The superdiffusion behavior, i.e. < x 2 (t) >∼ t 2ν , with ν > 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 >∼ t qν(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 the 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 −ν F (x/t ν ). This implies that the effective equation at large scale and long time for P (x, t), cannot obey neither the usual Fick equation nor other linear equations involving temporal and/or spatial fractional derivatives.PACS number(s): 05.45.+b, 05.60.+w;

While statistical mechanics describe the equilibrium state of systems with many degrees of freedom, and dynamical systems explain the irregular evolution of systems with few degrees of freedom, new tools are needed to study the evolution of systems with many degrees of freedom. This book presents the basic aspects of chaotic systems, with emphasis on systems composed by huge numbers of particles. Firstly, the basic concepts of chaotic dynamics are introduced, moving on to explore the role of ergodicity and chaos for the validity of statistical laws, and ending with problems characterized by the presence of more than one significant scale. Also discussed is the relevance of many degrees of freedom, coarse graining procedure, and instability mechanisms in justifying a statistical description of macroscopic bodies. Introducing the tools to characterize the non asymptotic behaviors of chaotic systems, this text will interest researchers and graduate students in statistical mechanics and chaos.

We report the first detailed experimental observation of the Batchelor regime [G. K. Batchelor, J. Fluid. Mech. 5, 113 (1959)], in which a passive scalar is dispersed by a large scale strain, at high Peclet numbers. The observation is performed in a controlled two-dimensional flow, forced at large scale, in conditions where a direct enstrophy cascade develops [J. Paret, M.-C. Jullien, and P. Tabeling, Phys. Rev. Lett. 83, 3418 (1999)]. The expected k(-1) spectrum is observed, along with exponential tails for the distributions of the concentration and concentration increments and logarithmlike behavior for the structure functions. These observations, confirmed by using simulated particles, provide a support to the theory.

We show that {\it strong} anomalous diffusion, i.e. $\mean{|x(t)|^q} \sim t^{q \nu(q)}$ where $q \nu(q)$ is a nonlinear function of $q$, is a generic phenomenon within a class of generalized continuous-time random walks. For such class of systems it is possible to compute analytically nu(2n) where n is an integer number. The presence of strong anomalous diffusion implies that the data collapse of the probability density function P(x,t)=t^{-nu}F(x/t^nu) cannot hold, a part (sometimes) in the limit of very small x/t^\nu, now nu=lim_{q to 0} nu(q). Moreover the comparison with previous numerical results shows that the shape of F(x/t^nu) is not universal, i.e., one can have systems with the same nu but different F.Comment: Final versio

As a turbulent flow advects a swarm of Lagrangian markers, the mutual separation between particles grows, and the shape of the swarm gets distorted. By following three points in an experimental turbulent two-dimensional flow with a k(-5/3) spectrum, we investigate the geometry of triangles, in a statistical sense. Two well-characterized shape distributions are identified. At long times when the average size of the triangles is larger than the integral scale, the distribution of shapes is Gaussian. When the size of the triangle is in the inertial range and grows as t(3/2) (Richardson's law), a plausibly self-similar, non-Gaussian probability distribution is observed, where very elongated triangles have a much larger probability than in the Gaussian regime. These results are discussed, and, in the latter case, compared with the predictions of a stochastic model recently introduced [A. Pumir et al., Phys. Rev. Lett. 85, 5324 (2000)].