Linear and fractal diffusion coefficients in a family of one dimensional chaotic maps

TL;DR

This paper derives exact analytical expressions for diffusion coefficients in a family of one-dimensional chaotic maps, revealing both fractal and linear parameter dependencies and linking these behaviors to the maps' topological and ergodic properties.

Contribution

It introduces a method to analytically compute diffusion coefficients for a family of chaotic maps, demonstrating the coexistence of fractal and linear behaviors in parameter space.

Findings

Diffusion coefficients are fractal functions of parameters in some maps.

Analytical expressions for diffusion coefficients are derived using generalized Takagi functions.

The structure of the diffusion coefficient depends on the topology of Markov partitions.

Abstract

We analyse deterministic diffusion in a simple, one-dimensional setting consisting of a family of four parameter dependent, chaotic maps defined over the real line. When iterated under these maps, a probability density function spreads out and one can define a diffusion coefficient. We look at how the diffusion coefficient varies across the family of maps and under parameter variation. Using a technique by which Taylor-Green-Kubo formulae are evaluated in terms of generalised Takagi functions, we derive exact, fully analytical expressions for the diffusion coefficients. Typically, for simple maps these quantities are fractal functions of control parameters. However, our family of four maps exhibits both fractal and linear behavior. We explain these different structures by looking at the topology of the Markov partitions and the ergodic properties of the maps.