectrum of Phonons: A Solvable Model

Abstract

The approach towards equilibrium of a slightly disturbed phonon gas is analyzed in terms of its normal decay modes. It is shown that a simplified model permits a complete solution of the linearized phonon Boltzmann equation. The model considers 3-phonon normal collisions as the relaxing process and assumes a nondispersive isotropic elastic continuum. In analogy with the Maxwellian molecule model, the coupling term is assumed independent of the frequency. The spectrum of relaxation rates found has no discrete modes except for one trivial solution of rate zero. Above this value there is a gap and then a continuum extending to infinity.