Diffusion of vibrations in disordered systems

We consider diffusion of vibrations in random lattices with translational invariance. Above some frequency $\omega_{\rm IR}$, corresponding to the Ioffe–Regel crossover (and depending on the strength of disorder), phonons cannot propagate through the lattice and transfer energy. On the other hand, most of the vibrations in this range are not localized. We show that these delocalized excitations are similar to diffusons introduced by Allen, Feldman et al. (see, e.g., Phil. Mag. B 79, 1715 (1999)) to describe heat transport in glasses. In this range the energy in the lattice is transferred by means of diffusion of vibrational excitations. We have calculated the diffusivity of the modes $D(\omega)$ using both the direct numerical solution of Newton equations and the formula of Edwards and Thouless. It is nearly a constant above $\omega_{\rm IR}$ and goes to zero at the localization threshold.