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## Power spectral density of a single Brownian trajectory

- The power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, whileThe power spectral density (PSD) of any time-dependent stochastic processX (t) is ameaningful feature of its spectral content. In its text-book definition, the PSD is the Fourier transform of the covariance function of X-t over an infinitely large observation timeT, that is, it is defined as an ensemble-averaged property taken in the limitT -> infinity. Alegitimate question is what information on the PSD can be reliably obtained from single-trajectory experiments, if one goes beyond the standard definition and analyzes the PSD of a single trajectory recorded for a finite observation timeT. In quest for this answer, for a d-dimensional Brownian motion (BM) we calculate the probability density function of a single-trajectory PSD for arbitrary frequency f, finite observation time T and arbitrary number k of projections of the trajectory on different axes. We show analytically that the scaling exponent for the frequency-dependence of the PSD specific to an ensemble of BM trajectories can be already obtained from a single trajectory, while the numerical amplitude in the relation between the ensemble-averaged and single-trajectory PSDs is afluctuating property which varies from realization to realization. The distribution of this amplitude is calculated exactly and is discussed in detail. Our results are confirmed by numerical simulations and single-particle tracking experiments, with remarkably good agreement. In addition we consider a truncated Wiener representation of BM, and the case of a discrete-time lattice random walk. We highlight some differences in the behavior of a single-trajectory PSD for BM and for the two latter situations. The framework developed herein will allow for meaningful physical analysis of experimental stochastic trajectories.…

Author details: | Diego KrapfORCiD, Enzo MarinariORCiD, Ralf MetzlerORCiDGND, Gleb OshaninORCiDGND, Xinran Xu, Alessio Squarcini |
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URN: | urn:nbn:de:kobv:517-opus4-424296 |

DOI: | https://doi.org/10.25932/publishup-42429 |

ISSN: | 1866-8372 |

Title of parent work (English): | Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe |

Subtitle (English): | what one can and cannot learn from it |

Publication series (Volume number): | Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe (655) |

Publication type: | Postprint |

Language: | English |

Date of first publication: | 2019/02/27 |

Completion year: | 2018 |

Publishing institution: | Universität Potsdam |

Release date: | 2019/02/27 |

Tag: | exact results; power spectral density; probability density function; single-trajectory analysis |

Issue: | 655 |

Page number: | 31 |

Source: | New Journal of Physics 20 (2018), Art. 023029 DOI 10.1088/1367-2630/aaa67c |

Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät |

DDC classification: | 5 Naturwissenschaften und Mathematik / 53 Physik / 530 Physik |

Peer review: | Referiert |

Publishing method: | Open Access |

License (German): | CC BY - Namensnennung, 4.0 International |