On Tsallis and Kaniadakis Divergences

In this paper we recall, extend and compute some information measures for the concomitants of the generalized order statistics (GOS) from the Farlie–Gumbel–Morgenstern (FGM) family. We focus on two types of information measures: some related to Shannon entropy, and some related to Tsallis entropy. Among the information measures considered are residual and past entropies which are important in a reliability context.

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“…We notice that this divergence is the divergence considered in [ 36 ], with

, it is the divergence analyzed in [ 56 ], with

, and it is Tsallis Relative Entropy from [ 57 ].…”

Section: Information Measures For the Concomitants From Fgm Family Ne...mentioning

confidence: 91%

Some Information Measures Properties of the GOS-Concomitants from the FGM Family

Suter

Cernat

Drăgan

2022

Entropy

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“…For more information about and applications of the Tsallis entropy, we recommend [8,[18][19][20][21][22][23][24][25].…”

Section: Preliminariesmentioning

confidence: 99%

“…The reader can find other results and applications concerning the Kaniadakis entropy in [23,[26][27][28][29][30][31].…”

Section: Preliminariesmentioning

confidence: 99%

Discrete Entropies of Chebyshev Polynomials

Sfetcu,

Preda

2024

Mathematics

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Because of its flexibility and multiple meanings, the concept of information entropy in its continuous or discrete form has proven to be very relevant in numerous scientific branches. For example, it is used as a measure of disorder in thermodynamics, as a measure of uncertainty in statistical mechanics as well as in classical and quantum information science, as a measure of diversity in ecological structures and as a criterion for the classification of races and species in population dynamics. Orthogonal polynomials are a useful tool in solving and interpreting differential equations. Lately, this subject has been intensively studied in many areas. For example, in statistics, by using orthogonal polynomials to fit the desired model to the data, we are able to eliminate collinearity and to seek the same information as simple polynomials. In this paper, we consider the Tsallis, Kaniadakis and Varma entropies of Chebyshev polynomials of the first kind and obtain asymptotic expansions. In the particular case of quadratic entropies, there are given concrete computations.

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“…In the last two decades, various authors have devoted themselves to the study of both the theoretical foundations of the theory and its applications not only in plasma physics but also in various other areas of the science of complex physical, natural, or artificial statistical systems. Some of these works deal with the H-theorem and the molecular chaos hypothesis [14,15], thermodynamic stability [16,17], Lesche stability [18][19][20][21], the Legendre structure of the resulting thermodynamics [22,23], the thermodynamics of nonequilibrium systems [24], quantum versions of the theory [25][26][27][28], the geometric structure of the theory [29,30], various mathematical aspects of the theory [31][32][33][34][35][36][37][38][39][40][41][42][43][44], etc. On the other hand, specific applications to physical systems have been considered, e.g., cosmic rays [5], relativistic [45] and classical [46] plasmas in presence of external electromagnetic fields, relaxation in relativistic plasmas under wave-particle interactions [47,48], electronic cooling [49], dark energy models , quantum gravity [71][72][73][74]…”

Section: Introductionmentioning

confidence: 99%

Relativistic Roots of κ-Entropy

Kaniadakis

2024

Entropy

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The axiomatic structure of the κ-statistcal theory is proven. In addition to the first three standard Khinchin–Shannon axioms of continuity, maximality, and expansibility, two further axioms are identified, namely the self-duality axiom and the scaling axiom. It is shown that both the κ-entropy and its special limiting case, the classical Boltzmann–Gibbs–Shannon entropy, follow unambiguously from the above new set of five axioms. It has been emphasized that the statistical theory that can be built from κ-entropy has a validity that goes beyond physics and can be used to treat physical, natural, or artificial complex systems. The physical origin of the self-duality and scaling axioms has been investigated and traced back to the first principles of relativistic physics, i.e., the Galileo relativity principle and the Einstein principle of the constancy of the speed of light. It has been shown that the κ-formalism, which emerges from the κ-entropy, can treat both simple (few-body) and complex (statistical) systems in a unified way. Relativistic statistical mechanics based on κ-entropy is shown that preserves the main features of classical statistical mechanics (kinetic theory, molecular chaos hypothesis, maximum entropy principle, thermodynamic stability, H-theorem, and Lesche stability). The answers that the κ-statistical theory gives to the more-than-a-century-old open problems of relativistic physics, such as how thermodynamic quantities like temperature and entropy vary with the speed of the reference frame, have been emphasized.

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On Tsallis and Kaniadakis Divergences

Cited by 9 publications

References 35 publications

Some Information Measures Properties of the GOS-Concomitants from the FGM Family

Some Information Measures Properties of the GOS-Concomitants from the FGM Family

Discrete Entropies of Chebyshev Polynomials

Relativistic Roots of κ-Entropy

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