Understanding the transport of orbits/particles in any dynamical system is a very difficult problem. However, recent work in the field of anomalous diffusion has shed light on this problem in an important limiting case. Here, we want to understand how the mean-squared displacement grows as a function of time in the long term/long range limit, i.e. <r2(t)> ~ ta. For different systems, the value of 'a' can change. Traditionally, this problem was approached from a PDE perspective. The most basic result comes from the standard diffusion equation where the a = 1. However, there are other PDE's which display non-diffusive behavior. For instance, a simple model for turbulence yields a = 3.
In many dynamical systems though, a PDE is not available. In these cases, the problem is attacked via continuous time random walk (CTRW) models. These models vary in compexity depending on the system of interest. For example, certain systems can alternate (chaotically) between phases of laminar growth (flights) and trapped phases where the orbits are localized for an extended period of time.
In this talk, I'll review CTRW models including their theoretical predictions on the value of a. Then, I'll review applications of this theory to nonlinear iterated maps, experimental fluids, and 3-dimensional systems displaying chaotic advection