Superdiffusion in self-reinforcing run-and-tumble model with rests

TL;DR

This paper models a self-reinforcing run-and-tumble process with rests, deriving conditions for superdiffusion, and shows how adding self-reinforcement and rests affects the diffusion behavior through theoretical analysis and simulations.

Contribution

It extends the telegrapher's equation by incorporating self-reinforcement and rests, revealing conditions for superdiffusion in such models.

Findings

Superdiffusion occurs when mean running time is at least 2/3 of mean resting time.

Self-reinforcement strength influences the transition between diffusion and superdiffusion.

Monte Carlo simulations confirm the theoretical criteria for superdiffusion.

Abstract

This paper introduces a run-and-tumble model with self-reinforcing directionality and rests. We derive a single governing hyperbolic partial differential equation for the probability density of random walk position, from which we obtain the second moment in the long time limit. We find the criteria for the transition between superdiffusion and diffusion caused by the addition of a rest state. The emergence of superdiffusion depends on both the parameter representing the strength of self-reinforcement and the ratio between mean running and resting times. The mean running time must be at least $2/3$ of the mean resting time for superdiffusion to be possible. Monte Carlo simulations validate this theoretical result. This work demonstrates the possibility of extending the telegrapher's (or Cattaneo) equation by adding self-reinforcing directionality so that superdiffusion occurs even when rests are introduced.