The theory of phonon relaxation in He(II) at low temperatures

Abstract

We have investigated analytically relaxation processes in the phonon system in helium(II) at sufficiently low temperatures where rotons are not exited. In accordance with the recent experimental data, the phonon velocity dispersion is supposed to be positive, though small. Two different relaxation processes exist in the phonon system in this case: (i) the fast longitudinal relaxation establishing equilibrium phonon distribution along each direction in the momentum space with the temperature and the drift velocity depending on the direction; (ii) the slow transverse relaxation setting up equilibrium between different directions. Using the energy and momentum conservation and general principles of the irreversible thermodynamics we have derived the expression for the transverse relaxation operator. It appears to be a differential operator of the fourth order and depends on a function of the “phonon temperature” Θ that cannot be determined from the general consideration. We have calculated this function for the case of three-phonon collisions. Physical properties of the transverse relaxation operator are discussed and the corresponding boundary conditions are formulated. Several typical physical problems, both linear and nonlinear, which can be formulated in terms of the transverse relaxation operator, are enumerated. With the help of the diagram method the contribution of multiphonon collisions both in the longitudional and in the transverse relaxation is evaluated.