Capturing correlations in chaotic diffusion by approximation methods

TL;DR

This paper compares three approximation methods for calculating the diffusion coefficient in chaotic systems, demonstrating their convergence and discussing their effectiveness in capturing complex correlations and potential fractal instabilities.

Contribution

It introduces and analyzes three systematic approximation schemes for the diffusion coefficient in chaotic maps, including new methods based on Taylor-Green-Kubo and Markov partition approaches.

Findings

All schemes converge to the exact diffusion coefficient.

Methods effectively capture irregular dynamical correlations.

Useful in identifying fractal instabilities in parameter-dependent diffusion.

Abstract

We investigate three different methods for systematically approximating the diffusion coefficient of a deterministic random walk on the line which contains dynamical correlations that change irregularly under parameter variation. Capturing these correlations by incorporating higher order terms, all schemes converge to the analytically exact result. Two of these methods are based on expanding the Taylor-Green-Kubo formula for diffusion, whilst the third method approximates Markov partitions and transition matrices by using the escape rate theory of chaotic diffusion. We check the practicability of the different methods by working them out analytically and numerically for a simple one-dimensional map, study their convergence and critically discuss their usefulness in identifying a possible fractal instability of parameter-dependent diffusion, in case of dynamics where exact results for the diffusion coefficient are not available.