Modeling the mobility with memory

TL;DR

This paper introduces a memory-dependent random walk model with parameters alpha and p, demonstrating how memory effects can produce diverse mobility patterns like sub-diffusion, trapping, and ultraslow diffusion, explaining empirical mobility behaviors.

Contribution

The study presents a novel random walk model incorporating memory effects with two parameters, revealing new mobility phenomena and transitions not previously characterized.

Findings

Memory induces sub-diffusion, trapping, and logarithmic diffusion behaviors.

Long-range anti-correlation from memory causes anomalous diffusion.

Impulse-driven jumps enable slow escape from traps, leading to ultraslow diffusion.

Abstract

We study a random walk model in which the jumping probability to a site is dependent on the number of previous visits to the site, as a model of the mobility with memory. To this end we introduce two parameters called the memory parameter alpha and the impulse parameter p. From extensive numerical simulations, we found that various limited mobility patterns such as sub-diffusion, trapping, and logarithmic diffusion could be observed. By the memory, a long-ranged directional anti-correlation kinetically-induces anomalous sub-diffusive and trapping behaviors, and transition between them. With random jumps by the impulse parameter, a trapped walker can escape from the trap very slowly, resulting in an ultraslow logarithmic diffusive behavior. Our results suggest that the memory of walker's has-beens can be one mechanism explaining many of empirical characteristics of the mobility of animated objects.