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

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 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 $r\simeq 0$, 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 $d_f\simeq 4/3$, 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 $d_f\simeq 4/3$, similar to the case of CR and RP configurations for small r.