Weak chaos, infinite ergodic theory, and anomalous dynamics

TL;DR

This chapter explores weak chaos and anomalous dynamics through simple models and experiments, highlighting their mathematical foundations and implications for biological systems and nonequilibrium physics.

Contribution

It introduces the concept of weak chaos, connects it with infinite ergodic theory, and applies stochastic models to explain anomalous diffusion and fluctuation relations.

Findings

Weak chaos is characterized by specific ergodic properties.

Anomalous diffusion can be modeled with fractional diffusion equations.

Experimental biological cell migration exhibits signatures of anomalous dynamics.

Abstract

This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau-Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau-Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.