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Special Issue "Nonadditive Entropies and Complex Systems"

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

Deadline for manuscript submissions: closed (30 November 2018)

Special Issue Editors

Guest Editor
Prof. Dr. Andrea Rapisarda

1. Dipartimento di Fisica e Astronomia and INFN sezione di Catania, Via S.Sofia 64, 95123 Catania, Italy
2. Complexity Science Hub Vienna, Josefstädter Straße 39, 1080 Vienna, Austria; http://csh.ac.at/
Website | E-Mail
Interests: complex systems; statistical mechanics; agent-based models; complex networks; econophysics; chaos and nonlinear dynamics
Guest Editor
Prof. Dr. Stefan Thurner

1 Medical University of Vienna, Spitalgasse 23, 1090 Vienna, Austria
2 Complexity Science Hub Vienna, Josefstädter Straße 39, 1080 Vienna, Austria; http://csh.ac.at/
Website | E-Mail
Interests: statistical mechanics of complex systems; theory of evolutionary processes; entropy formulations; network theory; scaling theory; anomalous diffusion
Guest Editor
Prof. Dr. Constantino Tsallis

1 Department of Theoretical Physics, Centro Brasileiro de Pesquisas Fisicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
3 Complexity Science Hub Vienna, Josefstädter Straße 39, 1080 Vienna, Austria; http://csh.ac.at/
Website | E-Mail
Fax: +55 21 2141 7190
Interests: nonextensive statistical mechanics; foundations and applications of statistical mechanics; complex systems

Special Issue Information

Dear Colleagues,

Complexity and complex systems emerge in natural, artificial, and social contexts, and have attracted strong and enthusiastic scientific attention all over the world during the last decades. The study of these fascinating systems focuses on concepts of emergent behavior, living organisms, languages, earthquakes, economics, ecology, social networks, and other fundamental problems of contemporary science and societies. Most of these systems are out-of-equilibrium and present weak chaos, long-range correlations, nonergodic behavior, multifractal hierarchical structures, for which standard equilibrium Boltzmann-Gibbs statistical mechanics is not applicable. In the last few decades, a large variety of complex systems, in various fields, has been successfully described using nonextensive generalized formalisms of statistical mechanics.

The aim of the present Special Issue is to solicit original and interdisciplinary contributions which cover new developments and original applications of generalized statistical mechanics to complex systems of various natures.

Prof. Dr. Andrea Rapisarda
Prof. Dr. Stefan Thurner
Prof. Dr. Constantino Tsallis
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 papers will be 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 1600 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

  • nonadditive entropies
  • nonextensive statistical mechanics
  • complex systems
  • complex networks
  • thermodynamics
  • applications

Published Papers (10 papers)

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Research

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Open AccessArticle d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies
Entropy 2019, 21(1), 31; https://doi.org/10.3390/e21010031
Received: 27 November 2018 / Revised: 22 December 2018 / Accepted: 2 January 2019 / Published: 4 January 2019
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Abstract
We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model (d=1,2,3) with interactions decaying with the distance rij as 1/rijα (
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We numerically study the first-principle dynamics and thermostatistics of a d-dimensional classical inertial Heisenberg ferromagnetic model ( d = 1 , 2 , 3 ) with interactions decaying with the distance r i j as 1 / r i j α ( α 0 ), where the limit α = 0 ( α ) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio α / d > 1 ( 0 α / d 1 ) characterizes the short-ranged (long-ranged) regime. By means of first-principle molecular dynamics we study: (i) The scaling with the system size N of the maximum Lyapunov exponent λ in the form λ N κ , where κ ( α / d ) depends only on the ratio α / d ; (ii) The time-averaged single-particle angular momenta probability distributions for a typical case in the long-range regime 0 α / d 1 (which turns out to be well fitted by q-Gaussians), and (iii) The time-averaged single-particle energies probability distributions for a typical case in the long-range regime 0 α / d 1 (which turns out to be well fitted by q-exponentials). Through the Lyapunov exponents we observe an intriguing, and possibly size-dependent, persistence of the non-Boltzmannian behavior even in the α / d > 1 regime. The universality that we observe for the probability distributions with regard to the ratio α / d makes this model similar to the α -XY and α -Fermi-Pasta-Ulam Hamiltonian models as well as to asymptotically scale-invariant growing networks. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessArticle Associating an Entropy with Power-Law Frequency of Events
Entropy 2018, 20(12), 940; https://doi.org/10.3390/e20120940
Received: 3 October 2018 / Revised: 19 November 2018 / Accepted: 23 November 2018 / Published: 6 December 2018
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Abstract
Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are
[...] Read more.
Events occurring with a frequency described by power laws, within a certain range of validity, are very common in natural systems. In many of them, it is possible to associate an energy spectrum and one can show that these types of phenomena are intimately related to Tsallis entropy S q . The relevant parameters become: (i) The entropic index q, which is directly related to the power of the corresponding distribution; (ii) The ground-state energy ε 0 , in terms of which all energies are rescaled. One verifies that the corresponding processes take place at a temperature T q with k T q ε 0 (i.e., isothermal processes, for a given q), in analogy with those in the class of self-organized criticality, which are known to occur at fixed temperatures. Typical examples are analyzed, like earthquakes, avalanches, and forest fires, and in some of them, the entropic index q and value of T q are estimated. The knowledge of the associated entropic form opens the possibility for a deeper understanding of such phenomena, particularly by using information theory and optimization procedures. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessFeature PaperArticle Emergence of Shear Bands in Confined Granular Systems: Singularity of the q-Statistics
Entropy 2018, 20(11), 862; https://doi.org/10.3390/e20110862
Received: 8 October 2018 / Revised: 31 October 2018 / Accepted: 1 November 2018 / Published: 9 November 2018
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Abstract
The statistics of grain displacements probability distribution function (pdf) during the shear of a granular medium displays an unusual dependence with the shear increment upscaling as recently evinced (see “experimental validation of a nonextensive scaling law in confined granular media”). Basically,
[...] Read more.
The statistics of grain displacements probability distribution function (pdf) during the shear of a granular medium displays an unusual dependence with the shear increment upscaling as recently evinced (see “experimental validation of a nonextensive scaling law in confined granular media”). Basically, the pdf of grain displacements has clear nonextensive (q-Gaussian) features at small scales, but approaches to Gaussian characteristics at large shear window scales—the granulence effect. Here, we extend this analysis studying a larger system (more grains considered in the experimental setup), which exhibits a severe shear band fault during the macroscopic straining. We calculate the pdf of grain displacements and the dependency of the q-statistics with the shear increment. This analysis has shown a singular behavior of q at large scales, displaying a non-monotonic dependence with the shear increment. By means of an independent image analysis, we demonstrate that this singular non-monotonicity could be associated with the emergence of a shear band within the confined system. We show that the exact point where the q-value inverts its tendency coincides with the emergence of a giant percolation cluster along the system, caused by the shear band. We believe that this original approach using Statistical Mechanics tools to identify shear bands can be a very useful piece to solve the complex puzzle of the rheology of dense granular systems. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessArticle Maximum Configuration Principle for Driven Systems with Arbitrary Driving
Entropy 2018, 20(11), 838; https://doi.org/10.3390/e20110838
Received: 11 September 2018 / Revised: 20 October 2018 / Accepted: 23 October 2018 / Published: 1 November 2018
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Abstract
Depending on context, the term entropy is used for a thermodynamic quantity, a measure of available choice, a quantity to measure information, or, in the context of statistical inference, a maximum configuration predictor. For systems in equilibrium or processes without memory, the mathematical
[...] Read more.
Depending on context, the term entropy is used for a thermodynamic quantity, a measure of available choice, a quantity to measure information, or, in the context of statistical inference, a maximum configuration predictor. For systems in equilibrium or processes without memory, the mathematical expression for these different concepts of entropy appears to be the so-called Boltzmann–Gibbs–Shannon entropy, H. For processes with memory, such as driven- or self- reinforcing-processes, this is no longer true: the different entropy concepts lead to distinct functionals that generally differ from H. Here we focus on the maximum configuration entropy (that predicts empirical distribution functions) in the context of driven dissipative systems. We develop the corresponding framework and derive the entropy functional that describes the distribution of observable states as a function of the details of the driving process. We do this for sample space reducing (SSR) processes, which provide an analytically tractable model for driven dissipative systems with controllable driving. The fact that a consistent framework for a maximum configuration entropy exists for arbitrarily driven non-equilibrium systems opens the possibility of deriving a full statistical theory of driven dissipative systems of this kind. This provides us with the technical means needed to derive a thermodynamic theory of driven processes based on a statistical theory. We discuss the Legendre structure for driven systems. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessArticle On Quantum Superstatistics and the Critical Behavior of Nonextensive Ideal Bose Gases
Entropy 2018, 20(10), 773; https://doi.org/10.3390/e20100773
Received: 31 August 2018 / Revised: 27 September 2018 / Accepted: 29 September 2018 / Published: 9 October 2018
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Abstract
We explore some important consequences of the quantum ideal Bose gas, the properties of which are described by a non-extensive entropy. We consider in particular two entropies that depend only on the probability. These entropies are defined in the framework of superstatistics, and
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We explore some important consequences of the quantum ideal Bose gas, the properties of which are described by a non-extensive entropy. We consider in particular two entropies that depend only on the probability. These entropies are defined in the framework of superstatistics, and in this context, such entropies arise when a system is exposed to non-equilibrium conditions, whose general effects can be described by a generalized Boltzmann factor and correspondingly by a generalized probability distribution defining a different statistics. We generalize the usual statistics to their quantum counterparts, and we will focus on the properties of the corresponding generalized quantum ideal Bose gas. The most important consequence of the generalized Bose gas is that the critical temperature predicted for the condensation changes in comparison with the usual quantum Bose gas. Conceptual differences arise when comparing our results with the ones previously reported regarding the q-generalized Bose–Einstein condensation. As the entropies analyzed here only depend on the probability, our results cannot be adjusted by any parameter. Even though these results are close to those of non-extensive statistical mechanics for q 1 , they differ and cannot be matched for any q. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessArticle Analytic Study of Complex Fractional Tsallis’ Entropy with Applications in CNNs
Entropy 2018, 20(10), 722; https://doi.org/10.3390/e20100722
Received: 29 July 2018 / Revised: 5 September 2018 / Accepted: 10 September 2018 / Published: 20 September 2018
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Abstract
In this paper, we study Tsallis’ fractional entropy (TFE) in a complex domain by applying the definition of the complex probability functions. We study the upper and lower bounds of TFE based on some special functions. Moreover, applications in complex neural networks (CNNs)
[...] Read more.
In this paper, we study Tsallis’ fractional entropy (TFE) in a complex domain by applying the definition of the complex probability functions. We study the upper and lower bounds of TFE based on some special functions. Moreover, applications in complex neural networks (CNNs) are illustrated to recognize the accuracy of CNNs. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessArticle Hedging for the Regime-Switching Price Model Based on Non-Extensive Statistical Mechanics
Entropy 2018, 20(4), 248; https://doi.org/10.3390/e20040248
Received: 13 March 2018 / Revised: 1 April 2018 / Accepted: 3 April 2018 / Published: 3 April 2018
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Abstract
To describe the movement of asset prices accurately, we employ the non-extensive statistical mechanics and the semi-Markov process to establish an asset price model. The model can depict the peak and fat tail characteristics of returns and the regime-switching phenomenon of macroeconomic system.
[...] Read more.
To describe the movement of asset prices accurately, we employ the non-extensive statistical mechanics and the semi-Markov process to establish an asset price model. The model can depict the peak and fat tail characteristics of returns and the regime-switching phenomenon of macroeconomic system. Moreover, we use the risk-minimizing method to study the hedging problem of contingent claims and obtain the explicit solutions of the optimal hedging strategies. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
Open AccessArticle Generalized Pesin-Like Identity and Scaling Relations at the Chaos Threshold of the Rössler System
Entropy 2018, 20(4), 216; https://doi.org/10.3390/e20040216
Received: 16 February 2018 / Revised: 15 March 2018 / Accepted: 20 March 2018 / Published: 23 March 2018
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Abstract
In this paper, using the Poincaré section of the flow we numerically verify a generalization of a Pesin-like identity at the chaos threshold of the Rössler system, which is one of the most popular three-dimensional continuous systems. As Poincaré section points of the
[...] Read more.
In this paper, using the Poincaré section of the flow we numerically verify a generalization of a Pesin-like identity at the chaos threshold of the Rössler system, which is one of the most popular three-dimensional continuous systems. As Poincaré section points of the flow show similar behavior to that of the logistic map, for the Rössler system we also investigate the relationships with respect to important properties of nonlinear dynamics, such as correlation length, fractal dimension, and the Lyapunov exponent in the vicinity of the chaos threshold. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Open AccessArticle Non-Gaussian Closed Form Solutions for Geometric Average Asian Options in the Framework of Non-Extensive Statistical Mechanics
Entropy 2018, 20(1), 71; https://doi.org/10.3390/e20010071
Received: 17 November 2017 / Revised: 30 December 2017 / Accepted: 16 January 2018 / Published: 18 January 2018
Cited by 2 | PDF Full-text (266 KB) | HTML Full-text | XML Full-text
Abstract
In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics
[...] Read more.
In this paper we consider pricing problems of the geometric average Asian options under a non-Gaussian model, in which the underlying stock price is driven by a process based on non-extensive statistical mechanics. The model can describe the peak and fat tail characteristics of returns. Thus, the description of underlying asset price and the pricing of options are more accurate. Moreover, using the martingale method, we obtain closed form solutions for geometric average Asian options. Furthermore, the numerical analysis shows that the model can avoid underestimating risks relative to the Black-Scholes model. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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Review

Jump to: Research

Open AccessReview Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory
Entropy 2018, 20(10), 804; https://doi.org/10.3390/e20100804
Received: 12 September 2018 / Revised: 9 October 2018 / Accepted: 10 October 2018 / Published: 19 October 2018
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Abstract
The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has
[...] Read more.
The entropy of Boltzmann-Gibbs, as proved by Shannon and Khinchin, is based on four axioms, where the fourth one concerns additivity. The group theoretic entropies make use of formal group theory to replace this axiom with a more general composability axiom. As has been pointed out before, generalised entropies crucially depend on the number of allowed degrees of freedom N. The functional form of group entropies is restricted (though not uniquely determined) by assuming extensivity on the equal probability ensemble, which leads to classes of functionals corresponding to sub-exponential, exponential or super-exponential dependence of the phase space volume W on N. We review the ensuing entropies, discuss the composability axiom and explain why group entropies may be particularly relevant from an information-theoretical perspective. Full article
(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)
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