A Semiclassical Condition for Chaos Based on Pesin Theorem

Gomez, Ignacio S.; Losada, Marcelo; Fortín, Sebastián; Castagnino, Mario; Portesi, Mariela

doi:10.1007/s10773-014-2437-6

Cited by 10 publications

(17 citation statements)

References 35 publications

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“…Using these approximations, and neglecting terms of order r 2 , in the formula (29) of F (p) one obtains that F (p) |r| holds for |r| 1. Replacing this inequality in (28) one has that…”

Section: Phase Transitions In a Pair Of Interacting Harmonic Oscillatorsmentioning

confidence: 99%

Notions of the ergodic hierarchy for curved statistical manifolds

Gomez

2017

Physica A: Statistical Mechanics and its Applications

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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.

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“…Using these approximations, and neglecting terms of order r 2 , in the formula (29) of F (p) one obtains that F (p) |r| holds for |r| 1. Replacing this inequality in (28) one has that…”

Section: Phase Transitions In a Pair Of Interacting Harmonic Oscillatorsmentioning

confidence: 99%

Notions of the ergodic hierarchy for curved statistical manifolds

Gomez

2017

Physica A: Statistical Mechanics and its Applications

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“…Taking this into account, we have developed an approach of the classical limit of quantum mechanics [13,[58][59][60][61][62] along with some of their consequences [63][64][65] according to which, under certain spectral conditions, a sufficiently macroscopic closed quantum system behaves as a classical statistical system. This approach can be related with quantum thermalization [50,51], in which the spectral condition is given by the Berry's conjecture.…”

Section: Quantum Chaos As Chaotic Classical Limitmentioning

confidence: 99%

About the Concept of Quantum Chaos

Gomez

Losada

Lombardi

2017

Entropy

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Abstract:The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations.

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“…Motivated by the characterization of chaotic dynamics made in [14,15] by means of a quantum extension of the ergodic hierarchy, now we study a generalization of the EH in the context of the information geometry. This allows to measure how independent the variables H i j are for the 2 × 2 correlated ensemble (3).…”

Section: Towards An Information Geometric Definition Of the Ergodic Hmentioning

confidence: 99%

“…In Gaussian ensembles theory one assumes that in a fully chaotic quantum system the interactions are neglected in such way that the Hamiltonian matrix elements can be considered statistically independent between them [13]. Related to this, in [14,15] a quantum extension of the EH was proposed, called the quantum ergodic hierarchy, which allowed to characterize the chaotic behaviors of the Casati-Prosen model [16] and the kicked rotator [12].…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by the characterizations of quantum chaotic systems made in [14,15] we discuss the 2 × 2 Gaussian Orthogonal Ensembles (GOE) within a 2D correlated statistical model and we propose a generalization of the ergodic, mixing and Bernoulli levels of the EH in the context of the IG and ED, called the information geometric ergodic hierarchy (IGEH). In order to study the correlation decay within IGEH we also define a distinguishability measure for a 2D correlated model that allows to give an upper bound for the information geometric correlation.…”

Section: Introductionmentioning

confidence: 99%

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Ergodic statistical models: Entropic dynamics and chaos

Gomez¹,

Portesi²

2017

AIP Conference Proceedings

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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 2×2 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.

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A Semiclassical Condition for Chaos Based on Pesin Theorem

Abstract: A semiclassical method to determine if the classical limit of a quantum system is chaotic or not, based on Pesin theorem, is presented. The method is applied to a phenomenological Gamow-type model and it is concluded that its classical limit is chaotic.

Cited by 10 publications

References 35 publications

Notions of the ergodic hierarchy for curved statistical manifolds

Notions of the ergodic hierarchy for curved statistical manifolds

About the Concept of Quantum Chaos

Ergodic statistical models: Entropic dynamics and chaos

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