Generalized Shannon–Khinchin axioms and uniqueness theorem for pseudo-additive entropies

Ilić, Vladimir; Stanković, Miomir

doi:10.1016/j.physa.2014.05.009

Cited by 35 publications

(36 citation statements)

References 14 publications

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“…Hanel and Thurner [8,9] classified the entropies according to their asymptotic scaling, Tempesta [10] studied the generalized entropies according to group properties, Biró and Barnaf [11] derived a new class of entropies from its interaction with heat reservoir. Ilić and Stanković [12] classified the pseudo-additive entropies by generalization of Khinchin axioms.…”

Section: Introductionmentioning

confidence: 99%

On statistical properties of Jizba–Arimitsu hybrid entropy

Çankaya

Korbel

2017

Physica A: Statistical Mechanics and its Applications

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Jizba-Arimitsu entropy (also called hybrid entropy) combines axiomatics of Rényi and Tsallis entropy. It has many common properties with them, on the other hand, some aspects as e.g., MaxEnt distributions, are completely different from the former two entropies. In this paper, we demonstrate the statistical properties of hybrid entropy, including the definition of hybrid entropy for continuous distributions, its relation to discrete entropy and calculation of hybrid entropy for some well-known distributions. Additionally, definition of hybrid divergence and its connection to Fisher metric is also discussed. Interestingly, the main properties of continuous hybrid entropy and hybrid divergence are completely different from measures based on Rényi and Tsallis entropy. This motivates us to introduce average hybrid entropy, which can be understood as an average between Tsallis and Rényi entropy

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Section: Introductionmentioning

confidence: 99%

On statistical properties of Jizba–Arimitsu hybrid entropy

Çankaya

Korbel

2017

Physica A: Statistical Mechanics and its Applications

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“…By employing the above f q -mapping we will simplify number of cumbersome mathematical steps and stay comparatively close to the reasonings used in the Ilić-Stanković article [1]. Under the f q -mapping the respective Aczél-Daróczy entropies read…”

Section: A) Additivity Rule -Independent Eventsmentioning

confidence: 78%

“…In their recent paper [1], Ilić and Stanković have suggested (in Remark 5.3) that there may be problem for the class of hybrid entropies introduced in [2] and elaborated in our recent paper [3]. In particular, they argued that the entropies in question do not satisfy the fourth axiom in generalized Shannon-Khinchin axioms (J-A axioms) introduced in [2].…”

Section: Introductionmentioning

confidence: 90%

Remarks on “Comments on ‘On q-non-extensive statistics with non-Tsallisian entropy’ ” [Physica A 466 (2017) 160]

Jizba¹,

Korbel²

2017

Physica A: Statistical Mechanics and its Applications

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Recently in [Physica A 411 (2014) 138] Ilić and Stanković have suggested that there may be problem for the class of hybrid entropies introduced in [P. Jizba and T. Arimitsu, Physica A 340 (2004) 110]. In this Comment we point out that the problem can be traced down to the q-additive entropic chain rule and to a peculiar behavior of the DeFinetti-Kolmogorov relation for escort distributions. However, despite this, one can still safely use the proposed hybrid entropies in most of the statistical-thermodynamics considerations.

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“…The latest property states that the entropy of a joint system can be represented as sum of the entropy of one system and the (generalized) conditional entropy of another, with respect to the first one [25]. An important generalization of Rényi entropies is the class of strongly pseudo-additive (SPA) entropies H α which can be represented as the h transformation of Rényi entropy [1]:…”

Section: Strongly Pseudo-additive Entropymentioning

confidence: 99%

“…, q m|i ) ∈ ∆ m , where q j|i = r i j /p i and q ∈ R + is some fixed parameter. Then, the strong pseudo-additivity states that [1]…”

Section: Strongly Pseudo-additive Entropymentioning

confidence: 99%

Equivalence between four versions of thermostatistics based on strongly pseudoadditive entropies

2019

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The class of strongly pseudo-additive entropies, which can be represented as an increasing continuous transformation of Shannon and Rényi entropies, have intensively been studied in previous decades. Although their mathematical structure has thoroughly been explored and established by generalized Shannon-Khinchin axioms, the analysis of their thermostatistical properties have mostly been limited to special cases which belong to two parameter Sharma-Mittal entropy class, such as Tsallis, Renyi and Gaussian entropies.In this paper we present a general analysis of the strongly pseudo-additive entropies thermostatistics by taking into account both linear and escort constraints on internal energy. We develop two types of dualities between the thermostatistics formalisms. By the first one, the formalism of Rényi entropy is transformed in the formalism of SPA entropy under general energy constraint and, by the second one, the generalized thermostatistics which corresponds to the linear constraint is transformed into the one which corresponds to the escort constraint. Thus, we establish the equivalence between four different thermostatistics formalisms based on Rényi and SPA entropies coupled with linear and escort constraints and we provide the transformation formulas. In this way we obtain a general framework which is applicable to the wide class of entropies and constraints previously discussed in the literature.As an example, we rederive maximum entropy distributions for Sharma-Mittal entropy and we establish new relationships between the corresponding thermodynamic potentials. We obtain, as special cases, previously developed expressions for maximum entropy distributions and thermodynamic quantities for Tsallis, Rényi and Gaussian entropies. In addition, the results are applied for derivation of thermostatistical relationships for supra-extensive entropy, which has not previously been considered.

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Generalized Shannon–Khinchin axioms and uniqueness theorem for pseudo-additive entropies

Cited by 35 publications

References 14 publications

On statistical properties of Jizba–Arimitsu hybrid entropy

On statistical properties of Jizba–Arimitsu hybrid entropy

Remarks on “Comments on ‘On q-non-extensive statistics with non-Tsallisian entropy’ ” [Physica A 466 (2017) 160]

Equivalence between four versions of thermostatistics based on strongly pseudoadditive entropies

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