Entropy

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 October 2018

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

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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

Open AccessArticle

The Middle-Income Trap and the Coping Strategies From Network-Based Perspectives

by Ming-Yang Zhou, Wen-Man Xiong, Xiao-Yu Li and Hao Liao

Entropy 2018, 20(10), 803; https://doi.org/10.3390/e20100803 - 18 October 2018

Abstract

When a developing country reaches a relatively average income level, it often stops growing further and its income does not improve. This is known as the middle-income trap. How to overcome this trap is a longstanding problem for developing countries, and has been

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When a developing country reaches a relatively average income level, it often stops growing further and its income does not improve. This is known as the middle-income trap. How to overcome this trap is a longstanding problem for developing countries, and has been studied in various research fields. In this work, we use the Fitness-Complexity method (FCM) to analyze the common characteristics of the countries that successfully get through the middle-income trap, and show the origin of the middle-income trap based on the international trade network. In the analysis, a novel method is proposed to characterize the interdependency between products. The results show that some middle-complexity products depend much on each other, which indicates that developing countries should focus on them simultaneously, implying high difficulty to escape the middle-income trap. To tackle the middle-income trap, developing countries should learn experiences from developed countries that share similar development history. we then design an effective method to evaluate the similarity between countries and recommend developed countries to a certain developing country. The effectiveness of our method is validated in the international trade network. Full article

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Open AccessArticle

Macroscopic Entropy of Non-Equilibrium Systems and Postulates of Extended Thermodynamics: Application to Transport Phenomena and Chemical Reactions in Nanoparticles

by Sergey I. Serdyukov

Entropy 2018, 20(10), 802; https://doi.org/10.3390/e20100802 - 18 October 2018

Abstract

In this work, we consider extended irreversible thermodynamics in assuming that the entropy density is a function of both common thermodynamic variables and their higher-order time derivatives. An expression for entropy production, and the linear phenomenological equations describing diffusion and chemical reactions, are

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In this work, we consider extended irreversible thermodynamics in assuming that the entropy density is a function of both common thermodynamic variables and their higher-order time derivatives. An expression for entropy production, and the linear phenomenological equations describing diffusion and chemical reactions, are found in the context of this approach. Solutions of the sets of linear equations with respect to fluxes and their higher-order time derivatives allows the coefficients of diffusion and reaction rate constants to be established as functions of size of the nanosystems in which these reactions occur. The Maxwell-Cattaneo and Jeffreys constitutive equations, as well as the higher-order constitutive equations, which describe the processes in reaction-diffusion systems, are obtained. Full article

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Open AccessArticle

Encryption Algorithm of Multiple-Image Using Mixed Image Elements and Two Dimensional Chaotic Economic Map

by A. A. Karawia

Entropy 2018, 20(10), 801; https://doi.org/10.3390/e20100801 - 18 October 2018

Abstract

To enhance the encryption proficiency and encourage the protected transmission of multiple images, the current work introduces an encryption algorithm for multiple images using the combination of mixed image elements (MIES) and a two-dimensional economic map. Firstly, the original images are grouped into

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To enhance the encryption proficiency and encourage the protected transmission of multiple images, the current work introduces an encryption algorithm for multiple images using the combination of mixed image elements (MIES) and a two-dimensional economic map. Firstly, the original images are grouped into one big image that is split into many pure image elements (PIES); secondly, the logistic map is used to shuffle the PIES; thirdly, it is confused with the sequence produced by the two-dimensional economic map to get MIES; finally, the MIES are gathered into a big encrypted image that is split into many images of the same size as the original images. The proposed algorithm includes a huge number key size space, and this makes the algorithm secure against hackers. Even more, the encryption results obtained by the proposed algorithm outperform existing algorithms in the literature. A comparison between the proposed algorithm and similar algorithms is made. The analysis of the experimental results and the proposed algorithm shows that the proposed algorithm is efficient and secure. Full article

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Open AccessArticle

Heat Transfer and Flow Structures of Laminar Confined Slot Impingement Jet with Power-Law Non-Newtonian Fluid

by Yan Qiang, Liejiang Wei, Xiaomei Luo, Hongchao Jian, Wenan Wang and Fenfen Li

Entropy 2018, 20(10), 800; https://doi.org/10.3390/e20100800 - 18 October 2018

Abstract

Heat transfer performances and flow structures of laminar impinging slot jets with power-law non-Newtonian fluids and corresponding typical industrial fluids (Carboxyl Methyl Cellulose (CMC) solutions and Xanthangum (XG) solutions) have been studied in this work. Investigations are performed for Reynolds number Re less

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Heat transfer performances and flow structures of laminar impinging slot jets with power-law non-Newtonian fluids and corresponding typical industrial fluids (Carboxyl Methyl Cellulose (CMC) solutions and Xanthangum (XG) solutions) have been studied in this work. Investigations are performed for Reynolds number Re less than 200, power-law index n ranging from 0.5 to 1.5 and consistency index K varying from 0.001 to 0.5 to explore heat transfer and flow structure of shear-thinning fluid and shear-thickening fluid. Results indicate that with the increase of n, K for a given Re, wall Nusselt number increases mainly attributing to the increase of inlet velocity U. For a given inlet velocity, wall Nusselt number decreases with the increase of n and K, which mainly attributes to the increase of apparent viscosity and the reduction of momentum diffusion. For the same Re, U and Pr, wall Nusselt number decreases with the increase of n. Among the study of industrial power-law shear-thinning fluid, CMC solution with 100 ppm shows the best heat transfer performance at a given velocity. Moreover, new correlation of Nusselt number about industrial fluid is proposed. In general, for the heat transfer of laminar confined impinging jet, it is best to use the working fluid with low viscosity. Full article

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Open AccessArticle

Long-Term Independence of Solar Wind Polytropic Index on Plasma Flow Speed

by George Livadiotis

Entropy 2018, 20(10), 799; https://doi.org/10.3390/e20100799 - 17 October 2018

Abstract

The paper derives the polytropic indices over the last two solar cycles (years 1995–2017) for the solar wind proton plasma near Earth (~1 AU). We use ~92-s datasets of proton plasma moments (speed, density, and temperature), measured from the Solar Wind Experiment instrument

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The paper derives the polytropic indices over the last two solar cycles (years 1995–2017) for the solar wind proton plasma near Earth (~1 AU). We use ~92-s datasets of proton plasma moments (speed, density, and temperature), measured from the Solar Wind Experiment instrument onboard Wind spacecraft, to estimate the moving averages of the polytropic index, as well as their weighted means and standard errors as a function of the solar wind speed and the year of measurements. The derived long-term behavior of the polytropic index agrees with the results of other previous methods. In particular, we find that the polytropic index remains quasi-constant with respect to the plasma flow speed, in agreement with earlier analyses of solar wind plasma. It is shown that most of the fluctuations of the polytropic index appear in the fast solar wind. The polytropic index remains quasi-constant, despite the frequent entropic variations. Therefore, on an annual basis, the polytropic index of the solar wind proton plasma near ~1 AU can be considered independent of the plasma flow speed. The estimated all-year weighted mean and its standard error is γ = 1.86 ± 0.09. Full article

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Open AccessArticle

Algorithmic Entropy and Landauer’s Principle Link Microscopic System Behaviour to the Thermodynamic Entropy

by Sean Devine

Entropy 2018, 20(10), 798; https://doi.org/10.3390/e20100798 - 17 October 2018

Abstract

Algorithmic information theory in conjunction with Landauer’s principle can quantify the cost of maintaining a reversible real-world computational system distant from equilibrium. As computational bits are conserved in an isolated reversible system, bit flows can be used to track the way a highly

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Algorithmic information theory in conjunction with Landauer’s principle can quantify the cost of maintaining a reversible real-world computational system distant from equilibrium. As computational bits are conserved in an isolated reversible system, bit flows can be used to track the way a highly improbable configuration trends toward a highly probable equilibrium configuration. In an isolated reversible system, all microstates within a thermodynamic macrostate have the same algorithmic entropy. However, from a thermodynamic perspective, when these bits primarily specify stored energy states, corresponding to a fluctuation from the most probable set of states, they represent “potential entropy”. However, these bits become “realised entropy” when, under the second law of thermodynamics, they become bits specifying the momentum degrees of freedom. The distance of a fluctuation from equilibrium is identified as the number of computational bits that move from stored energy states to momentum states to define a highly probable or typical equilibrium state. When reversibility applies, from Landauer’s principle, it costs

k_{B} l n 2 T

Joules to move a bit within the system from stored energy states to the momentum states. Full article

Open AccessArticle

Some Consequences of the Thermodynamic Cost of System Identification

by Chris Fields

Entropy 2018, 20(10), 797; https://doi.org/10.3390/e20100797 - 17 October 2018

Abstract

The concept of a “system” is foundational to physics, but the question of how observers identify systems is seldom addressed. Classical thermodynamics restricts observers to finite, finite-resolution observations with which to identify the systems on which “pointer state” measurements are to be made.

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The concept of a “system” is foundational to physics, but the question of how observers identify systems is seldom addressed. Classical thermodynamics restricts observers to finite, finite-resolution observations with which to identify the systems on which “pointer state” measurements are to be made. It is shown that system identification is at best approximate, even in a finite world, and that violations of the Leggett–Garg and Bell/CHSH (Clauser-Horne-Shimony-Holt) inequalities emerge naturally as requirements for successful system identification. Full article

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Open AccessCorrection

Correction: Naudts, J. Quantum Statistical Manifolds. Entropy 2018, 20, 472

by Jan Naudts

Entropy 2018, 20(10), 796; https://doi.org/10.3390/e20100796 - 17 October 2018

Abstract

Section 4 of “Naudts J. Quantum Statistical Manifolds. Entropy2018, 20, 472” contains errors. They have limited consequences for the remainder of the paper. A new version of this Section is found here. Some smaller shortcomings of the paper are taken care

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Section 4 of “Naudts J. Quantum Statistical Manifolds. Entropy2018, 20, 472” contains errors. They have limited consequences for the remainder of the paper. A new version of this Section is found here. Some smaller shortcomings of the paper are taken care of as well. In particular, the proof of Theorem 3 was not complete, and is therefore amended. Also, a few missing references are added. Full article

Open AccessArticle

The Interaction Analysis between the Sympathetic and Parasympathetic Systems in CHF by Using Transfer Entropy Method

by Daiyi Luo, Weifeng Pan, Yifan Li, Kaicheng Feng and Guanzheng Liu

Entropy 2018, 20(10), 795; https://doi.org/10.3390/e20100795 - 16 October 2018

Abstract

Congestive heart failure (CHF) is a cardiovascular disease associated with autonomic dysfunction, where sympathovagal imbalance was reported in many studies using heart rate variability (HRV). To learn more about the dynamic interaction in the autonomic nervous system (ANS), we explored the directed interaction

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Congestive heart failure (CHF) is a cardiovascular disease associated with autonomic dysfunction, where sympathovagal imbalance was reported in many studies using heart rate variability (HRV). To learn more about the dynamic interaction in the autonomic nervous system (ANS), we explored the directed interaction between the sympathetic nervous system (SNS) and the parasympathetic nervous system (PNS) with the help of transfer entropy (TE). This article included 24-h RR interval signals of 54 healthy subjects (31 males and 23 females, 61.38 ± 11.63 years old) and 44 CHF subjects (8 males and 2 females, 19 subjects’ gender were unknown, 55.51 ± 11.44 years old, 4 in class I, 8 in class II and 32 in class III~IV, according to the New York Heart Association Function Classification), obtained from the PhysioNet database and then segmented into 5-min non-overlapping epochs using cubic spline interpolation. For each segment in the normal group and CHF group, frequency-domain features included low-frequency (LF) power, high-frequency (HF) power and LF/HF ratio were extracted as classical estimators of autonomic activity. In the nonlinear domain, TE between LF and HF were calculated to quantify the information exchanging between SNS and PNS. Compared with the normal group, an extreme decrease in LF/HF ratio (p = 0.000) and extreme increases in both TE(LF→HF) (p = 0.000) and TE(HF→LF) (p = 0.000) in the CHF group were observed. Moreover, both in normal and CHF groups, TE(LF→HF) was a lot greater than TE(HF→LF) (p = 0.000), revealing that TE was able to distinguish the difference in the amount of directed information transfer among ANS. Extracted features were further applied in discriminating CHF using IBM SPSS Statistics discriminant analysis. The combination of the LF/HF ratio, TE(LF→HF) and TE(HF→LF) reached the highest screening accuracy (83.7%). Our results suggested that TE could serve as a complement to traditional index LF/HF in CHF screening. Full article

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Open AccessEditorial

Evaluation of Systems’ Irregularity and Complexity: Sample Entropy, Its Derivatives, and Their Applications across Scales and Disciplines

by Anne Humeau-Heurtier

Entropy 2018, 20(10), 794; https://doi.org/10.3390/e20100794 - 16 October 2018

Abstract

n/a Full article

Open AccessArticle

An Information-Theoretic Approach to Self-Organisation: Emergence of Complex Interdependencies in Coupled Dynamical Systems

by Fernando Rosas, Pedro A.M. Mediano, Martín Ugarte and Henrik J. Jensen

Entropy 2018, 20(10), 793; https://doi.org/10.3390/e20100793 - 16 October 2018

Abstract

Self-organisation lies at the core of fundamental but still unresolved scientific questions, and holds the promise of de-centralised paradigms crucial for future technological developments. While self-organising processes have been traditionally explained by the tendency of dynamical systems to evolve towards specific configurations, or

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Self-organisation lies at the core of fundamental but still unresolved scientific questions, and holds the promise of de-centralised paradigms crucial for future technological developments. While self-organising processes have been traditionally explained by the tendency of dynamical systems to evolve towards specific configurations, or attractors, we see self-organisation as a consequence of the interdependencies that those attractors induce. Building on this intuition, in this work we develop a theoretical framework for understanding and quantifying self-organisation based on coupled dynamical systems and multivariate information theory. We propose a metric of global structural strength that identifies when self-organisation appears, and a multi-layered decomposition that explains the emergent structure in terms of redundant and synergistic interdependencies. We illustrate our framework on elementary cellular automata, showing how it can detect and characterise the emergence of complex structures. Full article

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Open AccessArticle

Identifying Systemically Important Companies by Using the Credit Network of an Entire Nation

by Sebastian Poledna, Abraham Hinteregger and Stefan Thurner

Entropy 2018, 20(10), 792; https://doi.org/10.3390/e20100792 - 16 October 2018

Abstract

The notions of systemic importance and systemic risk of financial institutions are closely related to the topology of financial liability networks. In this work, we reconstruct and analyze the financial liability network of an entire economy using data of 50,159 firms and banks.

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The notions of systemic importance and systemic risk of financial institutions are closely related to the topology of financial liability networks. In this work, we reconstruct and analyze the financial liability network of an entire economy using data of 50,159 firms and banks. Our analysis contains 80.2% of the total liabilities of firms towards banks and all interbank liabilities in the Austrian banking system. The combination of firm-bank networks and interbank networks allows us to extend the concept of systemic risk to the real economy. In particular, the systemic importance of individual companies can be assessed, and for the first time, the financial ties between the financial and the real economy become explicitly visible. We find that firms contribute to systemic risk in similar ways as banks do. We identify a set of mid-sized companies that carry substantial systemic risk. Their default would affect up to 40% of the Austrian financial market. We find that all firms together create more systemic risk than the entire financial sector. In 2008, the total systemic risk of the Austrian interbank network amounted to only 29% of the total systemic risk of the entire financial network consisting of firms and banks. The work demonstrates that the notions of systemically important financial institutions (SIFIs) can be directly extended to firms. Full article

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Open AccessEditorial

How to Teach Heat Transfer More Systematically by Involving Entropy

by Heinz Herwig

Entropy 2018, 20(10), 791; https://doi.org/10.3390/e20100791 - 15 October 2018

Abstract

In order to teach heat transfer systematically and with a clear physical background, it is recommended that entropy should not be ignored as a fundamental quantity. Heat transfer processes are characterized by introducing the so-called “entropic potential” of the transferred energy, and an

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In order to teach heat transfer systematically and with a clear physical background, it is recommended that entropy should not be ignored as a fundamental quantity. Heat transfer processes are characterized by introducing the so-called “entropic potential” of the transferred energy, and an assessment number is based on this new quantity. Full article

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Open AccessArticle

Maximum Entropy Probability Density Principle in Probabilistic Investigations of Dynamic Systems

by Jiří Náprstek and Cyril Fischer

Entropy 2018, 20(10), 790; https://doi.org/10.3390/e20100790 - 15 October 2018

Abstract

In this study, we consider a method for investigating the stochastic response of a nonlinear dynamical system affected by a random seismic process. We present the solution of the probability density of a single/multiple-degree of freedom (SDOF/MDOF) system with several statically stable equilibrium

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In this study, we consider a method for investigating the stochastic response of a nonlinear dynamical system affected by a random seismic process. We present the solution of the probability density of a single/multiple-degree of freedom (SDOF/MDOF) system with several statically stable equilibrium states and with possible jumps of the snap-through type. The system is a Hamiltonian system with weak damping excited by a system of non-stationary Gaussian white noise. The solution based on the Gibbs principle of the maximum entropy of probability could potentially be implemented in various branches of engineering. The search for the extreme of the Gibbs entropy functional is formulated as a constrained optimization problem. The secondary constraints follow from the Fokker–Planck equation (FPE) for the system considered or from the system of ordinary differential equations for the stochastic moments of the response derived from the relevant FPE. In terms of the application type, this strategy is most suitable for SDOF/MDOF systems containing polynomial type nonlinearities. Thus, the solution links up with the customary formulation of the finite elements discretization for strongly nonlinear continuous systems. Full article

Open AccessArticle

Alternation of Defects and Phase Turbulence Induces Extreme Events in an Extended Microcavity Laser

by Sylvain Barbay, Saliya Coulibaly and Marcel G. Clerc

Entropy 2018, 20(10), 789; https://doi.org/10.3390/e20100789 - 15 October 2018

Abstract

Out-of-equilibrium systems exhibit complex spatiotemporal behaviors when they present a secondary bifurcation to an oscillatory instability. Here, we investigate the complex dynamics shown by a pulsing regime in an extended, one-dimensional semiconductor microcavity laser whose cavity is composed by integrated gain and saturable

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Out-of-equilibrium systems exhibit complex spatiotemporal behaviors when they present a secondary bifurcation to an oscillatory instability. Here, we investigate the complex dynamics shown by a pulsing regime in an extended, one-dimensional semiconductor microcavity laser whose cavity is composed by integrated gain and saturable absorber media. This system is known to give rise experimentally and theoretically to extreme events characterized by rare and high amplitude optical pulses following the onset of spatiotemporal chaos. Based on a theoretical model, we reveal a dynamical behavior characterized by the chaotic alternation of phase and amplitude turbulence. The highest amplitude pulses, i.e., the extreme events, are observed in the phase turbulence zones. This chaotic alternation behavior between different turbulent regimes is at contrast to what is usually observed in a generic amplitude equation model such as the Ginzburg–Landau model. Hence, these regimes provide some insight into the poorly known properties of the complex spatiotemporal dynamics exhibited by secondary instabilities of an Andronov–Hopf bifurcation. Full article

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Open AccessArticle

A Fast Feature Selection Algorithm by Accelerating Computation of Fuzzy Rough Set-Based Information Entropy

by Xiao Zhang, Xia Liu and Yanyan Yang

Entropy 2018, 20(10), 788; https://doi.org/10.3390/e20100788 - 13 October 2018

Abstract

The information entropy developed by Shannon is an effective measure of uncertainty in data, and the rough set theory is a useful tool of computer applications to deal with vagueness and uncertainty data circumstances. At present, the information entropy has been extensively applied

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The information entropy developed by Shannon is an effective measure of uncertainty in data, and the rough set theory is a useful tool of computer applications to deal with vagueness and uncertainty data circumstances. At present, the information entropy has been extensively applied in the rough set theory, and different information entropy models have also been proposed in rough sets. In this paper, based on the existing feature selection method by using a fuzzy rough set-based information entropy, a corresponding fast algorithm is provided to achieve efficient implementation, in which the fuzzy rough set-based information entropy taking as the evaluation measure for selecting features is computed by an improved mechanism with lower complexity. The essence of the acceleration algorithm is to use iterative reduced instances to compute the lambda-conditional entropy. Numerical experiments are further conducted to show the performance of the proposed fast algorithm, and the results demonstrate that the algorithm acquires the same feature subset to its original counterpart, but with significantly less time. Full article

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Open AccessArticle

Variations à la Fourier-Weyl-Wigner on Quantizations of the Plane and the Half-Plane

by Hervé Bergeron and Jean-Pierre Gazeau

Entropy 2018, 20(10), 787; https://doi.org/10.3390/e20100787 - 13 October 2018

Abstract

Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary

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Any quantization maps linearly function on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with the symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all resources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. We first review this quantization scheme before revisiting the cases where symmetry covariance is described by the Weyl-Heisenberg group and the affine group respectively, and we emphasize the fundamental role played by Fourier transform in both cases. As an original outcome of our generalisations of the Wigner-Weyl transform, we show that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations. Full article

Open AccessArticle

A QUBO Formulation of the Stereo Matching Problem for D-Wave Quantum Annealers

by William Cruz-Santos, Salvador E. Venegas-Andraca and Marco Lanzagorta

Entropy 2018, 20(10), 786; https://doi.org/10.3390/e20100786 - 12 October 2018

Abstract

In this paper, we propose a methodology to solve the stereo matching problem through quantum annealing optimization. Our proposal takes advantage of the existing Min-Cut/Max-Flow network formulation of computer vision problems. Based on this network formulation, we construct a quadratic pseudo-Boolean function and

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In this paper, we propose a methodology to solve the stereo matching problem through quantum annealing optimization. Our proposal takes advantage of the existing Min-Cut/Max-Flow network formulation of computer vision problems. Based on this network formulation, we construct a quadratic pseudo-Boolean function and then optimize it through the use of the D-Wave quantum annealing technology. Experimental validation using two kinds of stereo pair of images, random dot stereograms and gray-scale, shows that our methodology is effective. Full article

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