Ergodic statistical models: entropic dynamics and chaos

TL;DR

This paper introduces the information geometric ergodic hierarchy (IGEH), extending classical ergodic levels using statistical models on curved manifolds, and applies it to Gaussian Orthogonal Ensembles to analyze their mixing properties.

Contribution

It develops the IGEH framework using information geometry and applies it to GOE, revealing new insights into their ergodic and mixing behavior based on scalar curvature.

Findings

GOE belong to the IG mixing level with maximum negative scalar curvature

Proposed a measure of distinguishability as an upper bound of IG correlation

Identified the relationship between correlation strength and geometric properties

Abstract

We present an extension of the ergodic, mixing and Bernoulli levels of the ergodic hierarchy in dynamical systems, the information geometric ergodic hierarchy (IGEH), making use of statistical models on curved manifolds in the context of information geometry. We discuss the $2\times 2$ Gaussian Orthogonal Ensembles (GOE) within a $2D$ correlated model. For $r$ vanishingly small, we find that GOE belong to the information geometric (IG) mixing level having a maximum negative value of scalar curvature. Moreover, we propose a measure of distinguishability for the family of distributions of the $2D$ correlated model that results to be an upper bound of the IG correlation.