Single-trajectory spectral analysis of scaled Brownian motion
A standard 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\to \infty $ . In many experimental...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2019-01-01
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| Series: | New Journal of Physics |
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| Online Access: | https://doi.org/10.1088/1367-2630/ab2f52 |
| _version_ | 1797750097775165440 |
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| author | Vittoria Sposini Ralf Metzler Gleb Oshanin |
| author_facet | Vittoria Sposini Ralf Metzler Gleb Oshanin |
| author_sort | Vittoria Sposini |
| collection | DOAJ |
| description | A standard 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\to \infty $ . 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\to \infty $ 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. We demonstrate 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. We also 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 single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement. |
| first_indexed | 2024-03-12T16:27:44Z |
| format | Article |
| id | doaj.art-a81db0102b654854bd0e1f9180f2bd44 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:27:44Z |
| publishDate | 2019-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-a81db0102b654854bd0e1f9180f2bd442023-08-08T15:38:14ZengIOP PublishingNew Journal of Physics1367-26302019-01-0121707304310.1088/1367-2630/ab2f52Single-trajectory spectral analysis of scaled Brownian motionVittoria Sposini0Ralf Metzler1https://orcid.org/0000-0002-6013-7020Gleb Oshanin2https://orcid.org/0000-0001-8467-3226Institute for Physics & Astronomy, University of Potsdam , D-14476 Potsdam-Golm, Germany; Basque Centre for Applied Mathematics , E-48009 Bilbao, SpainInstitute for Physics & Astronomy, University of Potsdam , D-14476 Potsdam-Golm, GermanySorbonne Université , CNRS, Laboratoire de Physique Théorique de la Matière Condensée (UMR 7600), 4 Place Jussieu, F-75252 Paris Cedex 05, France; Interdisciplinary Scientific Center Poncelet (ISCP) , 119002, Moscow, RussiaA standard 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\to \infty $ . 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\to \infty $ 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. We demonstrate 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. We also 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 single-trajectory PSDs as an alternative (or complement) to analyses based on the time-averaged mean squared displacement.https://doi.org/10.1088/1367-2630/ab2f52diffusionanomalous diffusionpower spectral analysissingle trajectory analysis |
| spellingShingle | Vittoria Sposini Ralf Metzler Gleb Oshanin Single-trajectory spectral analysis of scaled Brownian motion New Journal of Physics diffusion anomalous diffusion power spectral analysis single trajectory analysis |
| title | Single-trajectory spectral analysis of scaled Brownian motion |
| title_full | Single-trajectory spectral analysis of scaled Brownian motion |
| title_fullStr | Single-trajectory spectral analysis of scaled Brownian motion |
| title_full_unstemmed | Single-trajectory spectral analysis of scaled Brownian motion |
| title_short | Single-trajectory spectral analysis of scaled Brownian motion |
| title_sort | single trajectory spectral analysis of scaled brownian motion |
| topic | diffusion anomalous diffusion power spectral analysis single trajectory analysis |
| url | https://doi.org/10.1088/1367-2630/ab2f52 |
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