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17 pages, 292 KiB

Open AccessArticle

Fractal Structure and Non-Extensive Statistics

by Airton Deppman, Tobias Frederico, Eugenio Megías and Debora P. Menezes

Entropy 2018, 20(9), 633; https://0-doi-org.brum.beds.ac.uk/10.3390/e20090633 - 24 Aug 2018

Cited by 34 | Viewed by 4020

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most [...] Read more.

The role played by non-extensive thermodynamics in physical systems has been under intense debate for the last decades. With many applications in several areas, the Tsallis statistics have been discussed in detail in many works and triggered an interesting discussion on the most deep meaning of entropy and its role in complex systems. Some possible mechanisms that could give rise to non-extensive statistics have been formulated over the last several years, in particular a fractal structure in thermodynamic functions was recently proposed as a possible origin for non-extensive statistics in physical systems. In the present work, we investigate the properties of such fractal thermodynamical system and propose a diagrammatic method for calculations of relevant quantities related to such a system. It is shown that a system with the fractal structure described here presents temperature fluctuation following an Euler Gamma Function, in accordance with previous works that provided evidence of the connections between those fluctuations and Tsallis statistics. Finally, the scale invariance of the fractal thermodynamical system is discussed in terms of the Callan–Symanzik equation. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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9 pages, 260 KiB

Open AccessArticle

An Application of Maximal Exponential Models to Duality Theory

by Marina Santacroce, Paola Siri and Barbara Trivellato

Entropy 2018, 20(7), 495; https://0-doi-org.brum.beds.ac.uk/10.3390/e20070495 - 27 Jun 2018

Cited by 4 | Viewed by 2124

Abstract

We use maximal exponential models to characterize a suitable polar cone in a mathematical convex optimization framework. A financial application of this result is provided, leading to a duality minimax theorem related to portfolio exponential utility maximization. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

12 pages, 275 KiB

Open AccessArticle

Symmetric Logarithmic Derivative of Fermionic Gaussian States

by Angelo Carollo, Bernardo Spagnolo and Davide Valenti

Entropy 2018, 20(7), 485; https://0-doi-org.brum.beds.ac.uk/10.3390/e20070485 - 22 Jun 2018

Cited by 28 | Viewed by 3928

Abstract

In this article, we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications range from quantum Metrology with thermal states to non-equilibrium [...] Read more.

In this article, we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications range from quantum Metrology with thermal states to non-equilibrium steady states with Fermionic many-body systems. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

17 pages, 294 KiB

Open AccessArticle

Nonlinear Kinetics on Lattices Based on the Kinetic Interaction Principle

by Giorgio Kaniadakis and Dionissios T. Hristopulos

Entropy 2018, 20(6), 426; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060426 - 01 Jun 2018

Cited by 4 | Viewed by 3149

Abstract

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker–Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, [...] Read more.

Master equations define the dynamics that govern the time evolution of various physical processes on lattices. In the continuum limit, master equations lead to Fokker–Planck partial differential equations that represent the dynamics of physical systems in continuous spaces. Over the last few decades, nonlinear Fokker–Planck equations have become very popular in condensed matter physics and in statistical physics. Numerical solutions of these equations require the use of discretization schemes. However, the discrete evolution equation obtained by the discretization of a Fokker–Planck partial differential equation depends on the specific discretization scheme. In general, the discretized form is different from the master equation that has generated the respective Fokker–Planck equation in the continuum limit. Therefore, the knowledge of the master equation associated with a given Fokker–Planck equation is extremely important for the correct numerical integration of the latter, since it provides a unique, physically motivated discretization scheme. This paper shows that the Kinetic Interaction Principle (KIP) that governs the particle kinetics of many body systems, introduced in G. Kaniadakis, Physica A 296, 405 (2001), univocally defines a very simple master equation that in the continuum limit yields the nonlinear Fokker–Planck equation in its most general form. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

21 pages, 15460 KiB

Open AccessArticle

Analysis of Chaotic Behavior in a Novel Extended Love Model Considering Positive and Negative External Environment

by Linyun Huang and Youngchul Bae

Entropy 2018, 20(5), 365; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050365 - 14 May 2018

Cited by 8 | Viewed by 4683

Abstract

The aim of this study was to describe a novel extended dynamical love model with the external environments of the love story of Romeo and Juliet. We used the sinusoidal function as external environments as it could represent the positive and negative characteristics [...] Read more.

The aim of this study was to describe a novel extended dynamical love model with the external environments of the love story of Romeo and Juliet. We used the sinusoidal function as external environments as it could represent the positive and negative characteristics of humans. We considered positive and negative advice from a third person. First, we applied the same amount of positive and negative advice. Second, the amount of positive advice was greater than that of negative advice. Third, the amount of positive advice was smaller than that of negative advice in an external environment. To verify the chaotic phenomena in the proposed extended dynamic love affair with external environments, we used time series, phase portraits, power spectrum, Poincare map, bifurcation diagram, and the maximal Lyapunov exponent. With a variation of parameter “a”, we recognized that the novel extended dynamic love affairs with different three situations of external environments had chaotic behaviors. We showed 1, 2, 4 periodic motion, Rössler type attractor, and chaotic attractor when parameter “a” varied under the following conditions: the amount of positive advice = the amount of negative advice, the amount of positive advice > the amount of negative advice, and the amount of positive advice < the amount of negative advice. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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11 pages, 1304 KiB

Open AccessFeature PaperArticle

Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange

by Maria Letizia Bertotti, Amit K. Chattopadhyay and Giovanni Modanese

Entropy 2018, 20(3), 166; https://0-doi-org.brum.beds.ac.uk/10.3390/e20030166 - 05 Mar 2018

Viewed by 3368

Abstract

In our recently proposed stochastic version of discretized kinetic theory, the exchange of wealth in a society is modelled through a large system of Langevin equations. The deterministic part of the equations is based on non-linear transition probabilities between income classes. The noise [...] Read more.

In our recently proposed stochastic version of discretized kinetic theory, the exchange of wealth in a society is modelled through a large system of Langevin equations. The deterministic part of the equations is based on non-linear transition probabilities between income classes. The noise terms can be additive, multiplicative or mixed, both with white or Ornstein–Uhlenbeck spectrum. The most important measured correlations are those between Gini inequality index G and social mobility M, between total income and G, and between M and total income. We describe numerical results concerning these correlations and a quantity which gives average stochastic deviations from the equilibrium solutions in dependence on the noise amplitude. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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9 pages, 245 KiB

Open AccessArticle

Lagrangian Function on the Finite State Space Statistical Bundle

by Giovanni Pistone

Entropy 2018, 20(2), 139; https://0-doi-org.brum.beds.ac.uk/10.3390/e20020139 - 22 Feb 2018

Cited by 8 | Viewed by 3320

Abstract

The statistical bundle is the set of couples (

Q, W

) of a probability density Q and a random variable W such that

Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

15 pages, 6699 KiB

Open AccessArticle

Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation

by Nattagit Jiteurtragool, Tachibana Masayoshi and Wimol San-Um

Entropy 2018, 20(2), 136; https://0-doi-org.brum.beds.ac.uk/10.3390/e20020136 - 20 Feb 2018

Cited by 7 | Viewed by 4211

Abstract

The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic [...] Read more.

The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic form of the sigmoidal chaotic map with three terms, i.e., x_n₊₁ = ∓Af_NL(Bx_n) ± Cx_n ± D, where A, B, C, and D are real constants. The unification of modified sigmoid and hyperbolic tangent (tanh) functions reveals the existence of a “unified sigmoidal chaotic map” generically fulfilling the three terms, with robust chaos partially appearing in some parameter ranges. A simplified generic form, i.e., x_n₊₁ = ∓f_NL(Bx_n) ± Cx_n, through various S-shaped functions, has recently led to the possibility of linearization using (i) hardtanh and (ii) signum functions. This study finds a linearized sigmoidal chaotic map that potentially offers robust chaos over an entire range of parameters. Chaos dynamics are described in terms of chaotic waveforms, histogram, cobweb plots, fixed point, Jacobian, and a bifurcation structure diagram based on Lyapunov exponents. As a practical example, a true random bit generator using the linearized sigmoidal chaotic map is demonstrated. The resulting output is evaluated using the NIST SP800-22 test suite and TestU01. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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16 pages, 482 KiB

Open AccessFeature PaperArticle

Kinetic Energy of a Free Quantum Brownian Particle

by Paweł Bialas and Jerzy Łuczka

Entropy 2018, 20(2), 123; https://0-doi-org.brum.beds.ac.uk/10.3390/e20020123 - 12 Feb 2018

Cited by 17 | Viewed by 4148

Abstract

We consider a paradigmatic model of a quantum Brownian particle coupled to a thermostat consisting of harmonic oscillators. In the framework of a generalized Langevin equation, the memory (damping) kernel is assumed to be in the form of exponentially-decaying oscillations. We discuss a [...] Read more.

We consider a paradigmatic model of a quantum Brownian particle coupled to a thermostat consisting of harmonic oscillators. In the framework of a generalized Langevin equation, the memory (damping) kernel is assumed to be in the form of exponentially-decaying oscillations. We discuss a quantum counterpart of the equipartition energy theorem for a free Brownian particle in a thermal equilibrium state. We conclude that the average kinetic energy of the Brownian particle is equal to thermally-averaged kinetic energy per one degree of freedom of oscillators of the environment, additionally averaged over all possible oscillators’ frequencies distributed according to some probability density in which details of the particle-environment interaction are present via the parameters of the damping kernel. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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14 pages, 6244 KiB

Open AccessArticle

Chaotic Dynamics of the Fractional-Love Model with an External Environment

by Linyun Huang and Youngchul Bae

Entropy 2018, 20(1), 53; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010053 - 12 Jan 2018

Cited by 20 | Viewed by 4158

Abstract

Based on the fractional order of nonlinear system for love model with a periodic function as an external environment, we analyze the characteristics of the chaotic dynamic. We analyze the relationship between the chaotic dynamic of the fractional order love model with an [...] Read more.

Based on the fractional order of nonlinear system for love model with a periodic function as an external environment, we analyze the characteristics of the chaotic dynamic. We analyze the relationship between the chaotic dynamic of the fractional order love model with an external environment and the value of fractional order (α, β) when the parameters are fixed. Meanwhile, we also study the relationship between the chaotic dynamic of the fractional order love model with an external environment and the parameters (a, b, c, d) when the fractional order of the system is fixed. When the parameters of fractional order love model are fixed, the fractional order (α, β) of fractional order love model system exhibit segmented chaotic states with the different fractional orders of the system. When the fractional order (α = β) of the system is fixed, the system shows the periodic state and the chaotic state as the parameter is changing as a result. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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9 pages, 264 KiB

Open AccessCommunication

Polyadic Entropy, Synergy and Redundancy among Statistically Independent Processes in Nonlinear Statistical Physics with Microphysical Codependence

by Rui A. P. Perdigão

Entropy 2018, 20(1), 26; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010026 - 04 Jan 2018

Cited by 7 | Viewed by 3795

Abstract

The information shared among observables representing processes of interest is traditionally evaluated in terms of macroscale measures characterizing aggregate properties of the underlying processes and their interactions. Traditional information measures are grounded on the assumption that the observable represents a memoryless process without [...] Read more.

The information shared among observables representing processes of interest is traditionally evaluated in terms of macroscale measures characterizing aggregate properties of the underlying processes and their interactions. Traditional information measures are grounded on the assumption that the observable represents a memoryless process without any interaction among microstates. Generalized entropy measures have been formulated in non-extensive statistical mechanics aiming to take microphysical codependence into account in entropy quantification. By taking them into consideration when formulating information measures, the question is raised on whether and if so how much information permeates across scales to impact on the macroscale information measures. The present study investigates and quantifies the emergence of macroscale information from microscale codependence among microphysics. In order to isolate the information emergence coming solely from the nonlinearly interacting microphysics, redundancy and synergy are evaluated among macroscale variables that are statistically independent from each other but not necessarily so within their own microphysics. Synergistic and redundant information are found when microphysical interactions take place, even if the statistical distributions are factorable. These findings stress the added value of nonlinear statistical physics to information theory in coevolutionary systems. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

Review

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20 pages, 368 KiB

Open AccessFeature PaperReview

Information Geometry of κ-Exponential Families: Dually-Flat, Hessian and Legendre Structures

by Antonio M. Scarfone, Hiroshi Matsuzoe and Tatsuaki Wada

Entropy 2018, 20(6), 436; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060436 - 05 Jun 2018

Cited by 11 | Viewed by 3364

Abstract

In this paper, we present a review of recent developments on the

κ

-deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the

κ

-formalism which are obtained starting from three, not equivalent, divergence functions, [...] Read more.

In this paper, we present a review of recent developments on the

κ

-deformed statistical mechanics in the framework of the information geometry. Three different geometric structures are introduced in the

κ

-formalism which are obtained starting from three, not equivalent, divergence functions, corresponding to the

κ

-deformed version of Kullback–Leibler, “Kerridge” and Brègman divergences. The first statistical manifold derived from the

κ

-Kullback–Leibler divergence form an invariant geometry with a positive curvature that vanishes in the

κ \to 0

limit. The other two statistical manifolds are related to each other by means of a scaling transform and are both dually-flat. They have a dualistic Hessian structure endowed by a deformed Fisher metric and an affine connection that are consistent with a statistical scalar product based on the

κ

-escort expectation. These flat geometries admit dual potentials corresponding to the thermodynamic Massieu and entropy functions that induce a Legendre structure of

κ

-thermodynamics in the picture of the information geometry. Full article

(This article belongs to the Special Issue Theoretical Aspect of Nonlinear Statistical Physics)

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