Weak chaos from Tsallis entropy

Kalogeropoulos, Nikos

doi:10.5339/connect.2012.12

Cited by 13 publications

(17 citation statements)

References 34 publications

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“…In the present work, we attempted to justify why dynamical systems whose statistical behavior is described by the Tsallis entropy, have vanishing largest Lyapunov exponent. This was essentially ascribed to employing the effective negative curvature metric (10), which is the "hyperbolization" of the Euclidean initially employed metric (13), as was pointed out in [12], [42]. Moreover we made the very strong assumption that the additivity properties of the configuration/phase space of the system are directly reflected on its thermodynamic additivity (7) which is not emergent in any non-trivial manner.…”

Section: Discussionmentioning

confidence: 99%

Vanishing largest Lyapunov exponent and Tsallis entropy

Kalogeropoulos

2013

QScience Connect

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We present a geometric argument that explains why some systems having vanishing largest Lyapunov exponent have underlying dynamics aspects of which can be effectively described by the Tsallis entropy. We rely on a comparison of the generalised additivity of the Tsallis entropy versus the ordinary additivity of the BGS entropy. We translate this comparison in metric terms by using an effective hyperbolic metric on the configuration/phase space for the Tsallis entropy versus the Euclidean one in the case of the BGS entropy. Solving the Jacobi equation for such hyperbolic metrics effectively sets the largest Lyapunov exponent computed with respect to the corresponding Euclidean metric to zero. This conclusion is in agreement with all currently known results about systems that have a simple asymptotic behaviour and are described by the Tsallis entropy.Comment: 15 pages, No figures. LaTex2e. Some overlap with arXiv:1104.4869 Additional references and clarifications in this version. To be published in QScience Connec

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Section: Discussionmentioning

confidence: 99%

Vanishing largest Lyapunov exponent and Tsallis entropy

Kalogeropoulos

2013

QScience Connect

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“…A second implication of the logarithmic form of (22), is the following: such a map has the form log P (x), P (x) : polynomial (42) Such logarithmic functions, in general, do not belong to any of the Banach spaces L p (R n ) for two possible reasons: their potential sources of non-integrabilitiy can arise either from the zeroes of P (x) or from the unboundedness of the logarithmic function "at infinity". They do however belong to a more general functional space, which is that of the functions of bounded mean oscillation (BMO) whose definition is as follows [20], [25]: Let f : M → R n be locally integrable and let f B indicate the mean value…”

Section: 3mentioning

confidence: 99%

Long-range interactions, doubling measures and Tsallis entropy

Kalogeropoulos¹

2014

Eur. Phys. J. B

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We present a path toward determining the statistical origin of the thermodynamic limit for systems with long-range interactions. We assume throughout that the systems under consideration have thermodynamic properties given by the Tsallis entropy. We rely on the composition property of the Tsallis entropy for determining effective metrics and measures on their configuration/phase spaces. We point out the significance of Muckenhoupt weights, of doubling measures and of doubling measure-induced metric deformations of the metric. We comment on the volume deformations induced by the Tsallis entropy composition and on the significance of functional spaces for these constructions.

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“…So, in a way, the Tsallis entropy is a "hyperbolic counterpart" of the BGS "Euclidean" entropy. After establishing (10) through a solvable group construction, we have more recently examined some of its implications [17] - [19].…”

Section: Introductionmentioning

confidence: 99%

Almost additive entropy

Kalogeropoulos

2014

Int. J. Geom. Methods Mod. Phys.

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We explore consequences of a hyperbolic metric induced by the composition property of the Harvda-Charvat/Daróczy/Cressie-Read/Tsallis entropy. We address the special case of systems described by small deviations of the non-extensive parameter q ≈ 1 from the "ordinary" additive case which is described by the Boltzmann/Gibbs/Shannon entropy. By applying the Gromov/Ruh theorem for almost flat manifolds, we show that such systems have a power-law rate of expansion of their configuration/phase space volume. We explore the possible physical significance of some geometric and topological results of this approach.

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Weak chaos from Tsallis entropy

Cited by 13 publications

References 34 publications

Vanishing largest Lyapunov exponent and Tsallis entropy

Vanishing largest Lyapunov exponent and Tsallis entropy

Long-range interactions, doubling measures and Tsallis entropy

Almost additive entropy

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