Beyond the Shannon–Khinchin formulation: The composability axiom and the universal-group entropy

Tempesta, Piergiulio

doi:10.1016/j.aop.2015.08.013

Cited by 40 publications

(113 citation statements)

References 49 publications

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“…Also, when composing a system with another in a certainty state (zero entropy), the entropy of the composed system will remain unchanged. This natural generalization of the SK axiom allows an infinite number of admissible entropic forms [25,26].…”

Section: S(a ∪ B) = (S(a)s(b))mentioning

confidence: 99%

“…Second, along the lines of [25,26], we discuss an axiomatic formulation of entropy, generalizing the fourth SK axiom and requiring, instead of additivity, the composability property [31]. In this setting, we postulate that the entropy of a composite system be a function of the entropy of the two components only.…”

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confidence: 99%

“…The physical root of this group structure relies in a recent generalization of the classical axiomatic formulation [25,26], originally proposed by Shannon and by Khinchin to characterize the BG entropy. The first three postulates, nowadays called the Shannon-Khinchin (SK) axioms [1,2], define some essential properties that any entropic functional S should satisfy.…”

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confidence: 99%

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Groups, information theory, and Einstein's likelihood principle

Sicuro

Tempesta

2016

Phys. Rev. E

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We propose a unifying picture where the notion of generalized entropy is related to information theory by means of a group-theoretical approach. The group structure comes from the requirement that an entropy be well defined with respect to the composition of independent systems, in the context of a recently proposed generalization of the Shannon-Khinchin axioms. We associate to each member of a large class of entropies a generalized information measure, satisfying the additivity property on a set of independent systems as a consequence of the underlying group law. At the same time, we also show that Einstein's likelihood function naturally emerges as a byproduct of our informational interpretation of (generally nonadditive) entropies. These results confirm the adequacy of composable entropies both in physical and social science contexts. DOI: 10.1103/PhysRevE.93.040101 The study of the relations among statistical mechanics, information theory and the notion of entropy is at the heart of the science of complexity, and in the last decades has been widely explored. After the seminal works of Shannon and Tsallis [7], has been the prototype of the nonadditive entropies studied in the last decades [8][9][10][11][12][13]. These functionals are generalizations of the BG entropy; they depend on one or more parameters, in such a way that the BG entropy is recovered as a particular limit. Generalized entropies have been successfully adopted for the study of both classical and quantum systems. Rényi's entropy, for example, plays a central role in information theory and in the study of multifractality [14]; along with the von Neumann entropy, it has been also used extensively in the evaluation of the entanglement entropy of quantum systems [15][16][17][18].The study of new entropic forms has led to a new flow of ideas regarding the old problem of the probabilistic versus the dynamical foundations of the notion of entropy. It is well known [19] that Einstein's approach was very different with respect to the probabilistic methodology of Boltzmann (which eventually emerged as the predominant one). Indeed, Einstein argued that the probabilities of occupation of the various regions of the phase space associated with a physical system cannot be postulated a priori. Instead, only a knowledge of * sicuro@cbpf.br † p.tempesta@fis.ucm.es dynamics, obtained by solving the equations of motion, could provide this information. For this reason, in his theory of fluctuations, Einstein [20] introduced the likelihood function W ∝ exp (S BG ) as a fundamental statistical quantity (for the sake of simplicity, here and in the following, we put k B ≡ 1, k B being the Boltzmann constant). He observed that by composing two independent systems A and B, the fundamental relationholds. Equation (1) is epistemologically crucial: it expresses the fact that the physical description of system A does not depend on the physical description of system B, and vice versa. Moreover, it is related to the additivity requirement of the information content of ind...

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Section: S(a ∪ B) = (S(a)s(b))mentioning

confidence: 99%

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confidence: 99%

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Groups, information theory, and Einstein's likelihood principle

Sicuro

Tempesta

2016

Phys. Rev. E

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“…The general theory of non-additive thermodynamics has also advanced significantly in the past few decades (see e.g. [12,35] and the references therein), and it has been shown, that by relaxing the additivity requirement in the axiomatic approach to the entropy definition (given by Shannon [36,37] and Khinchin [38]) to the weaker composability requirement, new possible functional forms of the entropy may arise [39]. As a consequence, there exist certain parametric extensions of the Boltzmann-Gibbs statistical entropy formula, which seem to be more appropriate to describe systems with long-range type interactions.…”

Section: Introductionmentioning

confidence: 99%

Thermodynamics, stability and Hawking–Page transition of Kerr black holes from Rényi statistics

Czinner

Iguchi

2017

Eur. Phys. J. C

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Thermodynamics of rotating black holes described by the Rényi formula as equilibrium and zeroth law compatible entropy function is investigated. We show that similarly to the standard Boltzmann approach, isolated Kerr black holes are stable with respect to axisymmetric perturbations in the Rényi model. On the other hand, when the black holes are surrounded by a bath of thermal radiation, slowly rotating black holes can also be in stable equilibrium with the heat bath at a fixed temperature, in contrast to the Boltzmann description. For the question of possible phase transitions in the system, we show that a HawkingPage transition and a first order small black hole/large black hole transition occur, analogous to the picture of rotating black holes in AdS space. These results confirm the similarity between the Rényi-asymptotically flat and Boltzmann-AdS approaches to black hole thermodynamics in the rotating case as well. We derive the relations between the thermodynamic parameters based on this correspondence.

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“…[31], is to obtain generalized entropy expressions through the underlying group-theoretical structure [32][33][34]. The general entropy expression of this universal group character includes many known trace-form entropy measures such as Boltzmann-Gibbs, Tsallis [1], Kaniadakis [3] and BorgesRoditi [35] entropies.…”

Section: Introductionmentioning

confidence: 99%

Group theory, entropy and the third law of thermodynamics

2017

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Curado et al. [Ann. Phys. 366 (2016) 22] have recently studied the axiomatic structure and the universality of a three-parameter trace-form entropy inspired by the group-theoretical structure. In this work, we study the group-theoretical entropy S a,b,r in the context of the third law of thermodynamics where the parameters {a, b, r} are all independent. We show that this threeparameter entropy expression can simultaneously satisfy the third law of thermodynamics and the three Khinchin axioms, namely continuity, concavity and expansibility only when the parameter b is set to zero. In other words, it is thermodynamically valid only as a two-parameter generalization Sa,r. Moreover, the restriction set by the third law i.e., the condition b = 0, is important in the sense that the so obtained two-parameter group-theoretical entropy becomes extensive only when this condition is met. We also illustrate the interval of validity of the third law using the one-dimensional Ising model with no external field. Finally, we show that the Sa,r is in the same universality class as that of the Kaniadakis entropy for 0 < r < 1 while it has a distinct universality class in the interval −1 < r < 0.

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Beyond the Shannon–Khinchin formulation: The composability axiom and the universal-group entropy

Cited by 40 publications

References 49 publications

Groups, information theory, and Einstein's likelihood principle

Groups, information theory, and Einstein's likelihood principle

Thermodynamics, stability and Hawking–Page transition of Kerr black holes from Rényi statistics

Group theory, entropy and the third law of thermodynamics

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