On moments and scaling regimes in anomalous random walks

Abstract

More and more stochastic transport phenomena in various real-world systems prove tobelong to the class of anomalous diffusion. This paper is devoted to the scaling ofdiffusion—a very fundamental feature of this transport process. Our aim is to provide acomprehensive theoretical overview of scaling properties, but also to connect it to theanalysis of experimental data.Anomalous diffusion is commonly characterized by an exponent in the power law of themean square displacement as a function of time . On the other hand, it is known that the probabilitydistribution function of diffusing particles can be approximated by(1/tα)Φ(r/tα). While for classical normal diffusion this scaling relation is exact, it maynot be valid globally for anomalous diffusion. In general, the exponentα obtainedfrom the scaling of the central part of the probability distribution function differs from the exponentν given by the mean square displacement. In this paper we systematically study how thescaling of different moments and parts of the probability distribution function can bedetermined and characterized even when no global scaling exists. We consider threerigorous methods for finding, respectively, the mean square displacement exponentν, the scaling exponentα and the profile of thescaling function Φ. We also show that alternatively the scaling exponentα can be determined by analyzing fractional moments⟨|r|q⟩ with . All analytical results are obtained in the framework of continuous-time random walks.For a wide class of coupled random walks, including the famous Lévy walk model, weintroduce a new unifying description which allows straightforward generalizations to othersystems. Finally, we show how fractional moments help to analyze experimental orsimulation data consistently.