Abstract
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here, we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process.
Highlights
To our knowledge only a few deterministic dynamical systems are known exhibiting all three regimes of subdiffusion, normal diffusion, and superdiffusion under parameter variation
We show that our simple map model generates a surprisingly non-trivial spectrum of different diffusive properties under parameter variation
Periodic polygonal billiards have very long lasting dynamical correlations and poor statistical properties, which are associated with very sensitive dependence of their transport properties on the details of their geometry.12. These difficulties to understand diffusion in polygonal billiards on the basis of dynamical systems theory are paralleled by difficulties in attempts to approximate their diffusive properties by stochastic theory: There still appears to be a controversy in the literature of whether continuous time random walk theory and Levy walks, fractional Fokker-Planck equations or scaling arguments should be applied to understand their anomalous diffusive properties, with different approaches yielding different results for the above exponent c of the MSD.3,8,21,34. While all these theories are based on dynamics generated from temporal randomness, spatial randomness leads yet to another important class of stochastic models, called random walks in random environments, which yields related types of anomalous diffusion: An important example in one dimension is the Levy Lorentz gas where the scatterers are randomly distributed according to a Levy-stable probability distribution of the scatterer positions
Summary
To our knowledge only a few deterministic dynamical systems are known exhibiting all three regimes of subdiffusion, normal diffusion, and superdiffusion under parameter variation. These are onedimensional maps generalising circle rotations which cut the original interval into several subintervals by permuting them non-chaotically Both polygonal billiards and IETs are known to exhibit highly non-trivial ergodic properties, and in general there does not seem to exist any theory to understand the complicated diffusive dynamics of such systems from first principles. While all these theories are based on dynamics generated from temporal randomness, spatial randomness leads yet to another important class of stochastic models, called random walks in random environments, which yields related types of anomalous diffusion: An important example in one dimension is the Levy Lorentz gas where the scatterers are randomly distributed according to a Levy-stable probability distribution of the scatterer positions This model has been studied both numerically and analytically revealing a highly non-trivial superdiffusive dynamics that depends in an intricate way on initial conditions and the type of averaging..
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