Equilibrium Approach to Second Sound in Solids

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Abstract
In the Boltzmann-equation derivation, second sound is seen as the self-sustained propagation of a temperature wave, assuming the establishment of local equilibrium. Kwok and Martin have shown, by a study of the linear response to a fluctuating displacement field, that second sound occurs as an elementary excitation. In this paper, we examine the phonon self-energy for a weakly anharmonic solid in thermal equilibrium. For thermal phonons, the usual lowest-order perturbation term yields the frequency shift and width, which are small compared with the phonon frequency. For a phonon of frequency comparable with the frequency width of the thermal phonons, a sum of ladder diagrams for the self-energy is needed. The sum is shown to be governed by a Boltzmann-type equation. In an isotropic, dispersionless model and under the conditions of strong normal three-phonon scatterings and weak umklapp scatterings, the second-sound mode occurs in the same way as the phonon mode. This is shown by a completely equilibrium method without any concepts of nonequilibrium statistical mechanics.