Fractional Brownian Motion with mean-density interaction

Fractional Brownian motion (FBM) is a stochastic process with long-range correlations in-time, and has been shown to be a useful model of systems featuring anomalous diffusion. We investigate the effects of interactions in an ensemble of particles undergoing FBM. Specifically, we introduce a mean-density interaction by which each particle interacts with the cumulative density produced by the entire ensemble of particles. This model is motivated by a recent biological application of FBM for modeling the distribution of serotonergic neurons in the brain [1]. We report the results of extensive computer simulations, and compute the resultant mean-square displacements and probability distributions of particles subject to FBM with such mean-density interactions. We also develop a scaling theory explaining our findings.