Fractional Brownian Motion in a Finite Interval: Correlations Effect Depletion or Accretion Zones of Particles Near Boundaries

Author

T. Guggenberger
G. Pagnini
Thomas Vojta, Missouri University of Science and TechnologyFollow
R. Metzler

Abstract

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)⟩ ≃ t^α 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.

Recommended Citation

T. Guggenberger et al., "Fractional Brownian Motion in a Finite Interval: Correlations Effect Depletion or Accretion Zones of Particles Near Boundaries," New Journal of Physics, vol. 21, no. 2, Institute of Physics - IOP Publishing, Feb 2019.

The definitive version is available at https://doi.org/10.1088/1367-2630/ab075f

Department(s)

Physics

Research Center/Lab(s)

Center for High Performance Computing Research

Comments

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

International Standard Serial Number (ISSN)

1367-2630

Document Type

Article - Journal

Document Version

Final Version

File Type

text

Language(s)

English

Rights

© 2019 The Authors, All rights reserved.

Creative Commons Licensing

This work is licensed under a Creative Commons Attribution 4.0 License.

Publication Date

01 Feb 2019