Ergodicity detection algorithms: Scaling of ergodicity in random symbolic dynamics

TL;DR

This paper introduces a new framework and algorithms for detecting ergodicity in symbolic dynamical systems, clarifying misconceptions and proposing a novel concept called sufficiency of sparse visit to improve detection methods.

Contribution

It presents an accessible discrete framework for ergodicity detection, clarifies ergodic regimes, and introduces the concept of sufficiency of sparse visit for better analysis.

Findings

Distinction between Gibbs-Boltzmann and von Neumann-Birkhoff ergodic regimes clarified.

A new concept, sufficiency of sparse visit, is introduced for ergodicity detection.

Algorithmic approaches for identifying ergodic regimes are demonstrated using symbolic sequences.

Abstract

The mathematical definitions of distinct concepts that are needed in building an ergodicity detection algorithm are introduced in a framework. This algorithmic framework is expressed in a discrete setting in an accessible manner for broader quantitative practitioners without loss of generality. At the same time, the common misconceptions of the requirement of visiting all available states in the time-averaged quantities for physical systems and non-existence of an ergodic process are resolved by introducing the distinction between Gibbs-Boltzmann and von Neumann-Birkhoff ergodic regimes. For this purpose, we introduce a new concept which is called sufficiency of sparse visit. We use finite symbolic random sequences as a pedagogical tool in establishing the different approaches for the detection of ergodic regimes of dynamical systems with vector patterns. The simple example system conveys the different attitudes in ergodicity regimes and offers guidance for building computational tools for its algorithmic detection.