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Fractional Brownian Motion in a Finite Interval: Correlations Effect Depletion or Accretion Zones of Particles Near Boundaries
New Journal of Physics
  • T. Guggenberger
  • G. Pagnini
  • Thomas Vojta, Missouri University of Science and Technology
  • R. Metzler

Fractional Brownian motion (FBM) is a Gaussian stochastic process with stationary, long-time correlated increments and is frequently used to model anomalous diffusion processes. We study numerically FBM confined to a finite interval with reflecting boundary conditions. The probability density function of this reflected FBM at long times converges to a stationary distribution showing distinct deviations from the fully flat distribution of amplitude 1/L in an interval of length L found for reflected normal Brownian motion. While for superdiffusion, corresponding to a mean squared displacement (MSD) ⟨X2(t)⟩ ≃ tα with 1 < α < 2, the probability density function is lowered in the centre of the interval and rises towards the boundaries, for subdiffusion (0 < α < 1) this behaviour is reversed and the particle density is depleted close to the boundaries. The MSD in these cases at long times converges to a stationary value, which is, remarkably, monotonically increasing with the anomalous diffusion exponent α. Our a priori surprising results may have interesting consequences for the application of FBM for processes such as molecule or tracer diffusion in the confines of living biological cells or organelles, or other viscoelastic environments such as dense liquids in microfluidic chambers.

Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft.
Keywords and Phrases
  • anomalous diffusion,
  • fractional Brownian motion,
  • reflecting boundary conditions
Document Type
Article - Journal
Document Version
Final Version
File Type
© 2019 The Authors, All rights reserved.
Creative Commons Licensing
Creative Commons Attribution 4.0
Publication Date
Citation Information
T. Guggenberger, G. Pagnini, Thomas Vojta and R. Metzler. "Fractional Brownian Motion in a Finite Interval: Correlations Effect Depletion or Accretion Zones of Particles Near Boundaries" New Journal of Physics Vol. 21 Iss. 2 (2019) ISSN: 1367-2630
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