Towards a definition of the Quantum Ergodic Hierarchy: Kolmogorov and Bernoulli systems

Gomez, Ignacio S.; Castagnino, Mario

doi:10.1016/j.physa.2013.08.070

Cited by 17 publications

(36 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…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%

“…Thus, the spectrum can be approximated by a continuum (as in the case of chaotic quantum billiards, such as the Sinai billiards [1,13,66]) and the sum ∑ ND (t) can be replaced by a double integral:…”

Section: Formulation Of the Classical Limit In Terms Of Weak Limitmentioning

confidence: 99%

“…This section is devoted to the quantum ergodic hierarchy developed in [12,13]. In Section 8.1 we review the construction of the quantum ergodic hierarchy and we show the consistency between the languages of measurable sets, distribution functions and quantum operators.…”

Section: The Quantum Ergodic Hierarchymentioning

confidence: 99%

“…In this paper we propose a unified framework for describing quantum chaos using the quantum ergodic hierarchy, previously developed by two of us in [12,13]. We also characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

About the Concept of Quantum Chaos

Gomez

Losada

Lombardi

2017

Entropy

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Quantum Chaos As Chaotic Classical Limitmentioning

confidence: 99%

Section: Formulation Of the Classical Limit In Terms Of Weak Limitmentioning

confidence: 99%

Section: The Quantum Ergodic Hierarchymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

About the Concept of Quantum Chaos

Gomez

Losada

Lombardi

2017

Entropy

Self Cite

View full text Add to dashboard Cite

show abstract

“…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 [25]. Related to this, in [26,27,28] a quantum extension of the EH was proposed, called the quantum ergodic hierarchy, which allowed to provide a characterization of the chaotic behaviors of the Casati-Prosen model [24] and the kicked rotator [21,22,23].…”

Section: Introductionmentioning

confidence: 99%

Notions of the ergodic hierarchy for curved statistical manifolds

Gomez

2017

Physica A: Statistical Mechanics and its Applications

View full text Add to dashboard Cite

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.

show abstract