Diffusion without Spreading of a Wave Packet in Nonlinear Random Models

Abstract

We discuss the long time behaviour of a finite energy wave packet in nonlinear Hamiltonians on infinite lattices at arbitrary dimension, exhibiting linear Anderson localization. Strong arguments both mathematical and numerical, suggest for infinite models that small amplitude wave packets may generate stationary quasiperiodic solutions (KAM tori) almost undistinguishable from linear wave packets. The probability of this event is non vanishing at small enough amplitude and goes to unity at amplitude zero. Most other wave packets (non KAM tori) are chaotic. We discuss the Arnold diffusion conjecture (recently partially proven) and propose a modified Boltzmann statistics for wave packets valid in generic models. The consequence is that the probability that a chaotic wave packet spreads to zero amplitude is zero. It must always remain focused around one or few chaotic spots which moves randomly over the whole system and generates subdiffusion.We study a class of Ding Dong models also generating subdiffusion where the nonlinearities are replaced by hard core potentials. Then we prove rigorously that spreading is impossible for any initial wave packet.