Abstract
Fractional Brownian motion is a Gaussian stochastic process with long-range correlations in time; it has been shown to be a useful model of anomalous diffusion. Here, we investigate the effects of mutual interactions in an ensemble of particles undergoing fractional Brownian motion. Specifically, we introduce a mean-density interaction in which each particle in the ensemble is coupled to the gradient of the total, time-integrated density produced by the entire ensemble. We report the results of extensive computer simulations for the mean-squared displacements and the probability densities of particles undergoing one-dimensional fractional Brownian motion with such a mean-density interaction. We find two qualitatively different regimes, depending on the anomalous diffusion exponent α characterizing the fractional Gaussian noise. The motion is governed by the interactions for α< 4/3, whereas it is dominated by the fractional Gaussian noise for α>4/3. We develop a scaling theory explaining our findings. We also discuss generalizations to higher space dimensions and nonlinear interactions, the relation of our process to the "true" or myopic self-avoiding walk, as well as applications to the growth of strongly stochastic axons (e.g., serotonergic fibers) in vertebrate brains.
Recommended Citation
J. House et al., "Fractional Brownian Motion with Mean-density Interaction: A Myopic Self-avoiding Fractional Stochastic Process," Physical Review E, vol. 112, no. 3 thru 1, p. 34119, American Physical Society, Sep 2025.
The definitive version is available at https://doi.org/10.1103/w5pk-bw5r
Department(s)
Physics
International Standard Serial Number (ISSN)
2470-0053
Document Type
Article - Journal
Document Version
Final Version
File Type
text
Language(s)
English
Rights
© 2025 American Physical Society, All rights reserved.
Publication Date
01 Sep 2025
PubMed ID
41116387
