Anomalous stochastic transport of particles with self-reinforcement and Mittag-Leffler distributed rest times

TL;DR

This paper presents a new stochastic particle transport model with self-reinforcement and Mittag-Leffler rest times, showing superdiffusion at intermediate times and subdiffusion asymptotically, relevant for biological transport.

Contribution

It introduces a novel persistent random walk model with non-local switching and demonstrates the transient superdiffusive behavior through analytical and simulation methods.

Findings

Model exhibits superdiffusion at intermediate times

Long-term behavior is subdiffusive, confirmed analytically

Mittag-Leffler rest times dominate transport dynamics

Abstract

We introduce a persistent random walk model for the stochastic transport of particles involving self-reinforcement and a rest state with Mittag-Leffler distributed residence times. The model involves a system of hyperbolic partial differential equations with a non-local switching term described by the Riemann-Liouville derivative. From Monte Carlo simulations, this model generates superdiffusion at intermediate times but reverts to subdiffusion in the long time asymptotic limit. To confirm this result, we derive the equation for the second moment and find that it is subdiffusive in the long time limit. Analyses of two simpler models are also included, which demonstrate the dominance of the Mittag-Leffler rest state leading to subdiffusion. The observation that transient superdiffusion occurs in an eventually subdiffusive system is a useful feature for application in stochastic biological transport.