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Non-additive Entropy Formulas: Motivation and Derivations

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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 (15 May 2023) | Viewed by 15945

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Prof. Dr. Tamás Sándor Bíró

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Guest Editor

1. Wigner Research Centre for Physics, Konkoly-Thege M. 29-33, 1121 Budapest, Hungary
2. Department of Physics, Babeş-Bolyai University, Str. M. Kogalniceanu 1, 400084 Cluj-Napoca, Romania
Interests: biophysics; quark-gluon plasma; hadron dynamics; statistical models; entropy; confinement

Dr. Airton Deppman

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Guest Editor

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

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Dear Colleagues,

The concept of Entropy has been changing swiftly since C. Tsallis and before him a series of mathematicians and informatics experts have proposed various generalizations of the Boltzmann-Gibbs entropy. The most distinguishing property of these formulas is their non-additivity for factorizing probabilities.

A non-additive entropy formula motivates investigations in many directions. Some works considered the theoretical and axiomatic implications of the new entropy. Others proposed physical mechanisms that could lead to non-additive entropy. Evidence of the applicability of generalized thermodynamics appears in many branches of Physics and in other fields, such as biological systems, socio-economics environments, information theory, and complex networks, among many others.

This Special Issue is dedicated to reviewing those developments, sharing new results and opening new perspectives to the advancement of our knowledge about Entropy. We welcome contributions on topics such as:

1) Implications and applications of the non-additive entropy;

2) Advances in theoretical aspects of entropy;

3) Mathematical aspects of the non-additive entropy;

4) Thermodynamical consequences of the generalized entropy;

5) New applications of the non-additive entropy;

6) Origins of the non-additivity in the entropy of complex systems.

Prof. Dr. Tamás Sándor Bíró
Dr. Airton Deppman
Guest Editors

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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.

Published Papers (12 papers)

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Editorial

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2 pages, 163 KiB

Open AccessEditorial

Non-Additive Entropy Formulas: Motivation and Derivations

by Tamás Sándor Biró and Airton Deppman

Entropy 2023, 25(8), 1203; https://doi.org/10.3390/e25081203 - 13 Aug 2023

Cited by 1 | Viewed by 685

Abstract

Entropy is a great tool in thermodynamics and statistical physics [...] Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

Research

Jump to: Editorial, Review

16 pages, 366 KiB

Open AccessFeature PaperArticle

Non-Additive Entropic Forms and Evolution Equations for Continuous and Discrete Probabilities

by Evaldo M. F. Curado and Fernando D. Nobre

Entropy 2023, 25(8), 1132; https://doi.org/10.3390/e25081132 - 27 Jul 2023

Viewed by 651

Abstract

Increasing interest has been shown in the subject of non-additive entropic forms during recent years, which has essentially been due to their potential applications in the area of complex systems. Based on the fact that a given entropic form should depend only on a set of probabilities, its time evolution is directly related to the evolution of these probabilities. In the present work, we discuss some basic aspects related to non-additive entropies considering their time evolution in the cases of continuous and discrete probabilities, for which nonlinear forms of Fokker–Planck and master equations are considered, respectively. For continuous probabilities, we discuss an H-theorem, which is proven by connecting functionals that appear in a nonlinear Fokker–Planck equation with a general entropic form. This theorem ensures that the stationary-state solution of the Fokker–Planck equation coincides with the equilibrium solution that emerges from the extremization of the entropic form. At equilibrium, we show that a Carnot cycle holds for a general entropic form under standard thermodynamic requirements. In the case of discrete probabilities, we also prove an H-theorem considering the time evolution of probabilities described by a master equation. The stationary-state solution that comes from the master equation is shown to coincide with the equilibrium solution that emerges from the extremization of the entropic form. For this case, we also discuss how the third law of thermodynamics applies to equilibrium non-additive entropic forms in general. The physical consequences related to the fact that the equilibrium-state distributions, which are obtained from the corresponding evolution equations (for both continuous and discrete probabilities), coincide with those obtained from the extremization of the entropic form, the restrictions for the validity of a Carnot cycle, and an appropriate formulation of the third law of thermodynamics for general entropic forms are discussed. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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10 pages, 293 KiB

Open AccessArticle

Fractal Derivatives, Fractional Derivatives and q-Deformed Calculus

by Airton Deppman, Eugenio Megías and Roman Pasechnik

Entropy 2023, 25(7), 1008; https://doi.org/10.3390/e25071008 - 30 Jun 2023

Cited by 6 | Viewed by 1366

Abstract

This work presents an analysis of fractional derivatives and fractal derivatives, discussing their differences and similarities. The fractal derivative is closely connected to Haussdorff’s concepts of fractional dimension geometry. The paper distinguishes between the derivative of a function on a fractal domain and the derivative of a fractal function, where the image is a fractal space. Different continuous approximations for the fractal derivative are discussed, and it is shown that the q-calculus derivative is a continuous approximation of the fractal derivative of a fractal function. A similar version can be obtained for the derivative of a function on a fractal space. Caputo’s derivative is also proportional to a continuous approximation of the fractal derivative, and the corresponding approximation of the derivative of a fractional function leads to a Caputo-like derivative. This work has implications for studies of fractional differential equations, anomalous diffusion, information and epidemic spread in fractal systems, and fractal geometry. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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9 pages, 457 KiB

Open AccessArticle

A Note on the Connection between Non-Additive Entropy and h-Derivative

by Jin-Wen Kang, Ke-Ming Shen and Ben-Wei Zhang

Entropy 2023, 25(6), 918; https://doi.org/10.3390/e25060918 - 09 Jun 2023

Cited by 1 | Viewed by 843

Abstract

In order to study as a whole a wide part of entropy measures, we introduce a two-parameter non-extensive entropic form with respect to the h-derivative, which generalizes the conventional Newton–Leibniz calculus. This new entropy,

S_{h, h^{'}}

, is proved [...] Read more.

S_{h, h^{'}}

, is proved to describe the non-extensive systems and recover several types of well-known non-extensive entropic expressions, such as the Tsallis entropy, the Abe entropy, the Shafee entropy, the Kaniadakis entropy and even the classical Boltzmann–Gibbs one. As a generalized entropy, its corresponding properties are also analyzed. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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21 pages, 3897 KiB

Open AccessArticle

Generalizing the Wells–Riley Infection Probability: A Superstatistical Scheme for Indoor Infection Risk Estimation

by Markos N. Xenakis

Entropy 2023, 25(6), 896; https://doi.org/10.3390/e25060896 - 02 Jun 2023

Cited by 1 | Viewed by 1822

Abstract

Recent evidence supports that air is the main transmission pathway of the recently identified SARS-CoV-2 coronavirus that causes COVID-19 disease. Estimating the infection risk associated with an indoor space remains an open problem due to insufficient data concerning COVID-19 outbreaks, as well as, methodological challenges arising from cases where environmental (i.e., out-of-host) and immunological (i.e., within-host) heterogeneities cannot be neglected. This work addresses these issues by introducing a generalization of the elementary Wells-Riley infection probability model. To this end, we adopted a superstatistical approach where the exposure rate parameter is gamma-distributed across subvolumes of the indoor space. This enabled us to construct a susceptible (S)–exposed (E)–infected (I) dynamics model where the Tsallis entropic index q quantifies the degree of departure from a well-mixed (i.e., homogeneous) indoor-air-environment state. A cumulative-dose mechanism is employed to describe infection activation in relation to a host’s immunological profile. We corroborate that the six-foot rule cannot guarantee the biosafety of susceptible occupants, even for exposure times as short as 15 min. Overall, our work seeks to provide a minimal (in terms of the size of the parameter space) framework for more realistic indoor

S E I

dynamics explorations while highlighting their Tsallisian entropic origin and the crucial yet elusive role that the innate immune system can play in shaping them. This may be useful for scientists and decision makers interested in probing different indoor biosafety protocols more thoroughly and comprehensively, thus motivating the use of nonadditive entropies in the emerging field of indoor space epidemiology. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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14 pages, 5814 KiB

Open AccessArticle

Necessary Condition of Self-Organisation in Nonextensive Open Systems

by Ozgur Afsar and Ugur Tirnakli

Entropy 2023, 25(3), 517; https://doi.org/10.3390/e25030517 - 17 Mar 2023

Viewed by 1096

Abstract

In this paper, we focus on evolution from an equilibrium state in a power law form by means of q-exponentials to an arbitrary one. Introducing new q-Gibbsian equalities as the necessary condition of self-organization in nonextensive open systems, we theoretically show [...] Read more.

Δ {\tilde{S}}_{q}

) and q-relative entropies (

K L_{q}

) in both Bregman and Csiszar forms after we clearly explain the connection between renormalized entropy by Klimantovich and relative entropy by Kullback-Leibler without using any predefined effective Hamiltonian. This function, in our treatment, spontaneously comes directly from the calculations. We also explain the difference between using ordinary and normalized q-expectations in mean energy calculations of the states. To verify the results numerically, we use a toy model of complexity, namely the logistic map defined as

X_{t + 1} = 1 - a X_{t}^{2}

, where

a \in [0, 2]

is the map parameter. We measure the level of self-organization using two distinct forms of the q-renormalized entropy through period doublings and chaotic band mergings of the map as the number of periods/chaotic-bands increase/decrease. We associate the behaviour of the q-renormalized entropies with the emergence/disappearance of complex structures in the phase space as the control parameter of the map changes. Similar to Shiner-Davison-Landsberg (SDL) complexity, we categorize the tendencies of the q-renormalized entropies for the evaluation of the map for the whole control parameter space. Moreover, we show that any evolution between two states possesses a unique

q = q^{*}

value (not a range for q values) for which the q-Gibbsian equalities hold and the values are the same for the Bregmann and Csiszar forms. Interestingly, if the evolution is from

a = 0

a = a_{c} ≃ 1.4011

, this unique

q^{*}

value is found to be

q^{*} ≃ 0.2445

, which is the same value of

q_{s e n s i t i v i t y}

given in the literature. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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13 pages, 337 KiB

Open AccessArticle

Some Non-Obvious Consequences of Non-Extensiveness of Entropy

by Grzegorz Wilk and Zbigniew Włodarczyk

Entropy 2023, 25(3), 474; https://doi.org/10.3390/e25030474 - 09 Mar 2023

Cited by 2 | Viewed by 1023

Abstract

Non-additive (or non-extensive) entropies have long been intensively studied and used in various fields of scientific research. This was due to the desire to describe the commonly observed quasi-power rather than the exponential nature of various distributions of the variables of interest when considered in the full available space of their variability. In this work we will concentrate on the example of high energy multiparticle production processes and will limit ourselves to only one form of non-extensive entropy, namely the Tsallis entropy. We will discuss some points not yet fully clarified and present some non-obvious consequences of non-extensiveness of entropy when applied to production processes. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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13 pages, 1437 KiB

Open AccessArticle

Testing Nonlinearity with Rényi and Tsallis Mutual Information with an Application in the EKC Hypothesis

by Elif Tuna, Atıf Evren, Erhan Ustaoğlu, Büşra Şahin and Zehra Zeynep Şahinbaşoğlu

Entropy 2023, 25(1), 79; https://doi.org/10.3390/e25010079 - 31 Dec 2022

Cited by 1 | Viewed by 1457

Abstract

The nature of dependence between random variables has always been the subject of many statistical problems for over a century. Yet today, there is a great deal of research on this topic, especially focusing on the analysis of nonlinearity. Shannon mutual information has been considered to be the most comprehensive measure of dependence for evaluating total dependence, and several methods have been suggested for discerning the linear and nonlinear components of dependence between two variables. We, in this study, propose employing the Rényi and Tsallis mutual information measures for measuring total dependence because of their parametric nature. We first use a residual analysis in order to remove linear dependence between the variables, and then we compare the Rényi and Tsallis mutual information measures of the original data with that the lacking linear component to determine the degree of nonlinearity. A comparison against the values of the Shannon mutual information measure is also provided. Finally, we apply our method to the environmental Kuznets curve (EKC) and demonstrate the validity of the EKC hypothesis for Eastern Asian and Asia-Pacific countries. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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13 pages, 291 KiB

Open AccessFeature PaperArticle

Non-Additive Entropy Composition Rules Connected with Finite Heat-Bath Effects

by Tamás Sándor Biró

Entropy 2022, 24(12), 1769; https://doi.org/10.3390/e24121769 - 03 Dec 2022

Viewed by 1084

Abstract

Mathematical generalizations of the additive Boltzmann–Gibbs–Shannon entropy formula have been numerous since the 1960s. In this paper we seek an interpretation of the Rényi and Tsallis q-entropy formulas single parameter in terms of physical properties of a finite capacity heat-bath and fluctuations of [...] Read more.

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

13 pages, 356 KiB

Open AccessEditor’s ChoiceArticle

Entropy Optimization, Generalized Logarithms, and Duality Relations

by Angel R. Plastino, Constantino Tsallis, Roseli S. Wedemann and Hans J. Haubold

Entropy 2022, 24(12), 1723; https://doi.org/10.3390/e24121723 - 25 Nov 2022

Cited by 5 | Viewed by 1481

Abstract

Several generalizations or extensions of the Boltzmann–Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies [...] Read more.

S_{q} \equiv k \frac{1 - \sum_{i} p_{i}^{q}}{q - 1} (q \in R; S_{1} = S_{B G} \equiv - k \sum_{i} p_{i} \ln p_{i})

have harvested the largest number of successful applications. The specific structural features of the

S_{q}

thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function

\ln_{q} x \equiv \frac{x^{1 - q} - 1}{1 - q} (\ln_{1} x = \ln x)

associated with the

S_{q}

entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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11 pages, 3462 KiB

Open AccessFeature PaperArticle

Information Shift Dynamics Described by Tsallis q = 3 Entropy on a Compact Phase Space

by Jin Yan and Christian Beck

Entropy 2022, 24(11), 1671; https://doi.org/10.3390/e24111671 - 17 Nov 2022

Viewed by 1566

Abstract

Recent mathematical investigations have shown that under very general conditions, exponential mixing implies the Bernoulli property. As a concrete example of statistical mechanics that are exponentially mixing we consider the Bernoulli shift dynamics by Chebyshev maps of arbitrary order

N \geq 2

, [...] Read more.

N \geq 2

, which maximizes Tsallis

q = 3

entropy rather than the ordinary

q = 1

Boltzmann-Gibbs entropy. Such an information shift dynamics may be relevant in a pre-universe before ordinary space-time is created. We discuss symmetry properties of the coupled Chebyshev systems, which are different for even and odd N. We show that the value of the fine structure constant

α_{e l} = 1 / 137

is distinguished as a coupling constant in this context, leading to uncorrelated behaviour in the spatial direction of the corresponding coupled map lattice for

N = 3

. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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Review

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15 pages, 320 KiB

Open AccessReview

Brief Review on the Connection between the Micro-Canonical Ensemble and the S_q-Canonical Probability Distribution

by Angel R. Plastino and Angelo Plastino

Entropy 2023, 25(4), 591; https://doi.org/10.3390/e25040591 - 30 Mar 2023

Cited by 3 | Viewed by 980

Abstract

Non-standard thermostatistical formalisms derived from generalizations of the Boltzmann–Gibbs entropy have attracted considerable attention recently. Among the various proposals, the one that has been most intensively studied, and most successfully applied to concrete problems in physics and other areas, is the one associated [...] Read more.

S_{q}

non-additive entropies. The

S_{q}

-based thermostatistics exhibits a number of peculiar features that distinguish it from other generalizations of the Boltzmann–Gibbs theory. In particular, there is a close connection between the

S_{q}

-canonical distributions and the micro-canonical ensemble. The connection, first pointed out in 1994, has been subsequently explored by several researchers, who elaborated this facet of the

S_{q}

-thermo-statistics in a number of interesting directions. In the present work, we provide a brief review of some highlights within this line of inquiry, focusing on micro-canonical scenarios leading to

S_{q}

-canonical distributions. We consider works on the micro-canonical ensemble, including historical ones, where the

S_{q}

-canonical distributions, although present, were not identified as such, and also more resent works by researchers who explicitly investigated the

S_{q}

-micro-canonical connection. Full article

(This article belongs to the Special Issue Non-additive Entropy Formulas: Motivation and Derivations)

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