Notions of the ergodic hierarchy for curved statistical manifolds

- IGEH quantifies the statistical independence between the microvariables.
- Multivariate Gaussian distributions and quadratic Hamiltonians are one-to-one.
- IGEH is compatible with the criterium of global chaos given by the scalar curvature.
- Phase transitions in statistical models can be characterized by means of IGEH levels.
- Controlling phase transitions by external parameters of the covariance matrix.

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 families 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 2×2 Gaussian ensembles. In both examples the scalar curvature results a global indicator of the dynamics.