Anomalous Stochastic Transport of Particles with Self-Reinforcement and Mittag–Leffler Distributed Rest Times

Daniel Han,Anna Gavrilova,Sergei Fedotov,Dmitri V. Alexandrov

doi:10.3390/fractalfract5040221

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, we found that this model generates superdiffusion at intermediate times but reverts to subdiffusion in the long time asymptotic limit. To confirm this result, we derived 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 applications in stochastic biological transport.

Highlights

The stochastic movement of intracellular organelles, cells and animals very often exhibits anomalous diffusion, which has led to the widespread use of fractional diffusion equations and fractional derivatives in modeling [1,2]
There are several recent observations that emphasize the importance of fractional models in biological phenomena, such as cancer cell motility [3], polarized cell dynamics [4], intracellular transport of organelles [5]
A model based on the elephant random walk [14] with reinforcement exhibiting superdiffusion, diffusion and subdiffusion at the long time limit has been formulated in discrete time and space [15]

Summary

Introduction

The stochastic movement of intracellular organelles, cells and animals very often exhibits anomalous diffusion, which has led to the widespread use of fractional diffusion equations and fractional derivatives in modeling [1,2]. Superdiffusion was modeled by a persistent random walk model using the concept of self-reinforcing directionality [9,10]. It is natural to formulate a self-reinforcing, persistent random walk model with Mittag–Leffler distributed rest times, which have power-law tails. A model based on the elephant random walk [14] with reinforcement exhibiting superdiffusion, diffusion and subdiffusion at the long time limit has been formulated in discrete time and space [15]. 2021, 5, 221 is to explore the impact of an anomalous rest state on self-reinforced persistent random walks with finite velocity Fractal Fract. 2021, 5, 221 is to explore the impact of an anomalous rest state on self-reinforced persistent random walks with finite velocity

Stochastic Transport with Self-Reinforcement and Mittag–Leffler Distributed

Second Moment Calculations

Single Active State Model

Bi-Directional Transport Model

Monte Carlo Simulations

Conclusions and Summary