From Deterministic Chaos to Anomalous Diffusion

TL;DR

This paper provides an accessible overview of deterministic chaos, diffusion, and anomalous diffusion, linking chaos theory with diffusion processes and illustrating their relevance through biological cell migration examples.

Contribution

It introduces the foundational concepts of chaos and diffusion, and demonstrates how the escape rate formalism connects chaos measures to diffusion coefficients in simple maps.

Findings

Derivation of diffusion coefficient using chaos quantities

Explanation of anomalous diffusion via continuous time random walk

Application to biological cell migration dynamics

Abstract

This is an easy-to-read introduction to foundations of deterministic chaos, deterministic diffusion and anomalous diffusion. The first part introduces to deterministic chaos in one-dimensional maps in form of Ljapunov exponents and dynamical entropies. The second part outlines the concept of deterministic diffusion. Then the escape rate formalism for deterministic diffusion, which expresses the diffusion coefficient in terms of the above two chaos quantities, is worked out for a simple map. Part three explains basics of anomalous diffusion by demonstrating the stochastic approach of continuous time random walk theory for an intermittent map. As an example of experimental applications, the anomalous dynamics of biological cell migration is discussed.