On the strong anomalous diffusion

Abstract

The superdiffusion behavior, i.e. $<x^2(t)>\sim t^{2\nu }$, with $\nu >1/2$, in general is not completely characherized by a unique exponent. We study some systems exhibiting strong anomalous diffusion, i.e. $<|x(t){|}^q>\sim t^{q\nu (q)}$ where $\nu (2)>1/2$ and $q\nu (q)$ is not a linear function of $q$. This feature is different from the weak superdiffusion regime, i.e. $\nu (q)=const>1/2$, 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 $q\nu (q)$ 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.

27 pages, 14 figures