Simple stochastic models showing strong anomalous diffusion

K. H. Andersen; P. Castiglione; A. Mazzino; A. Vulpiani

doi:10.1007/s100510070032

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Condensed Matter and Complex Systems

Eur. Phys. J. B 18, 447-452 (2000)
https://doi.org/10.1007/s100510070032

Simple stochastic models showing strong anomalous diffusion

K. H. Andersen¹^a, P. Castiglione¹^,2, A. Mazzino³ and A. Vulpiani¹

¹ Dipartimento di Fisica, and INFM, Università "La Sapienza", P.le A. Moro 2, 00185 Roma, Italy
² Laboratoire de Physique Statistique, École Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex 05, France
³ INFM-Dipartimento di Fisica, Università di Genova, 16146 Genova, Italy

Corresponding author: ^a ken@isva.dtu.dk

Received: 14 April 2000
Published online: 15 December 2000

Abstract

We show that strong anomalous diffusion, i.e. $\left< |x(t)|^q \right> \sim t^{q \nu(q)}$ 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 $\nu(2 n)$ 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 ν but different F.