From normal to anomalous diffusion in simple dynamical systems (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

Abstract

We have introduced to the problem of chaotic diffusion generated by deterministic dynamical systems. As the simplest examples possible we chose piecewise linear one-dimensional maps defined on the whole real line. For these dynamical systems the parameter-dependent diffusion coefficient can be calculated exactly analytically. We outlined a straightforward method of how to do so, which is based on evaluating fractal generalised Takagi functions. Surprisingly, the resulting diffusion coefficient exhibits fractal structures under parameter variation, which can be explained with respect to non-trivial dynamical correlations on microscopic scales. For certain classes of such maps the Hausdorff and the box counting dimensions of the corresponding diffusion coefficient curves under parameter variation have been calculated rigorously mathematically. Interestingly, these curves belong to a very special type of fractals for which both dimensions are exactly equal to one. The fractal structure emerges from logarithmic corrections in continuity properties that display an intricate dependence on parameter variation [24]. These fractal diffusion coefficients are not an artefact specific to one-dimensional maps. A line of work demonstrated that they also appear in Hamiltonian particle billiards [l] and even for particles moving in soft potential landscapes [25]. The latter system relates to electronic transport in artificial graphene that can be studied experimentally. Yet another string of work investigated anomalous diffusion in intermittent maps of Pomeau-Manneville type, which also turned out to display fractal parameter dependencies of suitably generalised anomalous diffusion coefficients [l], In this brief review we only discussed a simple type of random dynamical system as a second example generating non-trivial diffusion. We showed that mixing normal diffusive with localised dynamics randomly in time yields a novel type of intermittent dynamics characterised by the emergence of subdiffusion. To which extent the recipe that we proposed for obtaining this kind of random diffusive dynamical system can generate other types of anomalous diffusion remains to be explored.