Scale-invariant motion in intermittent chaotic systems

TLDR

The study investigates dynamics from iterated maps using a probabilistic continuous‑time random‑walk framework. The authors model the motion with two continuous‑time random‑walk variants—jumping between sites or moving at constant velocity with random direction—analyzing mean‑squared displacement and propagator from iterated maps and applying scaling arguments. They find that iterated maps produce both dispersive and enhanced diffusion, and for dispersive motion they highlight the stationary‑state problem and its relevance.

Abstract

We investigate the dynamics generated from iterated maps and analyze the motion in terms of the probabilistic continuous-time random-walk (CTRW) approach. Two different CTRW models are considered: (i) Particles jump between sites (turning points) or (ii) particles move at a constant velocity between sites and choose a new direction at random. For both models we study the mean-squared displacement 〈${\mathit{r}}^{2}$(t)〉 and the propagator P(r,t), the probability to be at location r at time t having started at the origin at t=0. Iterated maps are used to generate both dispersive and enhanced diffusion and the results are analyzed using the CTRW framework and scaling arguments. For the case of dispersive motion we discuss the problem of the stationary state and point out its relevance.