Simple stochastic models showing strong anomalous diffusion

Abstract

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

Cages and Anomalous Diffusion in Vibrated Dense Granular Media

“…A possible mechanism for superdiffusion is the continuous time random walk model (ctrw) [30,32]. In the ctrw the variable (here the velocity) performs a random walk taking discrete values ω i : it remains constant for a random time t and then jumps to a new value.…”

“…Such an approach is capable, in principle, of describing both caging and superdiffusion within a single model equation, at the price of losing immediate interpretation and plain calculations. In a complementary approach [32] a coarse-grained value of ω s (t) (where only the sign of this quantity is traced) follows a continuous time random walk (ctrw): It takes discrete values with random transition times extracted from a given distribution. A simplified version of the ctrw is discussed in the SM [21].…”

CTRW pathways to the fractional diffusion equation

“…By definition [35] strong anomalous diffusion behavior exhibits M 2j ðtÞ / t f ðjÞ where f ðjÞ is a non-linear function of j (see related work [47,[56][57][58][59]). Castiglione et al [35] point out that strong anomalous diffusion implies the failure of the standard scaling assumption equation (5), since this equation predicts M 2j ðtÞ / t aj a behavior called weak anomalous diffusion.…”

Infinite densities for Lévy walks

“…Surprisingly, in many cases the continuous spectrum qν(q) exhibits a bi-linear scaling (see details below). Examples for this piecewise linear behavior of qν(q) include nonlinear dynamical systems [2,[6][7][8][9], stochastic models with quenched and annealed disorder, in particular, the Lévy walk [16][17][18][19][20][21] and sand pile models [22]. Recent experiments on the active transport of polymers in the cell [15], theoretical investigation of the momentum [23] and the spatial [24] spreading of cold atoms in optical lattices and flows in porous media [25] further confirmed the generality of strong anomalous diffusion of the bi-linear type.…”