On strong anomalous diffusion

Superdiffusive behavior, i.e., 〈x2(t)〉∼t2ν, with ν>1/2, is in general 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 superdiffusive regime, i.e., ν(q)=const>1/2, occurring in random shear flows. 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 is due 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 P(x,t)=t−νF(x/tν) of the probability density. This implies that the effective equation at large scale and long time for P(x,t), obeys neither the usual Fick equation nor other linear equations involving temporal and/or spatial fractional derivatives.