Chaos and transport in coupled maps: from normal to anomalous diffusion

A fundamental problem of statistical mechanics is to understand transport processes such as diffusion on the basis of dynamical systems theory. In my talk I will outline an approach by which physical quantities like diffusion coefficients can be calculated exactly in terms of dynamical systems properties. Applying this theory to piecewise linear maps deterministically coupled on a one-dimensional lattice, the diffusion coefficient is found to be a fractal function of control parameters. This fractality is a consequence of memory effects that are topologically unstable under parameter variation. Another type of dynamical correlations occurs in intermittent variants of this model leading to anomalous diffusion. In a first approximation subdiffusion in such maps is well described by continuous time random walk theory and fractional diffusion equations. However, on fine scales the same fractal parameter dependencies show up as discussed for normal diffusion asking for more refined theories.