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Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday

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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 27160

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Special Issue Editors

Prof. Dr. Ugur Tirnakli

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Department of Physics, Faculty of Arts and Sciences, Izmir University of Economics, 35330 Izmir, Turkey
Interests: statistical physics; dynamical systems; chaotic dynamics; time series analysis
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Christian Beck

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Guest Editor

School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, UK
Interests: stochastic modelling; statistical physics; complex systems; chaotic dynamics

Prof. Dr. Hans J. Herrmann

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1. PMMH, ESPCI, 7 Quai Saint Bernard, 75005 Paris, France
2. Departamento de Fisica, Universidade Federal do Ceará, Fortaleza 60451-970, Ceará, Brazil
Interests: complex systems; critical phenomena; granular matter; statistical physics

Dr. Airton Deppman

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Institute of Physics, The São Paulo University, São Paulo, Brazil
Interests: nuclear reactions; Monte Carlo method; hadron physics; high energy collisions; non-extensive statistic
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Henrik Jeldtoft Jensen

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Centre for Complexity Science and Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Interests: complex systems; statistical mechanics; evolutionary dynamics

Prof. Dr. Evaldo M. F. Curado

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Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
Interests: statistical physics; complex systems; quantum mechanics

Prof. Dr. Fernando D. Nobre

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Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Urca, Rio de Janeiro 22290-180, Brazil
Interests: nonextensive statistical mechanics

Prof. Dr. Angelo Plastino

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Instituto de Física La Plata—CCT-CONICET, Universidad Nacional de La Plata, C.C. 727, La Plata 1900, Argentina
Interests: informaton theory; statistical mechanics; quantum information
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Dr. Astero Provata

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Institute of Nanoscience and Nanotechnology, National Center for Scientific Research “Demokritos”, 15310 Athens, Greece
Interests: non-linear dynamics; statistical physics; computational neuroscience; neuron network dynamics; complex networks; bioinformatics; reactions-diffusion systems; fractals and multifractals

Prof. Dr. Andrea Rapisarda

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1. Dipartimento di Fisica e Astronomia “E. Majorana”, University of Catania, 95123 Catania, Italy
2. Complexity Science Hub Vienna, Josefstädter Straße 39, 1080 Vienna, Austria
Interests: statistical mechanics; complex systems; chaos; complex networks; agent-based models
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to collect original research articles on the most recent research in nonadditive entropies and nonextensive statistical mechanics with their applications in physics and elsewhere, as well as comprehensive review articles covering these topics from a theoretical, experimental, or computational viewpoint.

This generalization of the centennial Boltzmann-Gibbs statistical mechanics and of the entropy upon which it is based were proposed in 1988 and have received, since then, many applications in natural, artificial, and social sciences. The undeniable success of the Boltzmann-Gibbs theory is deeply related to strongly chaotic nonlinear dynamical systems. In particular, for classical systems, the standard requirement is that the maximal Lyapunov exponent is positive. At the edge of chaos, where the maximal Lyapunov exponent vanishes, the need emerges for nonadditive entropies and consistent generalizations of quantities such as the Maxwellian distributions of velocities, the celebrated Boltzmann-Gibbs weight for energies, and Pesin-like identities. This generalized theory has received uncountable validations in complex systems.

Professor Constantino Tsallis has had an outstanding global impact on physics, astrophysics, geophysics, economics, mathematics, and computational sciences, among others. In recognition of his extraordinarily creative and productive scientific life and innumerable contributions to the field of statistical physics of complex systems, this Special Issue is dedicated to him on the occasion of his 80th birthday (5 November 2023).

Prof. Dr. Ugur Tirnakli
Prof. Dr. Christian Beck
Prof. Dr. Hans J. Herrmann
Dr. Airton Deppman
Prof. Dr. Henrik Jeldtoft Jensen
Prof. Dr. Evaldo M. F. Curado
Prof. Dr. Fernando D. Nobre
Prof. Dr. Angelo Plastino
Dr. Astero Provata
Prof. Dr. Andrea Rapisarda
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

nonextensive statistical mechanics
nonadditive entropies
complex systems
long-range interactions
generalized central limit theorem
generalized large deviation theory
dissipative systems
mesoscopic systems
Hamiltonian systems
quantum entanglement
earthquakes
fracture engineering
cosmology
astronomy
solar wind
high-energy particle collisions
plasma physics
information theory
economics

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Published Papers (21 papers)

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11 pages, 5644 KiB

Open AccessArticle

Tsallis q-Statistics Fingerprints in Precipitation Data across Sicily

by Vera Pecorino, Alessandro Pluchino and Andrea Rapisarda

Entropy 2024, 26(8), 623; https://doi.org/10.3390/e26080623 - 24 Jul 2024

Viewed by 1268

Abstract

Precipitation patterns are critical for understanding the hydrological and climatological dynamics of any region. Sicily, the largest island in the Mediterranean sea, with its diverse topography and climatic conditions, serves as an ideal case study for analyzing precipitation data, to gain insights into regional water resources, agricultural productivity, and climate change impacts. This paper employs advanced statistical physics methods, particularly Tsallis q-statistics, to analyze sub-hourly precipitation data from 2002 to 2023, provided by the Sicilian Agrometeorological Informative System (SIAS). We investigate several critical variables related to rainfall events, including duration, depth, maximum record, and inter-event time. The study spans two decades (2002–2012 and 2013–2023), analyzing the distributions of relevant variables. Additionally, we examine the simple returns of these variables to identify significant temporal changes, fitting these returns with q-Gaussian distributions. Our findings reveal the scale-invariant nature of precipitation events, the presence of long-range interactions, and memory effects, characteristic of complex environmental processes. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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12 pages, 636 KiB

Open AccessArticle

The Statistics of q-Statistics

by Deniz Eroglu, Bruce M. Boghosian, Ernesto P. Borges and Ugur Tirnakli

Entropy 2024, 26(7), 554; https://doi.org/10.3390/e26070554 - 28 Jun 2024

Viewed by 950

Abstract

Almost two decades ago, Ernesto P. Borges and Bruce M. Boghosian embarked on the intricate task of composing a manuscript to honor the profound contributions of Constantino Tsallis to the realm of statistical physics, coupled with a concise exploration of q-Statistics. Fast-forward to Constantino Tsallis’ illustrious 80th birthday celebration in 2023, where Deniz Eroglu and Ugur Tirnakli delved into Constantino’s collaborative network, injecting renewed vitality into the project. With hearts brimming with appreciation for Tsallis’ enduring inspiration, Eroglu, Boghosian, Borges, and Tirnakli proudly present this meticulously crafted manuscript as a token of their gratitude. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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12 pages, 996 KiB

Open AccessArticle

Characterizing Complex Spatiotemporal Patterns from Entropy Measures

by Luan Orion Barauna, Rubens Andreas Sautter, Reinaldo Roberto Rosa, Erico Luiz Rempel and Alejandro C. Frery

Entropy 2024, 26(6), 508; https://doi.org/10.3390/e26060508 - 12 Jun 2024

Viewed by 889

Abstract

In addition to their importance in statistical thermodynamics, probabilistic entropy measurements are crucial for understanding and analyzing complex systems, with diverse applications in time series and one-dimensional profiles. However, extending these methods to two- and three-dimensional data still requires further development. In this study, we present a new method for classifying spatiotemporal processes based on entropy measurements. To test and validate the method, we selected five classes of similar processes related to the evolution of random patterns: (i) white noise; (ii) red noise; (iii) weak turbulence from reaction to diffusion; (iv) hydrodynamic fully developed turbulence; and (v) plasma turbulence from MHD. Considering seven possible ways to measure entropy from a matrix, we present the method as a parameter space composed of the two best separating measures of the five selected classes. The results highlight better combined performance of Shannon permutation entropy (

S_{H}^{p}

) and a new approach based on Tsallis Spectral Permutation Entropy (

S_{q}^{s}

). Notably, our observations reveal the segregation of reaction terms in this

S_{H}^{p} \times S_{q}^{s}

space, a result that identifies specific sectors for each class of dynamic process, and it can be used to train machine learning models for the automatic classification of complex spatiotemporal patterns. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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21 pages, 1613 KiB

Open AccessFeature PaperArticle

Non-Thermal Solar Wind Electron Velocity Distribution Function

by Peter H. Yoon, Rodrigo A. López, Chadi S. Salem, John W. Bonnell and Sunjung Kim

Entropy 2024, 26(4), 310; https://doi.org/10.3390/e26040310 - 30 Mar 2024

Viewed by 1241

Abstract

The quiet-time solar wind electrons feature non-thermal characteristics when viewed from the perspective of their velocity distribution functions. They typically have an appearance of being composed of a denser thermal “core” population plus a tenuous energetic “halo” population. At first, such a feature was empirically fitted with the kappa velocity space distribution function, but ever since the ground-breaking work by Tsallis, the space physics community has embraced the potential implication of the kappa distribution as reflecting the non-extensive nature of the space plasma. From the viewpoint of microscopic plasma theory, the formation of the non-thermal electron velocity distribution function can be interpreted in terms of the plasma being in a state of turbulent quasi-equilibrium. Such a finding brings forth the possible existence of a profound inter-relationship between the non-extensive statistical state and the turbulent quasi-equilibrium state. The present paper further develops the idea of solar wind electrons being in the turbulent equilibrium, but, unlike the previous model, which involves the electrostatic turbulence near the plasma oscillation frequency (i.e., Langmuir turbulence), the present paper considers the impact of transverse electromagnetic turbulence, particularly, the turbulence in the whistler-mode frequency range. It is found that the coupling of spontaneously emitted thermal fluctuations and the background turbulence leads to the formation of a non-thermal electron velocity distribution function of the type observed in the solar wind during quiet times. This demonstrates that the whistler-range turbulence represents an alternative mechanism for producing the kappa-like non-thermal distribution, especially close to the Sun and in the near-Earth space environment. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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12 pages, 456 KiB

Open AccessArticle

Tsallis Distribution as a Λ-Deformation of the Maxwell–Jüttner Distribution

by Jean-Pierre Gazeau

Entropy 2024, 26(3), 273; https://doi.org/10.3390/e26030273 - 21 Mar 2024

Viewed by 1139

Abstract

Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as

β

, is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the q [...] Read more.

β

, is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the q-dependent Tsallis distribution can be conceptualized as a de Sitterian deformation of the relativistic Maxwell–Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by

\sqrt{Λ / 3}

, where

Λ

represents the cosmological constant within the

Λ

CDM standard model for cosmology. For a simple gas composed of particles with proper mass m, and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter q exhibits a dependence on the cosmological constant given by

q = 1 + ℓ_{c} \sqrt{Λ} / n

, where

ℓ_{c} = ℏ / m c

is the Compton length of the particle and

n

is a positive numerical factor, the determination of which awaits observational confirmation. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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13 pages, 281 KiB

Open AccessArticle

Group Structure as a Foundation for Entropies

by Henrik Jeldtoft Jensen and Piergiulio Tempesta

Entropy 2024, 26(3), 266; https://doi.org/10.3390/e26030266 - 18 Mar 2024

Cited by 2 | Viewed by 1360

Abstract

Entropy can signify different things. For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced, and it can be difficult to ascertain their respective importance and merits. Here, we consider entropy in an abstract sense, as a functional on a probability space, and we review how being able to handle the trivial case of non-interacting systems, together with the subtle requirement of extensivity, allows for a systematic classification of the functional form. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

9 pages, 288 KiB

Open AccessArticle

Analogies and Relations between Non-Additive Entropy Formulas and Gintropy

by Tamás S. Biró, András Telcs and Antal Jakovác

Entropy 2024, 26(3), 185; https://doi.org/10.3390/e26030185 - 22 Feb 2024

Cited by 1 | Viewed by 1089

Abstract

We explore formal similarities and mathematical transformation formulas between general trace-form entropies and the Gini index, originally used in quantifying income and wealth inequalities. We utilize the notion of gintropy introduced in our earlier works as a certain property of the Lorenz curve drawn in the map of the tail-integrated cumulative population and wealth fractions. In particular, we rediscover Tsallis’ q-entropy formula related to the Pareto distribution. As a novel result, we express the traditional entropy in terms of gintropy and reconstruct further non-additive formulas. A dynamical model calculation of the evolution of Gini index is also presented. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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13 pages, 3459 KiB

Open AccessArticle

A Nonlinear Dynamical View of Kleiber’s Law on the Metabolism of Plants and Animals

by Luis Jovanny Camacho-Vidales and Alberto Robledo

Entropy 2024, 26(1), 32; https://doi.org/10.3390/e26010032 - 28 Dec 2023

Cited by 2 | Viewed by 1572

Abstract

Kleiber’s empirical law, which describes that metabolism increases as the mass to the power

3 / 4

, has arguably remained life sciences’ enigma since its formal uncovering in 1930. Why is this behavior sustained over many orders of magnitude? There have been [...] Read more.

Kleiber’s empirical law, which describes that metabolism increases as the mass to the power

3 / 4

, has arguably remained life sciences’ enigma since its formal uncovering in 1930. Why is this behavior sustained over many orders of magnitude? There have been quantitative rationalizations put forward for both plants and animals based on realistic mechanisms. However, universality in scaling laws of this kind, like in critical phenomena, has not yet received substantiation. Here, we provide an account, with quantitative reproduction of the available data, of the metabolism for these two biology kingdoms by means of broad arguments based on statistical mechanics and nonlinear dynamics. We consider iterated renormalization group (RG) fixed-point maps that are associated with an extensive generalized (Tsallis) entropy. We find two unique universality classes that satisfy the

3 / 4

power law. One corresponds to preferential attachment processes—rich gets richer—and the other to critical processes that suppress the effort for motion. We discuss and generalize our findings to other empirical laws that exhibit similar situations, using data based on general but different concepts that form a conjugate pair that gives rise to the same power-law exponents. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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13 pages, 2598 KiB

Open AccessFeature PaperArticle

First-Principle Validation of Fourier’s Law: One-Dimensional Classical Inertial Heisenberg Model

by Henrique Santos Lima, Constantino Tsallis and Fernando Dantas Nobre

Entropy 2024, 26(1), 25; https://doi.org/10.3390/e26010025 - 25 Dec 2023

Cited by 1 | Viewed by 1588

Abstract

The thermal conductance of a one-dimensional classical inertial Heisenberg model of linear size L is computed, considering the first and last particles in thermal contact with heat baths at higher and lower temperatures,

T_{h}

and

T_{l}

( [...] Read more.

T_{h}

and

T_{l}

(

T_{h} > T_{l}

), respectively. These particles at the extremities of the chain are subjected to standard Langevin dynamics, whereas all remaining rotators (

i = 2, \dots, L - 1

) interact by means of nearest-neighbor ferromagnetic couplings and evolve in time following their own equations of motion, being investigated numerically through molecular-dynamics numerical simulations. Fourier’s law for the heat flux is verified numerically, with the thermal conductivity becoming independent of the lattice size in the limit

L \to \infty

, scaling with the temperature, as

κ (T) \sim T^{- 2.25}

, where

T = (T_{h} + T_{l}) / 2

. Moreover, the thermal conductance,

σ (L, T) \equiv κ (T) / L

, is well-fitted by a function, which is typical of nonextensive statistical mechanics, according to

σ (L, T) = A {exp}_{q} (- B x^{η})

, where A and B are constants,

x = L^{0.475} T

q = 2.28 \pm 0.04

, and

η = 2.88 \pm 0.04

. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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14 pages, 2003 KiB

Open AccessArticle

Results for Nonlinear Diffusion Equations with Stochastic Resetting

by Ervin K. Lenzi, Rafael S. Zola, Michely P. Rosseto, Renio S. Mendes, Haroldo V. Ribeiro, Luciano R. da Silva and Luiz R. Evangelista

Entropy 2023, 25(12), 1647; https://doi.org/10.3390/e25121647 - 12 Dec 2023

Viewed by 1245

Abstract

In this study, we investigate a nonlinear diffusion process in which particles stochastically reset to their initial positions at a constant rate. The nonlinear diffusion process is modeled using the porous media equation and its extensions, which are nonlinear diffusion equations. We use analytical and numerical calculations to obtain and interpret the probability distribution of the position of the particles and the mean square displacement. These results are further compared and shown to agree with the results of numerical simulations. Our findings show that a system of this kind exhibits non-Gaussian distributions, transient anomalous diffusion (subdiffusion and superdiffusion), and stationary states that simultaneously depend on the nonlinearity and resetting rate. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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10 pages, 515 KiB

Open AccessArticle

Centrality and System Size Dependence among Freezeout Parameters and the Implications for EOS and QGP in High-Energy Collisions

by Muhammad Waqas, Abd Haj Ismail, Haifa I. Alrebdi and Muhammad Ajaz

Entropy 2023, 25(12), 1586; https://doi.org/10.3390/e25121586 - 26 Nov 2023

Cited by 1 | Viewed by 1159

Abstract

Utilizing the Modified Hagedorn function with embedded flow, we analyze the transverse momenta (

p_{T}

) and transverse mass (

m_{T}

) spectra of

π^{+}

in Au–Au, Cu–Cu, and d–Au collisions at

\sqrt{s_{N N}}

= 200 GeV across various [...] Read more.

Utilizing the Modified Hagedorn function with embedded flow, we analyze the transverse momenta (

p_{T}

) and transverse mass (

m_{T}

) spectra of

π^{+}

in Au–Au, Cu–Cu, and d–Au collisions at

\sqrt{s_{N N}}

= 200 GeV across various centrality bins. Our study reveals the centrality and system size dependence of key freezeout parameters, including kinetic freezeout temperature

(T_{0})

, transverse flow velocity

(β_{T})

, entropy-related parameter

(n)

, and kinetic freezeout volume (V). Specifically,

T_{0}

and n increase from central to peripheral collisions, while

β_{T}

and V show the opposite trend. These parameters also exhibit system size dependence;

T_{0}

and

β_{T}

are smaller in larger collision systems, whereas V is larger. Importantly, central collisions correspond to a stiffer Equation of State (EOS), characterized by larger

β_{T}

and smaller

T_{0}

, while peripheral collisions indicate a softer EOS. These insights are crucial for understanding the properties of Quark–Gluon Plasma (QGP) and offer valuable constraints for Quantum Chromodynamics (QCD) models at high temperatures and densities. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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12 pages, 313 KiB

Open AccessArticle

Some New Results Involving Past Tsallis Entropy of Order Statistics

by Mansour Shrahili and Mohamed Kayid

Entropy 2023, 25(12), 1581; https://doi.org/10.3390/e25121581 - 24 Nov 2023

Cited by 1 | Viewed by 932

Abstract

This work focuses on exploring the properties of past Tsallis entropy as it applies to order statistics. The relationship between the past Tsallis entropy of an ordered variable in the context of any continuous probability law and the past Tsallis entropy of the ordered variable resulting from a uniform continuous probability law is worked out. For order statistics, this method offers important insights into the characteristics and behavior of the dynamic Tsallis entropy, which is associated with past events. In addition, we investigate how to find a bound for the new dynamic information measure related to the lifetime unit under various conditions and whether it is monotonic with respect to the time when the device is idle. By exploring these properties and also investigating the monotonic behavior of the new dynamic information measure, we contribute to a broader understanding of order statistics and related entropy quantities. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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32 pages, 1215 KiB

Open AccessArticle

Rapidity and Energy Dependencies of Temperatures and Volume Extracted from Identified Charged Hadron Spectra in Proton–Proton Collisions at a Super Proton Synchrotron (SPS)

by Pei-Pin Yang, Fu-Hu Liu and Khusniddin K. Olimov

Entropy 2023, 25(12), 1571; https://doi.org/10.3390/e25121571 - 22 Nov 2023

Cited by 3 | Viewed by 965

Abstract

The standard (Bose–Einstein/Fermi–Dirac, or Maxwell–Boltzmann) distribution from the relativistic ideal gas model is used to study the transverse momentum (

p_{T}

) spectra of identified charged hadrons (

π^{-}

π^{+}

K^{-}

K^{+}

\bar{p}

[...] Read more.

The standard (Bose–Einstein/Fermi–Dirac, or Maxwell–Boltzmann) distribution from the relativistic ideal gas model is used to study the transverse momentum (

p_{T}

) spectra of identified charged hadrons (

π^{-}

π^{+}

K^{-}

K^{+}

\bar{p}

, and p) with different rapidities produced in inelastic proton–proton (

p p

) collisions at a Super Proton Synchrotron (SPS). The experimental data measured using the NA61/SHINE Collaboration at the center-of-mass (c.m.) energies

\sqrt{s} = 6.3

, 7.7, 8.8, 12.3, and 17.3 GeV are fitted well with the distribution. It is shown that the effective temperature (

T_{e f f}

or T), kinetic freeze-out temperature (

T_{0}

), and initial temperature (

T_{i}

) decrease with the increase in rapidity and increase with the increase in c.m. energy. The kinetic freeze-out volume (V) extracted from the

π^{-}

π^{+}

K^{-}

K^{+}

, and

\bar{p}

spectra decreases with the rapidity and increase with the c.m. energy. The opposite tendency of V, extracted from the p spectra, is observed to be increasing with the rapidity and decreasing with the c.m. energy due to the effect of leading protons. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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14 pages, 300 KiB

Open AccessArticle

On Complex Matrix-Variate Dirichlet Averages and Its Applications in Various Sub-Domains

by Princy Thankamani, Nicy Sebastian and Hans J. Haubold

Entropy 2023, 25(11), 1534; https://doi.org/10.3390/e25111534 - 10 Nov 2023

Viewed by 927

Abstract

This paper is about Dirichlet averages in the matrix-variate case or averages of functions over the Dirichlet measure in the complex domain. The classical power mean contains the harmonic mean, arithmetic mean and geometric mean (Hardy, Littlewood and Polya), which is generalized to the y-mean by de Finetti and hypergeometric mean by Carlson; see the references herein. Carlson’s hypergeometric mean averages a scalar function over a real scalar variable type-1 Dirichlet measure, which is known in the current literature as the Dirichlet average of that function. The idea is examined when there is a type-1 or type-2 Dirichlet density in the complex domain. Averages of several functions are computed in such Dirichlet densities in the complex domain. Dirichlet measures are defined when the matrices are Hermitian positive definite. Some applications are also discussed. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

16 pages, 3210 KiB

Open AccessFeature PaperArticle

Tsallis Entropy and Mutability to Characterize Seismic Sequences: The Case of 2007–2014 Northern Chile Earthquakes

by Denisse Pasten, Eugenio E. Vogel, Gonzalo Saravia, Antonio Posadas and Oscar Sotolongo

Entropy 2023, 25(10), 1417; https://doi.org/10.3390/e25101417 - 5 Oct 2023

Cited by 4 | Viewed by 1277

Abstract

Seismic data have improved in quality and quantity over the past few decades, enabling better statistical analysis. Statistical physics has proposed new ways to deal with these data to focus the attention on specific matters. The present paper combines these two progressions to find indicators that can help in the definition of areas where seismic risk is developing. Our data comes from the IPOC catalog for 2007 to 2014. It covers the intense seismic activity near Iquique in Northern Chile during March/April 2014. Centered in these hypocenters we concentrate on the rectangle

L a t_{- 22}^{- 18}

and

L o n_{- 68}^{- 72}

and deepness between 5 and 70 km, where the major earthquakes originate. The analysis was performed using two complementary techniques: Tsallis entropy and mutability (dynamical entropy). Two possible forecasting indicators emerge: (1) Tsallis entropy (mutability) increases (decreases) broadly about two years before the main

M_{W} 8.1

earthquake. (2) Tsallis entropy (mutability) sharply decreases (increases) a few weeks before the

M_{W} 8.1

earthquake. The first one is about energy accumulation, and the second one is because of energy relaxation in the parallelepiped of interest. We discuss the implications of these behaviors and project them for possible future studies. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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16 pages, 5713 KiB

Open AccessArticle

Nonadditive Entropy Application to Detrended Force Sensor Data to Indicate Balance Disorder of Patients with Vestibular System Dysfunction

by Harun Yaşar Köse and Serhat İkizoğlu

Entropy 2023, 25(10), 1385; https://doi.org/10.3390/e25101385 - 27 Sep 2023

Viewed by 1099

Abstract

The healthy function of the vestibular system (VS) is of vital importance for individuals to carry out their daily activities independently and safely. This study carries out Tsallis entropy (TE)-based analysis on insole force sensor data in order to extract features to differentiate between healthy and VS-diseased individuals. Using a specifically developed algorithm, we detrend the acquired data to examine the fluctuation around the trend curve in order to consider the individual’s walking habit and thus increase the accuracy in diagnosis. It is observed that the TE value increases for diseased people as an indicator of the problem of maintaining balance. As one of the main contributions of this study, in contrast to studies in the literature that focus on gait dynamics requiring extensive walking time, we directly process the instantaneous pressure values, enabling a significant reduction in the data acquisition period. The extracted feature set is then inputted into fundamental classification algorithms, with support vector machine (SVM) demonstrating the highest performance, achieving an average accuracy of 95%. This study constitutes a significant step in a larger project aiming to identify the specific VS disease together with its stage. The performance achieved in this study provides a strong motivation to further explore this topic. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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12 pages, 7072 KiB

Open AccessArticle

Effects of Nonextensive Electrons on Dust–Ion Acoustic Waves in a Collisional Dusty Plasma with Negative Ions

by Zhipeng Liu

Entropy 2023, 25(9), 1363; https://doi.org/10.3390/e25091363 - 21 Sep 2023

Cited by 1 | Viewed by 1286

Abstract

The effects of nonextensive electrons on nonlinear ion acoustic waves in dusty negative ion plasmas with ion–dust collisions are investigated. Analytical results show that both solitary and shock waves are supported in this system. The wave propagation is governed by a Korteweg–de Vries Burgers-type equation. The coefficients of this equation are modified by the nonextensive parameter q. Numerical calculations indicate that the amplitude of solitary wave and oscillatory shock can be obviously modified by the nonextensive electrons, but the monotonic shock is little affected. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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11 pages, 577 KiB

Open AccessArticle

Magic Numbers and Mixing Degree in Many-Fermion Systems

by D. Monteoliva, A. Plastino and A. R. Plastino

Entropy 2023, 25(8), 1206; https://doi.org/10.3390/e25081206 - 14 Aug 2023

Viewed by 1001

Abstract

We consider an N fermion system at low temperature T in which we encounter special particle number values

N_{m}

exhibiting special traits. These values arise when focusing attention upon the degree of mixture (DM) of the pertinent quantum states. Given the coupling [...] Read more.

We consider an N fermion system at low temperature T in which we encounter special particle number values

N_{m}

exhibiting special traits. These values arise when focusing attention upon the degree of mixture (DM) of the pertinent quantum states. Given the coupling constant of the Hamiltonian, the DMs stay constant for all N-values but experience sudden jumps at the

N_{m}

. For a quantum state described by the matrix

ρ

, its purity is expressed by

T r ρ^{2}

and then the degree of mixture is given by

1 - T r ρ^{2}

, a quantity that coincides with the entropy

S_{q}

for

q = 2

. Thus, Tsallis entropy of index two faithfully represents the degree of mixing of a state, that is, it measures the extent to which the state departs from maximal purity. Macroscopic manifestations of the degree of mixing can be observed through various physical quantities. Our present study is closely related to properties of many-fermion systems that are usually manipulated at zero temperature. Here, we wish to study the subject at finite temperature. The Gibbs ensemble is appealed to. Some interesting insights are thereby gained. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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11 pages, 318 KiB

Open AccessArticle

Tsallis Entropy of a Used Reliability System at the System Level

by Mohamed Kayid and Mashael A. Alshehri

Entropy 2023, 25(4), 550; https://doi.org/10.3390/e25040550 - 23 Mar 2023

Cited by 5 | Viewed by 1302

Abstract

Measuring the uncertainty of the lifetime of technical systems has become increasingly important in recent years. This criterion is useful to measure the predictability of a system over its lifetime. In this paper, we assume a coherent system consisting of n components and [...] Read more.

t,

all components of the system are alive. We then apply the system signature to determine and use the Tsallis entropy of the remaining lifetime of a coherent system. It is a useful criterion for measuring the predictability of the lifetime of a system. Various results, such as bounds and order properties for the said entropy, are investigated. The results of this work can be used to compare the predictability of the remaining lifetime between two coherent systems with known signatures. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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Review

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22 pages, 16652 KiB

Open AccessReview

Reminiscences of Half a Century of Life in the World of Theoretical Physics

by Constantino Tsallis

Entropy 2024, 26(2), 158; https://doi.org/10.3390/e26020158 - 11 Feb 2024

Cited by 1 | Viewed by 1316

Abstract

Selma Lagerlöf said that culture is what remains when one has forgotten everything we had learned. Without any warranty, through ongoing research tasks, that I will ever attain this high level of wisdom, I simply share here reminiscences that have played, during my life, an important role in my incursions in science, mainly in theoretical physics. I end by presenting some perspectives for future developments. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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15 pages, 553 KiB

Open AccessReview

Open Problems within Nonextensive Statistical Mechanics

by Kenric P. Nelson

Entropy 2024, 26(2), 118; https://doi.org/10.3390/e26020118 - 29 Jan 2024

Cited by 1 | Viewed by 1323

Abstract

Nonextensive statistical mechanics has developed into an important framework for modeling the thermodynamics of complex systems and the information of complex signals. To mark the 80th birthday of the field’s founder, Constantino Tsallis, a review of open problems that can stimulate future research is provided. Over the thirty-year development of NSM, a variety of criticisms have been published ranging from questions about the justification for generalizing the entropy function to the interpretation of the generalizing parameter q. While these criticisms have been addressed in the past and the breadth of applications has demonstrated the utility of the NSM methodologies, this review provides insights into how the field can continue to improve the understanding and application of complex system models. The review starts by grounding q-statistics within scale-shape distributions and then frames a series of open problems for investigation. The open problems include using the degrees of freedom to quantify the difference between entropy and its generalization, clarifying the physical interpretation of the parameter q, improving the definition of the generalized product using multidimensional analysis, defining a generalized Fourier transform applicable to signal processing applications, and re-examining the normalization of nonextensive entropy. This review concludes with a proposal that the shape parameter is a candidate for defining the statistical complexity of a system. Full article

(This article belongs to the Special Issue Nonadditive Entropies and Nonextensive Statistical Mechanics—Dedicated to Professor Constantino Tsallis on the Occasion of His 80th Birthday)

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