The world according to Rényi: thermodynamics of multifractal systems

Jizba, Petr; Arimitsu, Toshihico

doi:10.1016/j.aop.2004.01.002

Cited by 255 publications

(310 citation statements)

References 43 publications

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“…It should be noted how P. Jizba and T.Arimitsu (5) showed that Rényi's measure can be obtained also extending the Shannon-Khinchin axioms to a quasi-linear conditional information 12) with f as given in 1.10.…”

Section: Rényi's Entropymentioning

confidence: 99%

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A step beyond Tsallis and Rényi entropies

2005

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J., B 30, 543-549) that provides one possible unification. We will show how this generalization that unifies Rényi and Tsallis entropy in a coherent picture naturally comes into being if the q-formalism of generalized logarithm and exponential functions is used, how together with Sharma-Mittal's measure another possible extension emerges which however does not obey a pseudo-additive law and lacks of other properties relevant for a generalized thermostatistics, and how the relation between all these information measures is best understood when described in terms of a particular logarithmic Kolmogorov-Nagumo average.

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Section: Rényi's Entropymentioning

confidence: 99%

“…Its inverse is the generalized q-exponential function 5) which becomes the exponential function for q = 1. The importance of the q-logarithm is that it satisfies a pseudo-additive law log q xy = log q x + log q y + (1 − q)(log q x)(log q y) .…”

Section: Tsallis' Entropymentioning

confidence: 99%

A step beyond Tsallis and Rényi entropies

2005

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“…In fact, when entropies I q or S q are used, it is necessary to know them for all q in order to obtain a full information on a given statistical system [35]. For ensuing applications in strange attractors the reader may consult Ref.…”

Section: For a Givenmentioning

confidence: 99%

“…Typical examples are provided by chaotic dynamical systems and multifractal statistical systems (see e.g., [35] and citations therein). Fully developed turbulence, earthquake analysis and generalized dimensions of strange attractors provide examples.…”

Section: Rény's Entropy: Entropy Of Multifractal Systemsmentioning

confidence: 99%

On q-non-extensive statistics with non-Tsallisian entropy

Jizba

Korbel

2016

Physica A: Statistical Mechanics and its Applications

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We combine an axiomatics of Rényi with the q-deformed version of Khinchin axioms to obtain a measure of information (i.e., entropy) which accounts both for systems with embedded self-similarity and non-extensivity. We show that the entropy thus obtained is uniquely solved in terms of a one-parameter family of information measures. The ensuing maximal-entropy distribution is phrased in terms of a special function known as the Lambert W-function. We analyze the corresponding "high" and "lowtemperature" asymptotics and reveal a non-trivial structure of the parameter space.

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“…Napier's logarithm assures additivity while (18) imposes the exponential average. It should be noted how P. Jizba and T.Arimitsu [18] showed that Renyi's measure can be obtained also from the extension of the Shannon-Khinchin axioms [19] to a quasi-linear conditional information…”

Section: How To Construct Generalized Kn-averaged Information-entropymentioning

confidence: 99%

On the extended Kolmogorov–Nagumo information-entropy theory, the duality and its possible implications for a non-extensive two-dimensional Ising model

Masi¹

2007

Physica A: Statistical Mechanics and its Applications

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The aim of this paper is to investigate the q → 1 q duality in an information-entropy theory of all q-generalized entropy functionals (Tsallis, Renyi and Sharma-Mittal measures) in the light of a representation based on generalized exponential and logarithm functions subjected to Kolmogorov's and Nagumo's averaging. We show that it is precisely in this representation that the form invariance of all entropy functionals is maintained under the action of this duality. The generalized partition function also results to be a scalar invariant under the q → 1 q transformation which can be interpreted as a non-extensive two dimensional Ising model duality between systems governed by two different power law long-range interactions and temperatures. This does not hold only for Tsallis statistics, but is a characteristic feature of all stationary distributions described by q-exponential Boltzmann factors.

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The world according to Rényi: thermodynamics of multifractal systems

Cited by 255 publications

References 43 publications

A step beyond Tsallis and Rényi entropies

A step beyond Tsallis and Rényi entropies

On q-non-extensive statistics with non-Tsallisian entropy

On the extended Kolmogorov–Nagumo information-entropy theory, the duality and its possible implications for a non-extensive two-dimensional Ising model

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