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

摘要

We investigate the motion of a single particle moving on a two-dimensionalsquare lattice whose sites are occupied by right and left rotators. These leftand right rotators deterministically rotate the particle's velocity to theright or left, respectively and \emph{flip} orientation from right to left orfrom left to right after scattering the particle. We study three types ofconfigurations of left and right rotators, which we think of as types of media,through with the particle moves. These are completely random (CR), randomperiodic (RP), and completely periodic (CP) configurations. For CRconfigurations the particle's dynamics depends on the ratio $r$ of right toleft scatterers in the following way. For small $r\simeq0$, when theconfiguration is nearly homogeneous, the particle subdiffuses with an exponentof 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 $r\simeq1$, the particle's dynamics transitions from subdiffusion to anomalousdiffusion with a fractal dimension of $d_f\simeq 7/4$, similar to that of apercolating cluster in 2-d. In RP configurations, which are more structuredthan CR configurations but also randomly generated, we find that the particlehas the same statistic as in the CR case. In contrast, CP configurations, whichare highly structured, typically will cause the particle to go through atransient stage of subdiffusion, which then abruptly changes to propagation.Interestingly, the subdiffusive stage has an exponent of approximately 2/3 anda fractal dimension of $d_f\simeq4/3$, similar to the case of CR and RPconfigurations for small $r$.