Abstract Klages:

I will first review `normal' deterministic diffusion in a simple piecewise linear chaotic map, where the mean square displacement of an ensemble of particles grows linearly in time. Analytical results yield a diffusion coefficient that is a fractal function of a control parameter, a phenomenon that should be seen in experiments [1]. As a second example, I will discuss diffusion in the nonlinear climbing sine map. The bifurcation scenario exhibited by this map leads to a complicated scenario between normal and anomalous diffusion, where the mean square displacement grows nonlinearly in time [2], depending on control parameters. Finally I will study the Pomeau-Manneville map, which displays intermittency by exhibiting weak chaos. Crosslinks to the new mathematical field of infinite ergodic theory will be briefly outlined. Stochastic continuous time random walk theory predicts subdiffusion for this map, that is, anomalous diffusion with a mean square displacement that grows less than linearly in time [2,3], as is confirmed by computer simulations. In a scaling limit a fractional diffusion equation will be derived. However, as in case of normal diffusion there are complicated fractal, possibly discontinuous parameter dependencies deviating from the predictions of stochastic theory.

[1] R.Klages, Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics (World Scientific, Singapore, 2007).

[2] R. Klages, G.Radons, I.M.Sokolov (Eds.), Anomalous transport (Wiley-VCH, Weinheim, 2008).

[3] R.Klages, From Deterministic Chaos to Anomalous Diffusion (book chapter in Reviews of Nonlinear Dynamics and Complexity Vol. 3, H.G.Schuster (Ed.), Wiley-VCH, Weinheim, March 2010).