Mean-field theory of myopic self-avoiding fractional Brownian motion

Abstract

Myopic self-avoiding fractional Brownian motion (FBM) is a stochastic process in which an ensemble of particles is driven by fractional Gaussian noise while being repelled by the gradient of the time-integrated ensemble density [J. House, R. Bakhshizada, S. Janu\v{s}onis, R. Metzler, and T. Vojta, Phys. Rev. E 112, 034119 (2025)]. Depending on the anomalous diffusion exponent $\alpha$ characterizing the noise, the process features two dynamical regimes: an interaction-dominated regime ($\alpha < \alpha_c=4/(d+2)$) where the mean-density interaction governs long-time dynamics, and a noise-dominated regime ($\alpha > \alpha_c$) where FBM correlations prevail. In the interaction-dominated regime, the mean-squared displacement grows as $\langle r^2(t) \rangle \sim t^{4/(d+2)}$ regardless of $\alpha$, while for $\alpha > \alpha_c$ the standard FBM scaling $\langle r^2(t) \rangle \sim t^{\alpha}$ is recovered. Here, we develop an analytical mean-field theory of myopic self-avoiding FBM, based on a Fokker-Planck approach to the interaction-dominated regime. This allows us to derive closed-form polynomial solutions for the probability density. To compare with computer simulations, we develop an efficient radial binning algorithm that significantly reduces the computational complexity, making large-scale three-dimensional simulations feasible. Extensive simulations in one, two, and three dimensions confirm the analytical predictions. We also discuss the application of the process to the self-organization of serotonergic axons (fibers) in vertebrate brains, where FBM paths with self-avoidance provide a natural framework for understanding spatial heterogeneities of fiber densities.