Notions of the ergodic hierarchy for curved statistical manifolds

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

Hermite–Gaussian model for quantum states

“…Superpositions of eigenstates of the form |ψ = n α n |n are particular cases of states given in equation (24), with λ nm = α n α * m , i.e.,ρ = |ψ ψ| = nm α n α * m |n m|. Therefore, the PDF of the position operator, the Fisher-Rao metric and the scalar curvature can be obtained from the general expressions (25), (29) and (30), considering λ nm = α n α * m…”

“…where in the last equation we have used the recurrence relations of the Hermite polynomials (A.2). Replacing expres-sions (36) in the Fisher-Rao metric (29), and taking into account relations (A.1) and (A.2), we obtain…”

“…The proof is similar to the case of mixtures of eigenstates. The diagonal elements are given in equation (29),…”

“…Motivated by previous works of some of us [29,30], we propose a generalization of the Gaussian model which we call the Hermite-Gaussian model, and we show its relation with the one-dimensional quantum harmonic oscillator. The application of information geometry techniques to the study of quantum harmonic oscillators can be useful in many applications.…”

“…Moreover, generalized extensions of the information geometry approach to the non-extensive formulation of statistical mechanics [19] have been also considered [20][21][22][23]. Applications of information geometry to chaos can be also performed by considering complexity on curved manifolds [24][25][26][27][28], leading to a criterion for characterizing global chaos on statistical manifolds: the more negative is the curvature, the more chaotic is the dynamics; from which some consequences concerning dynamical systems have been explored [29]. More generally, the curvature has been proved to be a quantifier which measures interactions in thermodynamical systems, where the positive or negative sign corresponds to repulsive or attractive correlations, respectively [7].…”