Numerical study of strong anomalous diffusion

The superdiffusion behavior, i.e., 〈x2(t)〉∼t2ν, with ν>1/2, in general is not completely characterized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e., 〈|x(t)|q〉∼tqν(q) where ν(2)>1/2 and qν(q) is not a linear function of q. This feature is different from the weak superdiffusion regime, i.e., ν(q)=const>1/2, as in random shear flows. The strong anomalous diffusion can be generated by nontrivial chaotic dynamics, e.g. Lagrangian motion in 2d time-dependent incompressible velocity fields, 2d symplectic maps and 1d intermittent maps. Typically the function qν(q) is piecewise linear. This corresponds to two mechanisms: a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions. In order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space. In the presence of strong anomalous diffusion one does not have a unique exponent and therefore one has the failure of the usual scaling of the probability distribution, i.e., P(x,t)=t−νF(x/tν). This implies that the effective equation at large scale and long time for P(x,t), can obey neither the usual Fick equation nor other linear equations involving temporal and/or spatial fractional derivatives.