The Symplectic Camel and Poincaré Superrecurrence: Open Problems

Gosson, Maurice de

doi:10.3390/e20070499

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…This rigidity of symplectic maps however, is only applicable to projections of phase space volumes on symplectic 2-planes of the phase space and does not hold for sections of phase space volumes [85]. It may be worth mentioning at this point, that the implications for Statistical Mechanics, if any, of the distinction between symplectic and volume-preserving maps is currently largely unknown [86,87,88]. Addressing this question may have far-reaching consequences for a better understanding of the foundations of Statistical Mechanics especially as it applies to "complex systems" or to systems out of equilibrium.…”

Section: Discussionmentioning

confidence: 99%

Toward a relative q-entropy

Kalogeropoulos

2020

Physica A: Statistical Mechanics and its Applications

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We address the question and related controversy of the formulation of the q-entropy, and its relative entropy counterpart, for models described by continuous (non-discrete) sets of variables. We notice that an L p normalized functional proposed by Lutwak-Yang-Zhang (LYZ), which is essentially a variation of a properly normalized relative Rényi entropy up to a logarithm, has extremal properties that make it an attractive candidate which can be used to construct such a relative q-entropy. We comment on the extremizing probability distributions of this LYZ functional, its relation to the escort distributions, a generalized Fisher information and the corresponding Cramér-Rao inequality. We point out potential physical implications of the LYZ entropic functional and of its extremal distributions.

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Section: Discussionmentioning

confidence: 99%

Toward a relative q-entropy

Kalogeropoulos

2020

Physica A: Statistical Mechanics and its Applications

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“…Another topic which might be worth exploring using the methods outlined in this paper is the study of Poincaré recurrence for subsystems. As we have explained elsewhere [22] the notion of symplectic capacity seems to play a fundamental role in recurrence (it was one of the motivations of Gromov in his study [29] of symplectic non-squeezing properties). It is clear from Theorem 3 that recurrence in the subsystems A and B is liable to occur faster than in the total system A ∪ B.…”

Section: Perspectives and Speculationsmentioning

confidence: 96%

Symplectic Coarse-Grained Classical and Semi-Classical Evolution of Subsystems: New Theoretical Aspects

de Gosson

2020

Preprint

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We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn.The key tool in our study is an extension of Gromov's "principle of the symplectic camel". This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to √ and represent ellipsoids of minimum uncertainty, which we have called "quantum blobs" in previous work.

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“…Maurice De Gosson next introduces the mathematics of Poincare’s recurrence theorem, and the associated notion of ‘superrecurrence’, in relation to the properties of symplectic topology, as applied to quantum mechanics [ 3 ]. De Gosson suggests that these recurrence properties “are closely related to Emergent Quantum Mechanics since they belong to the twilight zone between classical (Hamiltonian) mechanics and its quantization”, and he views these properties “as imprints of the quantum world on classical mechanics in its Hamiltonian formulation”.…”

Section: Quantum Ontology and Foundational Principlesmentioning

confidence: 99%

Emergent Quantum Mechanics: David Bohm Centennial Perspectives

Walleczek¹,

Grössing

Pylkkänen

et al. 2019

Entropy

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Emergent quantum mechanics (EmQM) explores the possibility of an ontology for quantum mechanics. The resurgence of interest in realist approaches to quantum mechanics challenges the standard textbook view, which represents an operationalist approach. The possibility of an ontological, i.e., realist, quantum mechanics was first introduced with the original de Broglie–Bohm theory, which has also been developed in another context as Bohmian mechanics. This Editorial introduces a Special Issue featuring contributions which were invited as part of the David Bohm Centennial symposium of the EmQM conference series (www.emqm17.org). Questions directing the EmQM research agenda are: Is reality intrinsically random or fundamentally interconnected? Is the universe local or nonlocal? Might a radically new conception of reality include a form of quantum causality or quantum ontology? What is the role of the experimenter agent in ontological quantum mechanics? The Special Issue also includes research examining ontological propositions that are not based on the Bohm-type nonlocality. These include, for example, local, yet time-symmetric, ontologies, such as quantum models based upon retrocausality. This Editorial provides topical overviews of thirty-one contributions which are organized into seven categories to provide orientation.

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The Symplectic Camel and Poincaré Superrecurrence: Open Problems

Cited by 5 publications

References 23 publications

Toward a relative q-entropy

Toward a relative q-entropy

Symplectic Coarse-Grained Classical and Semi-Classical Evolution of Subsystems: New Theoretical Aspects

Emergent Quantum Mechanics: David Bohm Centennial Perspectives

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