Breakdown of the Fluctuation-Dissipation Theorem for fast superdiffusion

We study anomalous diffusion for one-dimensional systems described by a generalized Langevin equation. We show that superdiffusion can be classified in slow superdiffusion and fast superdiffusion. For fast superdiffusion we prove that the Fluctuation-Dissipation Theorem does not hold. We show as well that the asymptotic behavior of the response function is a stretched exponential for anomalous diffusion and an exponential only for normal diffusion.