On strong anomalous diffusion

Superdiffusive behavior, i.e., [x(2)(t)] similar to t(2 nu), with nu > 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)] similar to t(q nu(q)) where nu(2) > 1/2 and q nu(q) is not a Linear function of q. This feature is different from the weak superdiffusive regime, i.e., nu(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 q nu(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(-nu)F(x/t(nu)) 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. (C) 1999 Published by Elsevier Science B.V. All rights reserved.