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Quantifying non-ergodicity of anomalous diffusion with higher order moments

  • Anomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. WeAnomalous diffusion is being discovered in a fast growing number of systems. The exact nature of this anomalous diffusion provides important information on the physical laws governing the studied system. One of the central properties analysed for finite particle motion time series is the intrinsic variability of the apparent diffusivity, typically quantified by the ergodicity breaking parameter EB. Here we demonstrate that frequently EB is insufficient to provide a meaningful measure for the observed variability of the data. Instead, important additional information is provided by the higher order moments entering by the skewness and kurtosis. We analyse these quantities for three popular anomalous diffusion models. In particular, we find that even for the Gaussian fractional Brownian motion a significant skewness in the results of physical measurements occurs and needs to be taken into account. Interestingly, the kurtosis and skewness may also provide sensitive estimates of the anomalous diffusion exponent underlying the data. We also derive a new result for the EB parameter of fractional Brownian motion valid for the whole range of the anomalous diffusion parameter. Our results are important for the analysis of anomalous diffusion but also provide new insights into the theory of anomalous stochastic processes.show moreshow less

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Metadaten
Author details:Maria Schwarzl, Aljaž Godec, Ralf MetzlerORCiDGND
DOI:https://doi.org/10.1038/s41598-017-03712-x
Pubmed ID:https://pubmed.ncbi.nlm.nih.gov/28634366
Title of parent work (English):Scientific reports
Publisher:Macmillan Publishers Limited
Place of publishing:London
Publication type:Article
Language:English
Date of first publication:2017/06/20
Publication year:2017
Publishing institution:Universität Potsdam
Release date:2017/10/19
Volume:7
Number of pages:18
Funding institution:Universität Potsdam, Publikationsfonds
Funding number:PA 2017_26
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
DDC classification:0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 000 Informatik, Informationswissenschaft, allgemeine Werke
5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
Peer review:Referiert
Grantor:Publikationsfonds der Universität Potsdam
Publishing method:Open Access
License (German):License LogoCC-BY - Namensnennung 4.0 International
External remark:Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 382
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