Numerical evidence for the bottleneck frequency of quasidiffusive acoustic phonons

A kinetic equation on quasidiffusion of phonons was recently analyzed by Esipov and he predicted the existence of a bottleneck frequency $\left({\nu }_{\mathrm{BN}}\right)$ which separates the phonons decaying from those diffusing to a detector. We have solved numerically the kinetic equation and obtained the temporal evolution of phonon concentration excited at the center of a spherical sample. We have also performed Monte Carlo simulations of phonon propagation in the same geometry. At a time much later than the ballistic arrival time of phonons, both sets of results exhibit a sharp peak in the phonon concentration around the predicted ${\nu }_{\mathrm{BN}}.$ With Monte Carlo simulations we have also confirmed the same relaxation rate for the phonons of frequencies $\nu <{\nu }_{\mathrm{BN}}.$