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Single-trajectory spectral analysis of scaled Brownian motion

  • Astandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon canAstandard approach to study time-dependent stochastic processes is the power spectral density (PSD), an ensemble-averaged property defined as the Fourier transform of the autocorrelation function of the process in the asymptotic limit of long observation times, T → ∞. In many experimental situations one is able to garner only relatively few stochastic time series of finite T, such that practically neither an ensemble average nor the asymptotic limit T → ∞ can be achieved. To accommodate for a meaningful analysis of such finite-length data we here develop the framework of single-trajectory spectral analysis for one of the standard models of anomalous diffusion, scaled Brownian motion.Wedemonstrate that the frequency dependence of the single-trajectory PSD is exactly the same as for standard Brownian motion, which may lead one to the erroneous conclusion that the observed motion is normal-diffusive. However, a distinctive feature is shown to be provided by the explicit dependence on the measurement time T, and this ageing phenomenon can be used to deduce the anomalous diffusion exponent.Wealso compare our results to the single-trajectory PSD behaviour of another standard anomalous diffusion process, fractional Brownian motion, and work out the commonalities and differences. Our results represent an important step in establishing singletrajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.show moreshow less

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Metadaten
Author details:Vittoria SposiniORCiD, Ralf MetzlerORCiDGND, Gleb OshaninORCiDGND
DOI:https://doi.org/10.1088/1367-2630/ab2f52
ISSN:1367-2630
Title of parent work (English):New Journal of Physics
Publisher:Deutsche Physikalische Gesellschaft ; IOP, Institute of Physics
Place of publishing:Bad Honnef und London
Publication type:Article
Language:English
Date of first publication:2019/07/23
Publication year:2019
Release date:2019/10/15
Tag:anomalous diffusion; diffusion; power spectral analysis; single trajectory analysis
Volume:21
Number of pages:16
Funding institution:Universität Potsdam
Funding number:PA 2019_65
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
DDC classification:5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
Peer review:Referiert
Grantor:Publikationsfonds der Universität Potsdam
Publishing method:Open Access
License (German):License LogoCreative Commons - Namensnennung, 3.0 Deutschland
External remark:Zweitveröffentlichung in der Schriftenreihe Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe ; 753
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