Subdiffusion, Anomalous Diffusion and Propagation of a Particle Moving in Random and Periodic Media

TL;DR

This paper studies how a particle moves on a 2D lattice with rotators, revealing different diffusion behaviors depending on the configuration and ratio of rotators, including subdiffusion, anomalous diffusion, and propagation.

Contribution

It introduces a model of particle motion in various media configurations and characterizes the transition from subdiffusion to anomalous diffusion and propagation.

Findings

Subdiffusion with exponent 2/3 in nearly homogeneous random media.

Transition to anomalous diffusion with fractal dimension 7/4 as ratio increases.

Structured media can induce a transient subdiffusive stage before propagation.

Abstract

We investigate the motion of a single particle moving on a two-dimensional square lattice whose sites are occupied by right and left rotators. These left and right rotators deterministically rotate the particle's velocity to the right or left, respectively and \emph{flip} orientation from right to left or from left to right after scattering the particle. We study three types of configurations of left and right rotators, which we think of as types of media, through with the particle moves. These are completely random (CR), random periodic (RP), and completely periodic (CP) configurations. For CR configurations the particle's dynamics depends on the ratio $r$ of right to left scatterers in the following way. For small $r\simeq0$, when the configuration is nearly homogeneous, the particle subdiffuses with an exponent of 2/3, similar to the diffusion of a macromolecule in a crowded environment. Also, the particle's trajectory has a fractal dimension of $d_f\simeq4/3$, comparable to that of a self-avoiding walk. As the ratio increases to $r\simeq 1$, the particle's dynamics transitions from subdiffusion to anomalous diffusion with a fractal dimension of $d_f\simeq 7/4$, similar to that of a percolating cluster in 2-d. In RP configurations, which are more structured than CR configurations but also randomly generated, we find that the particle has the same statistic as in the CR case. In contrast, CP configurations, which are highly structured, typically will cause the particle to go through a transient stage of subdiffusion, which then abruptly changes to propagation. Interestingly, the subdiffusive stage has an exponent of approximately 2/3 and a fractal dimension of $d_f\simeq4/3$, similar to the case of CR and RP configurations for small $r$.