On moments and scaling regimes in anomalous random walks
Schmiedeberg M, Zaburdaev V, Stark H (2009)
Publication Status: Published
Publication Type: Journal article
Publication year: 2009
Journal
Publisher: IOP PUBLISHING LTD
DOI: 10.1088/1742-5468/2009/12/P12020
Abstract
Anomalous diffusion is commonly characterized by an exponent in the power law of the mean square displacement as a function of time < r(2)(t)> proportional to t(2 nu). On the other hand, it is known that the probability distribution function of diffusing particles can be approximated by (1/t(alpha))Phi(r/t(alpha)). While for classical normal diffusion this scaling relation is exact, it may not be valid globally for anomalous diffusion. In general, the exponent a obtained from the scaling of the central part of the probability distribution function differs from the exponent nu 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 nu, the scaling exponent alpha and the profile of the scaling function Phi. We also show that alternatively the scaling exponent alpha can be determined by analyzing fractional moments with q << 1. All analytical results are obtained in the framework of continuous-time random walks. For a wide class of coupled random walks, including the famous Levy 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.
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APA:
Schmiedeberg, M., Zaburdaev, V., & Stark, H. (2009). On moments and scaling regimes in anomalous random walks. Journal of Statistical Mechanics-Theory and Experiment. https://doi.org/10.1088/1742-5468/2009/12/P12020
MLA:
Schmiedeberg, Michael, Vasily Zaburdaev, and Holger Stark. "On moments and scaling regimes in anomalous random walks." Journal of Statistical Mechanics-Theory and Experiment (2009).
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