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Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Statistical Physics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 29029

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Dr. Dionissios T. Hristopulos

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School of Electrical & Computer Engineering, Technical University of Crete, 73100 Crete, Greece
Interests: statistical physics; space–time statistics; machine learning; hydrology; climate change; brain connectivity
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Dr. Sergio Luiz E. F. da Silva

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Seism Invers & Imaging Grp, Universidade Federal Fluminense, Niteroi BR-24210346, RJ, Brazil
Interests: geophysics; statistical physics; image processing; inverse problems; data analysis
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Dr. Antonio M. Scarfone

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Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Interests: nonextensive statistical mechanics; nonlinear Fokker–Planck equations; geometry information; nonlinear Schroedinger equation; quantum groups and quantum algebras; complex systems
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Dear Colleagues,

Starting with three influential papers twenty years ago [Physica A 296, 405 (2001); Phys. Rev. E 66, 056125 (2002); Phys. Rev. E 72, 036108 (2005)], Giorgio Kaniadakis has pioneered the extension of Boltzmann's Stosszahlansatz (molecular chaos hypothesis) in the framework of special relativity by proposing a new entropy, which emerged as the relativistic generalization of the Boltzmann–Shannon entropy. The Kaniadakis entropy generates power-law tailed statistical distributions, which in the classical limit reduce to the Maxwell–Boltzmann exponential distribution.

This new entropy, also known as κ-entropy or κ-deformed entropy, is considered as one of the most viable candidates for explaining the experimentally observed power-law tailed statistical distributions in various physical, natural, and artificial, complex systems.

Following the introduction of the Kaniadakis entropy, more than 150 statistical physics papers contributed by more than 200 scientists have been published on the subject. Relevant advances have been made in the physical foundations and mathematical formalism of the theory, as well as its applications in statistical physics and thermodynamics, quantum statistics, quantum theory, plasma physics, nuclear fission, particle physics, astrophysics and cosmology, seismology and geophysics, waveform inversion, image processing, machine learning, networks, information theory and statistical sciences, fractal theory, genomics, biophysics, economics, finance, social sciences, and complex systems, among other topics.

The study of the Kaniadakis entropy and related functions is emerging as a rapidly developing research field which attracts a steadily increasing number of researchers from different countries and spans an ever-increasing domain of applications.

This Special Issue aims to collect high-quality review and original research papers, based on statistical physics and related fields, which focus on the Kaniadakis entropy and related probability distributions. The scope of this Special Issue includes papers focusing on mathematical formalism, theoretical foundations, and applications in all fields of science. Contributions that aim to provide synthesis of novel or recent results and/or address future prospects in this field are also welcome.

Dr. Dionissios T. Hristopulos
Dr. Sergio Luiz E. F. da Silva
Dr. Antonio M. Scarfone
Guest Editors

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Published Papers (15 papers)

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22 pages, 391 KiB

Open AccessFeature PaperArticle

Relativistic Roots of κ-Entropy

by Giorgio Kaniadakis

Entropy 2024, 26(5), 406; https://doi.org/10.3390/e26050406 - 7 May 2024

Cited by 1 | Viewed by 1352

Abstract

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 [...] Read more.

The axiomatic structure of the

κ

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. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

14 pages, 326 KiB

Open AccessFeature PaperArticle

Multi-Additivity in Kaniadakis Entropy

by Antonio M. Scarfone and Tatsuaki Wada

Entropy 2024, 26(1), 77; https://doi.org/10.3390/e26010077 - 17 Jan 2024

Cited by 1 | Viewed by 1436

Abstract

It is known that Kaniadakis entropy, a generalization of the Shannon–Boltzmann–Gibbs entropic form, is always super-additive for any bipartite statistically independent distributions. In this paper, we show that when imposing a suitable constraint, there exist classes of maximal entropy distributions labeled by a [...] Read more.

ℵ > 0

that makes Kaniadakis entropy multi-additive, i.e.,

S_{κ} [p^{A \cup B}] = (1 + ℵ) (S_{κ} [p^{A}] + S_{κ} [p^{B}])

, under the composition of two statistically independent and identically distributed distributions

p^{A \cup B} (x, y) = p^{A} (x) p^{B} (y)

, with reduced distributions

p^{A} (x)

and

p^{B} (y)

belonging to the same class. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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13 pages, 340 KiB

Open AccessArticle

Nonlinear Fokker–Planck Equations, H-Theorem and Generalized Entropy of a Composed System

by Luiz R. Evangelista and Ervin K. Lenzi

Entropy 2023, 25(9), 1357; https://doi.org/10.3390/e25091357 - 20 Sep 2023

Cited by 3 | Viewed by 1377

Abstract

We investigate the dynamics of a system composed of two different subsystems when subjected to different nonlinear Fokker–Planck equations by considering the H–theorem. We use the H–theorem to obtain the conditions required to establish a suitable dependence for the system’s interaction that agrees with the thermodynamics law when the nonlinearity in these equations is the same. In this framework, we also consider different dynamical aspects of each subsystem and investigate a possible expression for the entropy of the composite system. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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13 pages, 1538 KiB

Open AccessFeature PaperArticle

Kaniadakis’s Information Geometry of Compositional Data

by Giovanni Pistone and Muhammad Shoaib

Entropy 2023, 25(7), 1107; https://doi.org/10.3390/e25071107 - 24 Jul 2023

Cited by 1 | Viewed by 1289

Abstract

We propose to use a particular case of Kaniadakis’ logarithm for the exploratory analysis of compositional data following the Aitchison approach. The affine information geometry derived from Kaniadakis’ logarithm provides a consistent setup for the geometric analysis of compositional data. Moreover, the affine [...] Read more.

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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21 pages, 3706 KiB

Open AccessArticle

A Graph-Space Optimal Transport Approach Based on Kaniadakis κ-Gaussian Distribution for Inverse Problems Related to Wave Propagation

by Sérgio Luiz E. F. da Silva, João M. de Araújo, Erick de la Barra and Gilberto Corso

Entropy 2023, 25(7), 990; https://doi.org/10.3390/e25070990 - 28 Jun 2023

Cited by 2 | Viewed by 1432

Abstract

Data-centric inverse problems are a process of inferring physical attributes from indirect measurements. Full-waveform inversion (FWI) is a non-linear inverse problem that attempts to obtain a quantitative physical model by comparing the wave equation solution with observed data, optimizing an objective function. However, the FWI is strenuously dependent on a robust objective function, especially for dealing with cycle-skipping issues and non-Gaussian noises in the dataset. In this work, we present an objective function based on the Kaniadakis

κ

-Gaussian distribution and the optimal transport (OT) theory to mitigate non-Gaussian noise effects and phase ambiguity concerns that cause cycle skipping. We construct the

κ

-objective function using the probabilistic maximum likelihood procedure and include it within a well-posed version of the original OT formulation, known as the Kantorovich–Rubinstein metric. We represent the data in the graph space to satisfy the probability axioms required by the Kantorovich–Rubinstein framework. We call our proposal the

κ

-Graph-Space Optimal Transport FWI (

κ

-GSOT-FWI). The results suggest that the

κ

-GSOT-FWI is an effective procedure to circumvent the effects of non-Gaussian noise and cycle-skipping problems. They also show that the Kaniadakis

κ

-statistics significantly improve the FWI objective function convergence, resulting in higher-resolution models than classical techniques, especially when

κ = 0.6

. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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13 pages, 1606 KiB

Open AccessFeature PaperArticle

On the Kaniadakis Distributions Applied in Statistical Physics and Natural Sciences

by Tatsuaki Wada and Antonio Maria Scarfone

Entropy 2023, 25(2), 292; https://doi.org/10.3390/e25020292 - 4 Feb 2023

Cited by 4 | Viewed by 2120

Abstract

Constitutive relations are fundamental and essential to characterize physical systems. By utilizing the

κ

-deformed functions, some constitutive relations are generalized. We here show some applications of the Kaniadakis distributions, based on the inverse hyperbolic sine function, to some topics belonging to the [...] Read more.

Constitutive relations are fundamental and essential to characterize physical systems. By utilizing the

κ

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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17 pages, 349 KiB

Open AccessFeature PaperArticle

Gravity and Cosmology in Kaniadakis Statistics: Current Status and Future Challenges

by Giuseppe Gaetano Luciano

Entropy 2022, 24(12), 1712; https://doi.org/10.3390/e24121712 - 24 Nov 2022

Cited by 27 | Viewed by 2721

Abstract

Kaniadakis statistics is a widespread paradigm to describe complex systems in the relativistic realm. Recently, gravitational and cosmological scenarios based on Kaniadakis (

κ

-deformed) entropy have been considered, leading to generalized models that predict a richer phenomenology comparing to their standard Maxwell–Boltzmann [...] Read more.

Kaniadakis statistics is a widespread paradigm to describe complex systems in the relativistic realm. Recently, gravitational and cosmological scenarios based on Kaniadakis (

κ

-deformed) entropy have been considered, leading to generalized models that predict a richer phenomenology comparing to their standard Maxwell–Boltzmann counterparts. The purpose of the present effort is to explore recent advances and future challenges of Gravity and Cosmology in Kaniadakis statistics. More specifically, the first part of the work contains a review of

κ

-entropy implications on Holographic Dark Energy, Entropic Gravity, Black hole thermodynamics and Loop Quantum Gravity, among others. In the second part, we focus on the study of Big Bang Nucleosynthesis in Kaniadakis Cosmology. By demanding consistency between theoretical predictions of our model and observational measurements of freeze-out temperature fluctuations and primordial abundances of

^{4} H e

and D, we constrain the free

κ

-parameter, discussing to what extent the Kaniadakis framework can provide a successful description of the observed Universe. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

16 pages, 302 KiB

Open AccessArticle

The κ-Deformed Calogero–Leyvraz Lagrangians and Applications to Integrable Dynamical Systems

by Partha Guha

Entropy 2022, 24(11), 1673; https://doi.org/10.3390/e24111673 - 17 Nov 2022

Cited by 2 | Viewed by 1471

Abstract

The Calogero–Leyvraz Lagrangian framework, associated with the dynamics of a charged particle moving in a plane under the combined influence of a magnetic field as well as a frictional force, proposed by Calogero and Leyvraz, has some special features. It is endowed with a Shannon “entropic” type kinetic energy term. In this paper, we carry out the constructions of the 2D Lotka–Volterra replicator equations and the

N = 2

Relativistic Toda lattice systems using this class of Lagrangians. We take advantage of the special structure of the kinetic term and deform the kinetic energy term of the Calogero–Leyvraz Lagrangians using the

κ

-deformed logarithm as proposed by Kaniadakis and Tsallis. This method yields the new construction of the

κ

-deformed Lotka–Volterra replicator and relativistic Toda lattice equations. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

16 pages, 6525 KiB

Open AccessArticle

Doppler Broadening of Neutron Cross-Sections Using Kaniadakis Entropy

by Willian Vieira de Abreu, João Márcio Maciel, Aquilino Senra Martinez, Alessandro da Cruz Gonçalves and Lucas Schmidt

Entropy 2022, 24(10), 1437; https://doi.org/10.3390/e24101437 - 9 Oct 2022

Cited by 2 | Viewed by 2270

Abstract

In the last seven years, Kaniadakis statistics, or

κ

-statistics, have been applied in reactor physics to obtain generalized nuclear data, which can encompass, for instance, situations that lie outside thermal equilibrium. In this sense, numerical and analytical solutions were developed for the [...] Read more.

In the last seven years, Kaniadakis statistics, or

κ

κ

-statistics. However, the accuracy and robustness of the developed solutions contemplating the κ distribution can only be appropriately verified if applied inside an official nuclear data processing code to calculate neutron cross-sections. Hence, the present work inserts an analytical solution for the deformed Doppler broadening cross-section inside the nuclear data processing code FRENDY, developed by the Japan Atomic Energy Agency. To do that, we applied a new computational method called the Faddeeva package, developed by MIT, to calculate error functions present in the analytical function. With this deformed solution inserted in the code, we were able to calculate, for the first time, deformed radiative capture cross-section data for four different nuclides. The usage of the Faddeeva package brought more accurate results when compared to other standard packages, reducing the percentage errors in the tail zone in relation to the numerical solution. The deformed cross-section data agreed with the expected behavior compared to the Maxwell–Boltzmann. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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22 pages, 1284 KiB

Open AccessFeature PaperArticle

Kaniadakis Functions beyond Statistical Mechanics: Weakest-Link Scaling, Power-Law Tails, and Modified Lognormal Distribution

by Dionissios T. Hristopulos and Anastassia Baxevani

Entropy 2022, 24(10), 1362; https://doi.org/10.3390/e24101362 - 26 Sep 2022

Cited by 4 | Viewed by 2801

Abstract

Probabilistic models with flexible tail behavior have important applications in engineering and earth science. We introduce a nonlinear normalizing transformation and its inverse based on the deformed lognormal and exponential functions proposed by Kaniadakis. The deformed exponential transform can be used to generate skewed data from normal variates. We apply this transform to a censored autoregressive model for the generation of precipitation time series. We also highlight the connection between the heavy-tailed

κ

-Weibull distribution and weakest-link scaling theory, which makes the

κ

-Weibull suitable for modeling the mechanical strength distribution of materials. Finally, we introduce the

κ

-lognormal probability distribution and calculate the generalized (power) mean of

κ

-lognormal variables. The

κ

-lognormal distribution is a suitable candidate for the permeability of random porous media. In summary, the

κ

-deformations allow for the modification of tails of classical distribution models (e.g., Weibull, lognormal), thus enabling new directions of research in the analysis of spatiotemporal data with skewed distributions. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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15 pages, 1850 KiB

Open AccessEditor’s ChoiceArticle

A Bayesian Analysis of Plant DNA Length Distribution via κ-Statistics

by Maxsuel M. F. de Lima, Dory H. A. L. Anselmo, Raimundo Silva, Glauber H. S. Nunes, Umberto L. Fulco, Manoel S. Vasconcelos and Vamberto D. Mello

Entropy 2022, 24(9), 1225; https://doi.org/10.3390/e24091225 - 1 Sep 2022

Cited by 5 | Viewed by 1804

Abstract

We report an analysis of the distribution of lengths of plant DNA (exons). Three species of Cucurbitaceae were investigated. In our study, we used two distinct

κ

distribution functions, namely,

κ

-Maxwellian and double-

κ

, to fit the length distributions. To determine [...] Read more.

We report an analysis of the distribution of lengths of plant DNA (exons). Three species of Cucurbitaceae were investigated. In our study, we used two distinct

κ

distribution functions, namely,

κ

-Maxwellian and double-

κ

, to fit the length distributions. To determine which distribution has the best fitting, we made a Bayesian analysis of the models. Furthermore, we filtered the data, removing outliers, through a box plot analysis. Our findings show that the sum of

κ

-exponentials is the most appropriate to adjust the distribution curves and that the values of the

κ

parameter do not undergo considerable changes after filtering. Furthermore, for the analyzed species, there is a tendency for the

κ

parameter to lay within the interval

(0.27; 0.43)

. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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8 pages, 253 KiB

Open AccessFeature PaperArticle

Kaniadakis Entropy Leads to Particle–Hole Symmetric Distribution

by Tamás S. Biró

Entropy 2022, 24(9), 1217; https://doi.org/10.3390/e24091217 - 30 Aug 2022

Cited by 2 | Viewed by 1376

Abstract

We discuss generalized exponentials, whose inverse functions are at the core of generalized entropy formulas, with respect to particle–hole (KMS) symmetry. The latter is fundamental in field theory; so, possible statistical generalizations of the Boltzmann formula-based thermal field theory have to take this [...] Read more.

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

13 pages, 501 KiB

Open AccessArticle

The Longitudinal Plasma Modes of κ-Deformed Kaniadakis Distributed Plasmas Carrying Orbital Angular Momentum

by Ling Tan, Qiaoyun Yang, Hui Chen and Sanqiu Liu

Entropy 2022, 24(9), 1211; https://doi.org/10.3390/e24091211 - 29 Aug 2022

Cited by 3 | Viewed by 1805

Abstract

Based on plasma kinetic theory, the dispersion and Landau damping of Langmuir and ion-acoustic waves carrying finite orbital angular momentum (OAM) were investigated in the

κ

-deformed Kaniadakis distributed plasma system. The results showed that the peculiarities of the investigated subjects relied on [...] Read more.

Based on plasma kinetic theory, the dispersion and Landau damping of Langmuir and ion-acoustic waves carrying finite orbital angular momentum (OAM) were investigated in the

κ

-deformed Kaniadakis distributed plasma system. The results showed that the peculiarities of the investigated subjects relied on the deformation parameter

κ

and OAM parameter

η

. For both Langmuir and ion-acoustic waves, dispersion was enhanced with increased

κ

, while the Landau damping was suppressed. Conversely, both the dispersion and Landau damping were depressed by OAM. Moreover, the results coincided with the straight propagating plane waves in a Maxwellian plasma system when

κ = 0

and

η \to \infty

. It was expected that the present results would give more insight into the trapping and transportation of plasma particles and energy. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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Review

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18 pages, 819 KiB

Open AccessReview

The Kaniadakis Distribution for the Analysis of Income and Wealth Data

by Fabio Clementi

Entropy 2023, 25(8), 1141; https://doi.org/10.3390/e25081141 - 30 Jul 2023

Cited by 1 | Viewed by 1412

Abstract

The paper reviews the “

κ

-generalized distribution”, a statistical model for the analysis of income data. Basic analytical properties, interrelationships with other distributions, and standard measures of inequality such as the Gini index and the Lorenz curve are covered. An extension of [...] Read more.

The paper reviews the “

κ

κ

-generalized model of income/wealth is often in very good agreement with the observed data. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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16 pages, 2242 KiB

Open AccessReview

The Scientific Contribution of the Kaniadakis Entropy to Nuclear Reactor Physics: A Brief Review

by Aquilino Senra Martinez and Willian Vieira de Abreu

Entropy 2023, 25(3), 478; https://doi.org/10.3390/e25030478 - 9 Mar 2023

Cited by 2 | Viewed by 1566

Abstract

In nuclear reactors, tracking the loss and production of neutrons is crucial for the safe operation of such devices. In this regard, the microscopic cross section with the Doppler broadening function is a way to represent the thermal agitation movement in a reactor core. This function usually considers the Maxwell–Boltzmann statistics for the velocity distribution. However, this distribution cannot be applied on every occasion, i.e., in conditions outside the thermal equilibrium. In order to overcome this potential limitation, Kaniadakis entropy has been used over the last seven years to generate generalised nuclear data. This short review article summarises what has been conducted so far and what has to be conducted yet. Full article

(This article belongs to the Special Issue Twenty Years of Kaniadakis Entropy: Current Trends and Future Perspectives)

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