Fractional Brownian motion with mean-density interaction: a myopic self-avoiding fractional stochastic process

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.