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Spectral Content of a Single Non-Brownian Trajectory

  • Time-dependent processes are often analyzed using the power spectral density (PSD) calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble average. Frequently, the available experimental datasets are too small for such ensemble averages, and hence, it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from S(f, T), the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable parametrized by frequency f and observation time T, for a broad family of anomalous diffusions-fractional Brownian motion with Hurst index H-and derive exactly its probability density function. We show that S(f, T) is proportional-up to a random numerical factor whose universal distribution we determine-to the ensemble-averaged PSD. For subdiffusion (H < 1/2), we find that S(f, T) similar to A/f(2H+1) with random amplitude A. In sharp contrast, for superdiffusion (H > 1/2) S(f, T) similar toTime-dependent processes are often analyzed using the power spectral density (PSD) calculated by taking an appropriate Fourier transform of individual trajectories and finding the associated ensemble average. Frequently, the available experimental datasets are too small for such ensemble averages, and hence, it is of a great conceptual and practical importance to understand to which extent relevant information can be gained from S(f, T), the PSD of a single trajectory. Here we focus on the behavior of this random, realization-dependent variable parametrized by frequency f and observation time T, for a broad family of anomalous diffusions-fractional Brownian motion with Hurst index H-and derive exactly its probability density function. We show that S(f, T) is proportional-up to a random numerical factor whose universal distribution we determine-to the ensemble-averaged PSD. For subdiffusion (H < 1/2), we find that S(f, T) similar to A/f(2H+1) with random amplitude A. In sharp contrast, for superdiffusion (H > 1/2) S(f, T) similar to BT2H-1/f(2) with random amplitude B. Remarkably, for H > 1/2 the PSD exhibits the same frequency dependence as Brownian motion, a deceptive property that may lead to false conclusions when interpreting experimental data. Notably, for H > 1/2 the PSD is ageing and is dependent on T. Our predictions for both sub-and superdiffusion are confirmed by experiments in live cells and in agarose hydrogels and by extensive simulations.show moreshow less

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Author details:Diego KrapfORCiD, Nils Lukat, Enzo MarinariORCiDGND, Ralf MetzlerORCiDGND, Gleb OshaninORCiDGND, Christine Selhuber-UnkelORCiDGND, Alessio Squarcini, Lorenz StadlerGND, Matthias Weiss, Xinran Xu
DOI:https://doi.org/10.1103/PhysRevX.9.011019
ISSN:2160-3308
Title of parent work (English):Physical review : X, Expanding access
Publisher:American Physical Society
Place of publishing:College Park
Publication type:Article
Language:English
Date of first publication:2019/01/31
Publication year:2019
Release date:2021/04/19
Tag:Biological Physics; Interdisciplinary Physics; Statistical Physics
Volume:9
Issue:1
Number of pages:13
Funding institution:National Science FoundationNational Science Foundation (NSF) [1401432]; European Research Council under the European Unions Horizon 2020 Research and Innovation Program [694925]; German Research Foundation (DFG)German Research Foundation (DFG) [ME-1535/7-1]; Humboldt Polish Honorary Research Fellowship from the Foundation for Polish Science; DFG through the Collaborative Research Centre [CRC 1261]; VolkswagenStiftungVolkswagen [Az. 92738]
Organizational units:Mathematisch-Naturwissenschaftliche Fakultät / Institut für Physik und Astronomie
DDC classification:5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik
Peer review:Referiert
Publishing method:Open Access / Gold Open-Access
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License (German):License LogoCC-BY - Namensnennung 4.0 International
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