Rayleigh scattering, long-time tails, and the harmonic spectrum of topologically disordered systems

We show rigorously that a topologically disordered system interacting harmonically via force constants, which have a sufficiently short-ranged site-distance dependence, exhibits Rayleigh scattering in the low-frequency limit, i.e., a sound attenuation constant, which is proportional to omega(d+1), where omega is the frequency and d the dimensionality. This had been questioned in the literature. The corresponding nonanalyticity in the spectrum is related to a long-time tail in the velocity autocorrelation function of the analogous diffusion problem, which varies with time t as t(-(d+2)/2). A self-consistent theory for the spectrum is formulated, which has the correct analytical properties.