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Different behaviors of diffusing diffusivity dynamics based on three different definitions of fractional Brownian motion

Wei Wang1, Qing Wei2, Aleksei V. Chechkin1,3,4,5, and Ralf Metzler1,5,*

  • *Contact author: rmetzler@uni-potsdam.de
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Phys. Rev. E 112, 014108 – Published 7 July, 2025

DOI: https://doi.org/10.1103/w8gv-3fxt

Abstract

The effects of "diffusing diffusivity” (DD), a stochastically time-varying diffusion coefficient, are explored within the frameworks of three different forms of fractional Brownian motion (FBM): (i) the Langevin equation driven by fractional Gaussian noise (LE-FBM), (ii) the Weyl integral representation introduced by Mandelbrot and van Ness (MN-FBM), and (iii) the Riemann-Liouville fractional integral representation (RL-FBM) introduced by Lévy. The statistical properties of the three FBM-generalized DD models are examined, including the mean-squared displacement (MSD), mean-squared increment (MSI), autocovariance function (ACVF) of increments, and the probability density function (PDF). Despite the long-believed equivalence of MN-FBM and LE-FBM, their corresponding FBM-DD models exhibit distinct behavior in terms of MSD and MSI. In the MN-FBM-DD model, the statistical characteristics directly reflect an effective diffusivity equal to its mean value. In contrast, in LE-FBM-DD, correlations in the random diffusivity give rise to an unexpected crossover behavior in both MSD and MSI. We also find that the MSI and ACVF are nonstationary in RL-FBM-DD but stationary in the other two DD models. All DD models display a crossover from a short-time non-Gaussian PDF to a long-time Gaussian PDF. Our findings offer guidance for experimentalists in selecting appropriate FBM-generalized models to describe viscoelastic yet non-Gaussian dynamics in bio- and soft-matter systems with heterogeneous environments.

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Phys. Rev. E 112, 014108– Published 7 July, 2025
  • Received 13 April 2025
  • Accepted 9 June 2025
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