A trivial non-chaotic map lattice asymptotically indistiguishable from a L\'evy walk

Abstract

In search for mathematically tractable models of anomalous diffusion, we introduce a simple dynamical system consisting of a chain of coupled maps of the interval whose Lyapunov exponents vanish everywhere. The volume preserving property and the vanishing Lyapunov exponents are intended to mimic the dynamics of polygonal billiards, which are known to give rise to anomalous diffusion, but which are too complicated to be analyzed as thoroughly as desired. Depending on the value taken by a single parameter \alpha, our map experiences sub-diffusion, super-diffusion or normal diffusion. Therefore its transport properties can be compared with those of given L\'evy walks describing transport in quenched disordered media. Fixing \alpha\ so that the mean square displacement generated by our map and that generated by the corresponding L\'evy walk asymptotically coincide, we prove that all moments of the corresponding asymptotic distributions coincide as well, hence all observables which are expressed in terms of the moments coincide.