On moments and scaling regimes in anomalous random walks

More and more stochastic transport phenomena in various real-world systems prove to belong to the class of anomalous diffusion. This paper is devoted to the scaling of diffusion—a very fundamental feature of this transport process. Our aim is to provide a comprehensive theoretical overview of scaling properties, but also to connect it to the analysis of experimental data.

Anomalous diffusion is commonly characterized by an exponent in the power law of the mean square displacement as a function of time . On the other hand, it is known that the probability distribution function of diffusing particles can be approximated by (1/t^α)Φ(r/t^α). While for classical normal diffusion this scaling relation is exact, it may not be valid globally for anomalous diffusion. In general, the exponent α obtained from the scaling of the central part of the probability distribution function differs from the exponent ν given by the mean square displacement. In this paper we systematically study how the scaling of different moments and parts of the probability distribution function can be determined and characterized even when no global scaling exists. We consider three rigorous methods for finding, respectively, the mean square displacement exponent ν, the scaling exponent α and the profile of the scaling function Φ. We also show that alternatively the scaling exponent α 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, we introduce a new unifying description which allows straightforward generalizations to other systems. Finally, we show how fractional moments help to analyze experimental or simulation data consistently.