# Notions of the ergodic hierarchy for curved statistical manifolds

## TL;DR

This paper extends the ergodic hierarchy to curved statistical manifolds using information geometry, linking statistical independence with Hamiltonian dynamics and illustrating with examples like harmonic oscillators and Gaussian ensembles.

## Contribution

It introduces a geometric framework for ergodic properties on curved manifolds, connecting statistical models with physical systems and providing new tools for analyzing Hamiltonian dynamics.

## Key findings

- Scalar curvature acts as a global indicator of dynamics.
- Correlation between microvariables follows a power law in temperature.
- The geometric approach applies to systems with phase transitions.

## Abstract

We present an extension of the ergodic, mixing, and Bernoulli levels of the ergodic hierarchy for statistical models on curved manifolds, making use of elements of the information geometry. This extension focuses on the notion of statistical independence between the microscopical variables of the system. Moreover, we establish an intimately relationship between statistical models and family of probability distributions belonging to the canonical ensemble, which for the case of the quadratic Hamiltonian systems provides a closed form for the correlations between the microvariables in terms of the temperature of the heat bath as a power law. From this we obtain an information geometric method for studying Hamiltonian dynamics in the canonical ensemble. We illustrate the results with two examples: a pair of interacting harmonic oscillators presenting phase transitions and the 2x2 Gaussian ensembles. In both examples the scalar curvature results a global indicator of the dynamics.