Editorial

3 pages, 188 KiB

Open AccessEditorial

Nonadditive Entropies and Complex Systems

by Andrea Rapisarda, Stefan Thurner and Constantino Tsallis

Entropy 2019, 21(5), 538; https://doi.org/10.3390/e21050538 - 27 May 2019

Cited by 4 | Viewed by 3852

An entropic functional S is said additive if it satisfies, for any two probabilistically independent systems A and B, that

S (A + B) = S (A) + S (B)

[...] [...] Read more.

An entropic functional S is said additive if it satisfies, for any two probabilistically independent systems A and B, that

S (A + B) = S (A) + S (B)

[...] Full article

(This article belongs to the Special Issue Nonadditive Entropies and Complex Systems)

Research

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14 pages, 658 KiB

Open AccessArticle

d-Dimensional Classical Heisenberg Model with Arbitrarily-Ranged Interactions: Lyapunov Exponents and Distributions of Momenta and Energies

by Antonio Rodríguez, Fernando D. Nobre and Constantino Tsallis

Entropy 2019, 21(1), 31; https://doi.org/10.3390/e21010031 - 04 Jan 2019

Cited by 18 | Viewed by 3682

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

r_{i j}

as

1 / r_{i j}^{α}

( [...] Read more.

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 \geq 0

), where the limit

α = 0

(

α \to \infty

) corresponds to infinite-range (nearest-neighbour) interactions, and the ratio

α / d > 1

(

0 \leq α / d \leq 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 \sim 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 \leq α / d \leq 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 \leq α / d \leq 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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12 pages, 681 KiB

Open AccessArticle

Associating an Entropy with Power-Law Frequency of Events

by Evaldo M. F. Curado, Fernando D. Nobre and Angel Plastino

Entropy 2018, 20(12), 940; https://doi.org/10.3390/e20120940 - 06 Dec 2018

Cited by 7 | Viewed by 3429

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} \propto 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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14 pages, 5409 KiB

Open AccessFeature PaperArticle

Emergence of Shear Bands in Confined Granular Systems: Singularity of the q-Statistics

by Léo Viallon-Galinier, Gaël Combe, Vincent Richefeu and Allbens Picardi Faria Atman

Entropy 2018, 20(11), 862; https://doi.org/10.3390/e20110862 - 09 Nov 2018

Cited by 6 | Viewed by 3468

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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13 pages, 499 KiB

Open AccessArticle

Maximum Configuration Principle for Driven Systems with Arbitrary Driving

by Rudolf Hanel and Stefan Thurner

Entropy 2018, 20(11), 838; https://doi.org/10.3390/e20110838 - 01 Nov 2018

Cited by 5 | Viewed by 2823

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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11 pages, 860 KiB

Open AccessArticle

On Quantum Superstatistics and the Critical Behavior of Nonextensive Ideal Bose Gases

by Octavio Obregón, José Luis López and Marco Ortega-Cruz

Entropy 2018, 20(10), 773; https://doi.org/10.3390/e20100773 - 09 Oct 2018

Cited by 15 | Viewed by 2377

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 [...] Read more.

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 \sim 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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11 pages, 367 KiB

Open AccessArticle

Analytic Study of Complex Fractional Tsallis’ Entropy with Applications in CNNs

by Rabha W. Ibrahim and Maslina Darus

Entropy 2018, 20(10), 722; https://doi.org/10.3390/e20100722 - 20 Sep 2018

Cited by 14 | Viewed by 3429

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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9 pages, 740 KiB

Open AccessArticle

Hedging for the Regime-Switching Price Model Based on Non-Extensive Statistical Mechanics

by Pan Zhao, Jian Pan, Benda Zhou, Jixia Wang and Yu Song

Entropy 2018, 20(4), 248; https://doi.org/10.3390/e20040248 - 03 Apr 2018

Cited by 1 | Viewed by 3172

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)

12 pages, 3305 KiB

Open AccessArticle

Generalized Pesin-Like Identity and Scaling Relations at the Chaos Threshold of the Rössler System

by Kivanc Cetin, Ozgur Afsar and Ugur Tirnakli

Entropy 2018, 20(4), 216; https://doi.org/10.3390/e20040216 - 23 Mar 2018

Cited by 3 | Viewed by 3598

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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12 pages, 266 KiB

Open AccessArticle

Non-Gaussian Closed Form Solutions for Geometric Average Asian Options in the Framework of Non-Extensive Statistical Mechanics

by Pan Zhao, Benda Zhou and Jixia Wang

Entropy 2018, 20(1), 71; https://doi.org/10.3390/e20010071 - 18 Jan 2018

Cited by 6 | Viewed by 3306

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

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11 pages, 263 KiB

Open AccessReview

Group Entropies: From Phase Space Geometry to Entropy Functionals via Group Theory

by Henrik Jeldtoft Jensen and Piergiulio Tempesta

Entropy 2018, 20(10), 804; https://doi.org/10.3390/e20100804 - 19 Oct 2018

Cited by 17 | Viewed by 3176

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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