A simple non-chaotic map generating subdiffusive, diffusive and superdiffusive dynamics

TL;DR

This paper introduces a simple non-chaotic dynamical system called the slicer map that can produce a range of diffusion behaviors, providing insights into complex polygonal billiards and anomalous transport phenomena.

Contribution

The paper presents an analytically tractable non-chaotic model that exhibits subdiffusive, diffusive, and superdiffusive dynamics, linking deterministic systems to stochastic diffusion classes.

Findings

The slicer map transitions from subdiffusion to superdiffusion as parameters vary.

All moments of the position distribution are calculated analytically.

Transport properties match different classes of stochastic processes in various regimes.

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.