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Journal ArticleDOI

On the ergodic properties of nowhere dispersing billiards

01 Oct 1979-Communications in Mathematical Physics (Springer-Verlag)-Vol. 65, Iss: 3, pp 295-312
TL;DR: In this paper, the B-property for two-dimensional domains with focusing and neutral regular components is proved and some examples of three and more dimensional domains with billiards obeying this property are also considered.
Abstract: For billiards in two dimensional domains with boundaries containing only focusing and neutral regular components and satisfacting some geometrical conditionsB-property is proved. Some examples of three and more dimensional domains with billiards obeying this property are also considered.
Citations
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Journal ArticleDOI
TL;DR: The eigenstate thermalization hypothesis (ETH) as discussed by the authors is a natural extension of quantum chaos and random matrix theory (RMT) that allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath.
Abstract: This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the i...

1,536 citations

Book
01 Oct 2000
TL;DR: This book provides an overview of recent developments in large margin classifiers, examines connections with other methods, and identifies strengths and weaknesses of the method, as well as directions for future research.
Abstract: From the Publisher: The concept of large margins is a unifying principle for the analysis of many different approaches to the classification of data from examples, including boosting, mathematical programming, neural networks, and support vector machines. The fact that it is the margin, or confidence level, of a classification--that is, a scale parameter--rather than a raw training error that matters has become a key tool for dealing with classifiers. This book shows how this idea applies to both the theoretical analysis and the design of algorithms. The book provides an overview of recent developments in large margin classifiers, examines connections with other methods (e.g., Bayesian inference), and identifies strengths and weaknesses of the method, as well as directions for future research. Among the contributors are Manfred Opper, Vladimir Vapnik, and Grace Wahba.

1,059 citations


Cites background from "On the ergodic properties of nowher..."

  • ...However, by excluding from the billiard a spherical region inside the polyhedron, one can make the dynamics hyperbolic [Bunimovich, 1979]....

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Journal ArticleDOI
TL;DR: The eigenstate thermalization hypothesis (ETH) as mentioned in this paper is a natural extension of quantum chaos and random matrix theory (RMT) and it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath.
Abstract: This review gives a pedagogical introduction to the eigenstate thermalization hypothesis (ETH), its basis, and its implications to statistical mechanics and thermodynamics. In the first part, ETH is introduced as a natural extension of ideas from quantum chaos and random matrix theory (RMT). To this end, we present a brief overview of classical and quantum chaos, as well as RMT and some of its most important predictions. The latter include the statistics of energy levels, eigenstate components, and matrix elements of observables. Building on these, we introduce the ETH and show that it allows one to describe thermalization in isolated chaotic systems without invoking the notion of an external bath. We examine numerical evidence of eigenstate thermalization from studies of many-body lattice systems. We also introduce the concept of a quench as a means of taking isolated systems out of equilibrium, and discuss results of numerical experiments on quantum quenches. The second part of the review explores the implications of quantum chaos and ETH to thermodynamics. Basic thermodynamic relations are derived, including the second law of thermodynamics, the fundamental thermodynamic relation, fluctuation theorems, and the Einstein and Onsager relations. In particular, it is shown that quantum chaos allows one to prove these relations for individual Hamiltonian eigenstates and thus extend them to arbitrary stationary statistical ensembles. We then show how one can use these relations to obtain nontrivial universal energy distributions in continuously driven systems. At the end of the review, we briefly discuss the relaxation dynamics and description after relaxation of integrable quantum systems, for which ETH is violated. We introduce the concept of the generalized Gibbs ensemble, and discuss its connection with ideas of prethermalization in weakly interacting systems.

985 citations

Journal ArticleDOI
TL;DR: In this paper, the statistical properties of quantum chaos are considered on the basis of the well-known model of a kicked rotator and the quasienergy spectrum and the structure of the eigenfunctions in the case of strong classical chaos.

605 citations

Journal ArticleDOI
TL;DR: In this paper, the hydrogen atom in a uniform magnetic field is discussed as a real and physical example of a simple nonintegrable system and the quantum mechanical spectrum shows a region of approximate separability which breaks down as we approach the classical escape threshold.

423 citations

References
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Book ChapterDOI
01 Jan 2014
TL;DR: Ergodic theory concerns with the study of the long-time behavior of a dynamical system as mentioned in this paper, and it is known as Birkhoff's ergodic theorem, which states that the time average exists and is equal to the space average.
Abstract: Ergodic theory concerns with the study of the long-time behavior of a dynamical system. An interesting result known as Birkhoff’s ergodic theorem states that under certain conditions, the time average exists and is equal to the space average. The applications of ergodic theory are the main concern of this note. We will introduce fundamental concepts in ergodic theory, Birkhoff’s ergodic theorem and its consequences.

3,140 citations


"On the ergodic properties of nowher..." refers background in this paper

  • ...For example, it is known [ 20 ] that the billiard in ellips is a completely integrable system....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of proper degeneracy and prove the existence of a non-degeneracy of diffeomorphisms with respect to a constant number of vertices.
Abstract: CONTENTSIntroduction § 1. Results § 2. Preliminary results from mechanics § 3. Preliminary results from mathematics § 4. The simplest problem of stability § 5. Contents of the paperChapter I. Theory of perturbations § 1. Integrable and non-integrable problems of dynamics § 2. The classical theory of perturbations § 3. Small denominators § 4. Newton's method § 5. Proper degeneracy § 6. Remark 1 § 7. Remark 2 § 8. Application to the problem of proper degeneracy § 9. Limiting degeneracy. Birkhoff's transformation § 10. Stability of positions of equilibrium of Hamiltonian systemsChapter II. Adiabatic invariants § 1. The concept of an adiabatic invariant § 2. Perpetual adiabatic invariance of action with a slow periodic variation of the Hamiltonian § 3. Adiabatic invariants of conservative systems § 4. Magnetic traps § 5. The many-dimensional caseChapter III. The stability of planetary motions § 1. Picture of the motion § 2. Jacobi, Delaunay and Poincare variables §3. Birkhoff's transformation § 4. Calculation of the asymptotic behaviour of the coefficients in the expansion of § 5. The many-body problemChapter IV. The fundamental theorem § 1. Fundamental theorem § 2. Inductive theorem § 3. Inductive lemma § 4. Fundamental lemma § 5. Lemma on averaging over rapid variables § 6. Proof of the fundamental lemma § 7. Proof of the inductive lemma § 8. Proof of the inductive theorem § 9. Lemma on the non-degeneracy of diffeomorphisms § 10. Averaging over rapid variables § 11. Polar coordinates § 12. The applicability of the inductive theorem § 13. Passage to the limit § 14. Proof of the fundamental theoremChapter V. Technical lemmas § 1. Domains of type D § 2. Arithmetic lemmas § 3. Analytic lemmas § 4. Geometric lemmas § 5. Convergence lemmas § 6. NotationChapter VI. Appendix § 1. Integrable systems § 2. Unsolved problems § 3. Neighbourhood of an invariant manifold §4. Intermixing § 5. Smoothing techniquesReferences

1,057 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals.
Abstract: In this paper we consider dynamical systems resulting from the motion of a material point in domains with strictly convex boundary, that is, such that the operator of the second quadratic form is negative-definite at each point of the boundary, where the boundary is taken to be equipped with the field of inward normals. We prove that such systems are ergodic and are K-systems. The basic method of investigation is the construction of transversal foliations for such systems and the study of their properties.

914 citations


"On the ergodic properties of nowher..." refers background in this paper

  • ...The following three lemmas can be easily proved with the help of elementary geometrical considerations, the corresponding proofs can be found for instance in [1] (see also [2], where some corrections and specifications are given)....

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  • ...One can work out the following stages of the demonstration of the K-property for our billiards, in analogy with paper [5] (see also [1,2]), with some minor modifications, β-property can be concluded from K-property just in the same way as for dispersing billiards (see [17])....

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  • ...for almost every point xeύl/v The process of construction of these fibers is a variant of the proof of HadamarPerron's theorem for manifolds (see [1, 14])....

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  • ...In complete analogy with the case of dispersing billiards (see [1], pp....

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  • ...(The proof of convergence of κ\x) for dispersing billiards is trivial [1,2]....

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Journal ArticleDOI
TL;DR: In this paper, smooth dynamical systems having contracting and expanding invariant foliations (of not necessarily complementary dimensions) are investigated, and the K-property is established for such systems under additional assumptions.
Abstract: Smooth dynamical systems having contracting and expanding invariant foliations (of not necessarily complementary dimensions) are investigated. Ergodicity and the K-property are established for such dynamical systems under additional assumptions.

381 citations


"On the ergodic properties of nowher..." refers background in this paper

  • ...Such fibers play a central role in studying of ergodic properties of classical dynamical systems, such as Anosov systems, partially hyperbolic systems, dispersing billiards and so on (see [14-16])....

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Book
01 Jan 1976
TL;DR: In this paper, a rich collection of examples from Erevan's lectures is presented to introduce the reader to the main themes of the subject, such as the existence of invariant measures, geodesic flows on Riemannian manifolds, ergodic theory of an ideal gas, and entropy of dynamical systems.
Abstract: Based on lectures in Erevan, this exposition of ergodic theory contains a rich collection of examples well chosen to introduce the reader to the main themes of the subject. Topics discussed include existence of invariant measures, geodesic flows on Riemannian manifolds, ergodic theory of an ideal gas, and entropy of dynamical systems.

349 citations