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Inertia triggers nonergodicity of fractional Brownian motion

Andrey G. Cherstvy1,2,*, Wei Wang3,†, Ralf Metzler2,‡, and Igor M. Sokolov1,4,§

  • 1Institut für Physik, Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
  • 2Institute for Physics & Astronomy, University of Potsdam, Karl-Liebknecht-Straße 24/25, 14476 Potsdam-Golm, Germany
  • 3Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 Dresden, Germany
  • 4IRIS Adlershof, Zum Großen Windkanal 6, 12489 Berlin, Germany

  • *a.cherstvy@gmail.com
  • †weiwangnuaa@gmail.com
  • ‡rmetzler@uni-potsdam.de
  • §igor.sokolov@physik.hu-berlin.de
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Phys. Rev. E 104, 024115 – Published 13 August, 2021

DOI: https://doi.org/10.1103/PhysRevE.104.024115

Abstract

How related are the ergodic properties of the over- and underdamped Langevin equations driven by fractional Gaussian noise? We here find that for massive particles performing fractional Brownian motion (FBM) inertial effects not only destroy the stylized fact of the equivalence of the ensemble-averaged mean-squared displacement (MSD) to the time-averaged MSD (TAMSD) of overdamped or massless FBM, but also dramatically alter the values of the ergodicity-breaking parameter (EB). Our theoretical results for the behavior of EB for underdamped or massive FBM for varying particle mass m, Hurst exponent H, and trace length T are in excellent agreement with the findings of stochastic computer simulations. The current results can be of interest for the experimental community employing various single-particle-tracking techniques and aiming at assessing the degree of nonergodicity for the recorded time series (studying, e.g., the behavior of EB versus lag time). To infer FBM as a realizable model of anomalous diffusion for a set single-particle-tracking data when massive particles are being tracked, the EBs from the data should be compared to EBs of massive (rather than massless) FBM.

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Phys. Rev. E 104, 024115– Published 13 August, 2021
  • Received 17 June 2021
  • Accepted 29 July 2021
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