Ergodicity and large deviations in physical systems with stochastic dynamics

TL;DR

This paper reviews how large deviation theory applies to ergodic physical systems, providing insights into rare events, phase transitions, and connections to control theory, with implications for understanding entropy, fluctuations, and metastability.

Contribution

It offers a comprehensive review of large deviation theory in physical systems, highlighting new principles, phase transitions, and links to optimal control that advance understanding of complex dynamics.

Findings

Large deviation theory explains rare events in physical systems.

Connections between large deviations and optimal control are explored.

Insights into entropy production and phase transitions are provided.

Abstract

In ergodic physical systems, time-averaged quantities converge (for large times) to their ensemble-averaged values. Large deviation theory describes rare events where these time averages differ significantly from the corresponding ensemble averages. It allows estimation of the probabilities of these events, and their mechanisms. This theory has been applied to a range of physical systems, where it has yielded new insights into entropy production, current fluctuations, metastability, transport processes, and glassy behaviour. We review some of these developments, identifying general principles. We discuss a selection of dynamical phase transitions, and we highlight some connections between large-deviation theory and optimal control theory.