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22 pages, 464 KiB

Open AccessReview

Equilibrium and Non-Equilibrium Lattice Dynamics of Anharmonic Systems

by Keivan Esfarjani and Yuan Liang

Entropy 2022, 24(11), 1585; https://doi.org/10.3390/e24111585 - 1 Nov 2022

Cited by 1 | Viewed by 2100

Abstract

In this review, motivated by the recent interest in high-temperature materials, we review our recent progress in theories of lattice dynamics in and out of equilibrium. To investigate thermodynamic properties of anharmonic crystals, the self-consistent phonon theory was developed, mainly in the 1960s, [...] Read more.

In this review, motivated by the recent interest in high-temperature materials, we review our recent progress in theories of lattice dynamics in and out of equilibrium. To investigate thermodynamic properties of anharmonic crystals, the self-consistent phonon theory was developed, mainly in the 1960s, for rare gas atoms and quantum crystals. We have extended this theory to investigate the properties of the equilibrium state of a crystal, including its unit cell shape and size, atomic positions and lattice dynamical properties. Using the equation-of-motion method combined with the fluctuation–dissipation theorem and the Donsker–Furutsu–Novikov (DFN) theorem, this approach was also extended to investigate the non-equilibrium case where there is heat flow across a junction or an interface. The formalism is a classical one and therefore valid at high temperatures. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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40 pages, 1510 KiB

Open AccessFeature PaperReview

Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties—A Review

by Moisés García Chamorro, Rubén Gómez González and Vicente Garzó

Entropy 2022, 24(6), 826; https://doi.org/10.3390/e24060826 - 14 Jun 2022

Cited by 10 | Viewed by 2666

Abstract

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently [...] Read more.

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures

T_{i}

(measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows. Full article

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18 pages, 3026 KiB

Open AccessReview

Metastability of Synchronous and Asynchronous Dynamics

by Emilio Nicola Maria Cirillo, Vanessa Jacquier and Cristian Spitoni

Entropy 2022, 24(4), 450; https://doi.org/10.3390/e24040450 - 24 Mar 2022

Cited by 4 | Viewed by 2048

Abstract

Metastability is a ubiquitous phenomenon in nature, which interests several fields of natural sciences. Since metastability is a genuine non-equilibrium phenomenon, its description in the framework of thermodynamics and statistical mechanics has progressed slowly for a long time. Since the publication of the [...] Read more.

Metastability is a ubiquitous phenomenon in nature, which interests several fields of natural sciences. Since metastability is a genuine non-equilibrium phenomenon, its description in the framework of thermodynamics and statistical mechanics has progressed slowly for a long time. Since the publication of the first seminal paper in which the metastable behavior of the mean field Curie–Weiss model was approached by means of stochastic techniques, this topic has been largely studied by the scientific community. Several papers and books have been published in which many different spin models were studied and different approaches were developed. In this review, we focus on the comparison between the metastable behavior of synchronous and asynchronous dynamics, namely, stochastic processes in discrete time in which, at each time, either all the spins or one single spin is updated. In particular, we discuss how two different stochastic implementations of the very same Hamiltonian give rise to different metastable behaviors. Full article

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16 pages, 412 KiB

Open AccessReview

A Short Review on Minimum Description Length: An Application to Dimension Reduction in PCA

by Vittoria Bruni, Maria Lucia Cardinali and Domenico Vitulano

Entropy 2022, 24(2), 269; https://doi.org/10.3390/e24020269 - 13 Feb 2022

Cited by 40 | Viewed by 3736

Abstract

The minimun description length (MDL) is a powerful criterion for model selection that is gaining increasing interest from both theorists and practicioners. It allows for automatic selection of the best model for representing data without having a priori information about them. It simply [...] Read more.

The minimun description length (MDL) is a powerful criterion for model selection that is gaining increasing interest from both theorists and practicioners. It allows for automatic selection of the best model for representing data without having a priori information about them. It simply uses both data and model complexity, selecting the model that provides the least coding length among a predefined set of models. In this paper, we briefly review the basic ideas underlying the MDL criterion and its applications in different fields, with particular reference to the dimension reduction problem. As an example, the role of MDL in the selection of the best principal components in the well known PCA is investigated. Full article

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26 pages, 410 KiB

Open AccessReview

λ-Deformation: A Canonical Framework for Statistical Manifolds of Constant Curvature

by Jun Zhang and Ting-Kam Leonard Wong

Entropy 2022, 24(2), 193; https://doi.org/10.3390/e24020193 - 27 Jan 2022

Cited by 5 | Viewed by 2476

Abstract

This paper systematically presents the

λ

-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have [...] Read more.

This paper systematically presents the

λ

-deformation as the canonical framework of deformation to the dually flat (Hessian) geometry, which has been well established in information geometry. We show that, based on deforming the Legendre duality, all objects in the Hessian case have their correspondence in the

λ

-deformed case:

λ

-convexity,

λ

-conjugation,

λ

-biorthogonality,

λ

-logarithmic divergence,

λ

-exponential and

λ

-mixture families, etc. In particular,

λ

-deformation unifies Tsallis and Rényi deformations by relating them to two manifestations of an identical

λ

-exponential family, under subtractive or divisive probability normalization, respectively. Unlike the different Hessian geometries of the exponential and mixture families, the

λ

-exponential family, in turn, coincides with the

λ

-mixture family after a change of random variables. The resulting statistical manifolds, while still carrying a dualistic structure, replace the Hessian metric and a pair of dually flat conjugate affine connections with a conformal Hessian metric and a pair of projectively flat connections carrying constant (nonzero) curvature. Thus,

λ

-deformation is a canonical framework in generalizing the well-known dually flat Hessian structure of information geometry. Full article

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15 pages, 346 KiB

Open AccessReview

A Review of Bayesian Hypothesis Testing and Its Practical Implementations

by Zhengxiao Wei, Aijun Yang, Leno Rocha, Michelle F. Miranda and Farouk S. Nathoo

Entropy 2022, 24(2), 161; https://doi.org/10.3390/e24020161 - 21 Jan 2022

Cited by 7 | Viewed by 5252

Abstract

We discuss hypothesis testing and compare different theories in light of observed or experimental data as fundamental endeavors in the sciences. Issues associated with the p-value approach and null hypothesis significance testing are reviewed, and the Bayesian alternative based on the Bayes [...] Read more.

We discuss hypothesis testing and compare different theories in light of observed or experimental data as fundamental endeavors in the sciences. Issues associated with the p-value approach and null hypothesis significance testing are reviewed, and the Bayesian alternative based on the Bayes factor is introduced, along with a review of computational methods and sensitivity related to prior distributions. We demonstrate how Bayesian testing can be practically implemented in several examples, such as the t-test, two-sample comparisons, linear mixed models, and Poisson mixed models by using existing software. Caveats and potential problems associated with Bayesian testing are also discussed. We aim to inform researchers in the many fields where Bayesian testing is not in common use of a well-developed alternative to null hypothesis significance testing and to demonstrate its standard implementation. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

18 pages, 824 KiB

Open AccessReview

Design and Test of an Integrated Random Number Generator with All-Digital Entropy Source

by Luca Crocetti, Stefano Di Matteo, Pietro Nannipieri, Luca Fanucci and Sergio Saponara

Entropy 2022, 24(2), 139; https://doi.org/10.3390/e24020139 - 18 Jan 2022

Cited by 16 | Viewed by 3916

Abstract

In the cybersecurity field, the generation of random numbers is extremely important because they are employed in different applications such as the generation/derivation of cryptographic keys, nonces, and initialization vectors. The more unpredictable the random sequence, the higher its quality and the lower [...] Read more.

In the cybersecurity field, the generation of random numbers is extremely important because they are employed in different applications such as the generation/derivation of cryptographic keys, nonces, and initialization vectors. The more unpredictable the random sequence, the higher its quality and the lower the probability of recovering the value of those random numbers for an adversary. Cryptographically Secure Pseudo-Random Number Generators (CSPRNGs) are random number generators (RNGs) with specific properties and whose output sequence has such a degree of randomness that it cannot be distinguished from an ideal random sequence. In this work, we designed an all-digital RNG, which includes a Deterministic Random Bit Generator (DRBG) that meets the security requirements for cryptographic applications as CSPRNG, plus an entropy source that showed high portability and a high level of entropy. The proposed design has been intensively tested against both NIST and BSI suites to assess its entropy and randomness, and it is ready to be integrated into the European Processor Initiative (EPI) chip. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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26 pages, 401 KiB

Open AccessReview

Combining Measures of Signal Complexity and Machine Learning for Time Series Analyis: A Review

by Sebastian Raubitzek and Thomas Neubauer

Entropy 2021, 23(12), 1672; https://doi.org/10.3390/e23121672 - 13 Dec 2021

Cited by 15 | Viewed by 4250

Abstract

Measures of signal complexity, such as the Hurst exponent, the fractal dimension, and the Spectrum of Lyapunov exponents, are used in time series analysis to give estimates on persistency, anti-persistency, fluctuations and predictability of the data under study. They have [...] Read more.

Measures of signal complexity, such as the Hurst exponent, the fractal dimension, and the Spectrum of Lyapunov exponents, are used in time series analysis to give estimates on persistency, anti-persistency, fluctuations and predictability of the data under study. They have proven beneficial when doing time series prediction using machine and deep learning and tell what features may be relevant for predicting time-series and establishing complexity features. Further, the performance of machine learning approaches can be improved, taking into account the complexity of the data under study, e.g., adapting the employed algorithm to the inherent long-term memory of the data. In this article, we provide a review of complexity and entropy measures in combination with machine learning approaches. We give a comprehensive review of relevant publications, suggesting the use of fractal or complexity-measure concepts to improve existing machine or deep learning approaches. Additionally, we evaluate applications of these concepts and examine if they can be helpful in predicting and analyzing time series using machine and deep learning. Finally, we give a list of a total of six ways to combine machine learning and measures of signal complexity as found in the literature. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

28 pages, 2598 KiB

Open AccessReview

A Multi-Method Survey on the Use of Sentiment Analysis in Multivariate Financial Time Series Forecasting

by Charalampos M. Liapis, Aikaterini Karanikola and Sotiris Kotsiantis

Entropy 2021, 23(12), 1603; https://doi.org/10.3390/e23121603 - 29 Nov 2021

Cited by 15 | Viewed by 4134

Abstract

In practice, time series forecasting involves the creation of models that generalize data from past values and produce future predictions. Moreover, regarding financial time series forecasting, it can be assumed that the procedure involves phenomena partly shaped by the social environment. Thus, the [...] Read more.

In practice, time series forecasting involves the creation of models that generalize data from past values and produce future predictions. Moreover, regarding financial time series forecasting, it can be assumed that the procedure involves phenomena partly shaped by the social environment. Thus, the present work is concerned with the study of the use of sentiment analysis methods in data extracted from social networks and their utilization in multivariate prediction architectures that involve financial data. Through an extensive experimental process, 22 different input setups using such extracted information were tested, over a total of 16 different datasets, under the schemes of 27 different algorithms. The comparisons were structured under two case studies. The first concerns possible improvements in the performance of the forecasts in light of the use of sentiment analysis systems in time series forecasting. The second, having as a framework all the possible versions of the above configuration, concerns the selection of the methods that perform best. The results, as presented by various illustrations, indicate, on the one hand, the conditional improvement of predictability after the use of specific sentiment setups in long-term forecasts and, on the other, a universal predominance of long short-term memory architectures. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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63 pages, 779 KiB

Open AccessReview

A Review of the System-Intrinsic Nonequilibrium Thermodynamics in Extended Space (MNEQT) with Applications

by Purushottam D. Gujrati

Entropy 2021, 23(12), 1584; https://doi.org/10.3390/e23121584 - 26 Nov 2021

Cited by 2 | Viewed by 2734

Abstract

The review deals with a novel approach (MNEQT) to nonequilibrium thermodynamics (NEQT) that is based on the concept of internal equilibrium (IEQ) in an enlarged state space

S_{Z}

involving internal variables as additional state variables. The IEQ macrostates are unique in [...] Read more.

The review deals with a novel approach (MNEQT) to nonequilibrium thermodynamics (NEQT) that is based on the concept of internal equilibrium (IEQ) in an enlarged state space

S_{Z}

involving internal variables as additional state variables. The IEQ macrostates are unique in

S_{Z}

and have no memory just as EQ macrostates are in the EQ state space

S_{X} \subset S_{Z}

. The approach provides a clear strategy to identify the internal variables for any model through several examples. The MNEQT deals directly with system-intrinsic quantities, which are very useful as they fully describe irreversibility. Because of this, MNEQT solves a long-standing problem in NEQT of identifying a unique global temperature T of a system, thus fulfilling Planck’s dream of a global temperature for any system, even if it is not uniform such as when it is driven between two heat baths; T has the conventional interpretation of satisfying the Clausius statement that the exchange macroheat

d_{e} Q

flows from hot to cold, and other sensible criteria expected of a temperature. The concept of the generalized macroheat

d Q = d_{e} Q + d_{i} Q

converts the Clausius inequality

d S \geq d_{e} Q / T_{0}

for a system in a medium at temperature

T_{0}

into the Clausius equality

d S \equiv d Q / T

, which also covers macrostates with memory, and follows from the extensivity property. The equality also holds for a NEQ isolated system. The novel approach is extremely useful as it also works when no internal state variables are used to study nonunique macrostates in the EQ state space

S_{X}

at the expense of explicit time dependence in the entropy that gives rise to memory effects. To show the usefulness of the novel approach, we give several examples such as irreversible Carnot cycle, friction and Brownian motion, the free expansion, etc. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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24 pages, 858 KiB

Open AccessReview

Information Geometry, Fluctuations, Non-Equilibrium Thermodynamics, and Geodesics in Complex Systems

by Eun-jin Kim

Entropy 2021, 23(11), 1393; https://doi.org/10.3390/e23111393 - 24 Oct 2021

Cited by 53 | Viewed by 4402

Abstract

Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance [...] Read more.

Information theory provides an interdisciplinary method to understand important phenomena in many research fields ranging from astrophysical and laboratory fluids/plasmas to biological systems. In particular, information geometric theory enables us to envision the evolution of non-equilibrium processes in terms of a (dimensionless) distance by quantifying how information unfolds over time as a probability density function (PDF) evolves in time. Here, we discuss some recent developments in information geometric theory focusing on time-dependent dynamic aspects of non-equilibrium processes (e.g., time-varying mean value, time-varying variance, or temperature, etc.) and their thermodynamic and physical/biological implications. We compare different distances between two given PDFs and highlight the importance of a path-dependent distance for a time-dependent PDF. We then discuss the role of the information rate

Γ = \frac{d L}{d t}

and relative entropy in non-equilibrium thermodynamic relations (entropy production rate, heat flux, dissipated work, non-equilibrium free energy, etc.), and various inequalities among them. Here,

L

is the information length representing the total number of statistically distinguishable states a PDF evolves through over time. We explore the implications of a geodesic solution in information geometry for self-organization and control. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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28 pages, 11229 KiB

Open AccessReview

Information Entropy in Chemistry: An Overview

by Denis Sh. Sabirov and Igor S. Shepelevich

Entropy 2021, 23(10), 1240; https://doi.org/10.3390/e23101240 - 23 Sep 2021

Cited by 79 | Viewed by 8859

Abstract

Basic applications of the information entropy concept to chemical objects are reviewed. These applications deal with quantifying chemical and electronic structures of molecules, signal processing, structural studies on crystals, and molecular ensembles. Recent advances in the mentioned areas make information entropy a central [...] Read more.

Basic applications of the information entropy concept to chemical objects are reviewed. These applications deal with quantifying chemical and electronic structures of molecules, signal processing, structural studies on crystals, and molecular ensembles. Recent advances in the mentioned areas make information entropy a central concept in interdisciplinary studies on digitalizing chemical reactions, chemico-information synthesis, crystal engineering, as well as digitally rethinking basic notions of structural chemistry in terms of informatics. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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22 pages, 880 KiB

Open AccessEditor’s ChoiceReview

Three-Factor Kinetic Equation of Catalyst Deactivation

by Zoë Gromotka, Gregory Yablonsky, Nickolay Ostrovskii and Denis Constales

Entropy 2021, 23(7), 818; https://doi.org/10.3390/e23070818 - 27 Jun 2021

Cited by 9 | Viewed by 3259

Abstract

The three-factor kinetic equation of catalyst deactivation was obtained in terms of apparent kinetic parameters. The three factors correspond to the main cycle with a linear, detailed mechanism regarding the catalytic intermediates, a cycle of reversible deactivation, and a stage of irreversible deactivation [...] Read more.

The three-factor kinetic equation of catalyst deactivation was obtained in terms of apparent kinetic parameters. The three factors correspond to the main cycle with a linear, detailed mechanism regarding the catalytic intermediates, a cycle of reversible deactivation, and a stage of irreversible deactivation (aging), respectively. The rate of the main cycle is obtained for the fresh catalyst under a quasi-steady-state assumption. The phenomena of reversible and irreversible deactivation are presented as special separate factors (hierarchical separation). In this case, the reversible deactivation factor is a function of the kinetic apparent parameters of the reversible deactivation and of those of the main cycle. The irreversible deactivation factor is a function of the apparent kinetic parameters of the main cycle, of the reversible deactivation, and of the irreversible deactivation. The conditions of such separability are found. The obtained equation is applied successfully to describe the literature data on the reversible catalyst deactivation processes in the dehydration of acetaldehyde over

T i O_{2}

anatase and in crotonaldehyde hydrogenation on supported metal catalysts. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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35 pages, 1675 KiB

Open AccessEditor’s ChoiceReview

The Entropy Universe

by Maria Ribeiro, Teresa Henriques, Luísa Castro, André Souto, Luís Antunes, Cristina Costa-Santos and Andreia Teixeira

Entropy 2021, 23(2), 222; https://doi.org/10.3390/e23020222 - 11 Feb 2021

Cited by 59 | Viewed by 12698

Abstract

About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical [...] Read more.

About 160 years ago, the concept of entropy was introduced in thermodynamics by Rudolf Clausius. Since then, it has been continually extended, interpreted, and applied by researchers in many scientific fields, such as general physics, information theory, chaos theory, data mining, and mathematical linguistics. This paper presents The Entropy Universe, which aims to review the many variants of entropies applied to time-series. The purpose is to answer research questions such as: How did each entropy emerge? What is the mathematical definition of each variant of entropy? How are entropies related to each other? What are the most applied scientific fields for each entropy? We describe in-depth the relationship between the most applied entropies in time-series for different scientific fields, establishing bases for researchers to properly choose the variant of entropy most suitable for their data. The number of citations over the past sixteen years of each paper proposing a new entropy was also accessed. The Shannon/differential, the Tsallis, the sample, the permutation, and the approximate entropies were the most cited ones. Based on the ten research areas with the most significant number of records obtained in the Web of Science and Scopus, the areas in which the entropies are more applied are computer science, physics, mathematics, and engineering. The universe of entropies is growing each day, either due to the introducing new variants either due to novel applications. Knowing each entropy’s strengths and of limitations is essential to ensure the proper improvement of this research field. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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23 pages, 484 KiB

Open AccessReview

An Optimality Summary: Secret Key Agreement with Physical Unclonable Functions

by Onur Günlü and Rafael F. Schaefer

Entropy 2021, 23(1), 16; https://doi.org/10.3390/e23010016 - 24 Dec 2020

Cited by 10 | Viewed by 2912

Abstract

We address security and privacy problems for digital devices and biometrics from an information-theoretic optimality perspective to conduct authentication, message encryption/decryption, identification or secure and private computations by using a secret key. A physical unclonable function (PUF) provides local security to digital devices [...] Read more.

We address security and privacy problems for digital devices and biometrics from an information-theoretic optimality perspective to conduct authentication, message encryption/decryption, identification or secure and private computations by using a secret key. A physical unclonable function (PUF) provides local security to digital devices and this review gives the most relevant summary for information theorists, coding theorists, and signal processing community members who are interested in optimal PUF constructions. Low-complexity signal processing methods are applied to simplify information-theoretic analyses. The best trade-offs between the privacy-leakage, secret-key, and storage rates are discussed. Proposed optimal constructions that jointly design the vector quantizer and error-correction code parameters are listed. These constructions include modern and algebraic codes such as polar codes and convolutional codes, both of which can achieve small block-error probabilities at short block lengths, corresponding to a small number of PUF circuits. Open problems in the PUF literature from signal processing, information theory, coding theory, and hardware complexity perspectives and their combinations are listed to stimulate further advancements in the research on local privacy and security. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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26 pages, 466 KiB

Open AccessReview

A Survey of Information Entropy Metrics for Complex Networks

by Yamila M. Omar and Peter Plapper

Entropy 2020, 22(12), 1417; https://doi.org/10.3390/e22121417 - 15 Dec 2020

Cited by 50 | Viewed by 5921

Abstract

Information entropy metrics have been applied to a wide range of problems that were abstracted as complex networks. This growing body of research is scattered in multiple disciplines, which makes it difficult to identify available metrics and understand the context in which they [...] Read more.

Information entropy metrics have been applied to a wide range of problems that were abstracted as complex networks. This growing body of research is scattered in multiple disciplines, which makes it difficult to identify available metrics and understand the context in which they are applicable. In this work, a narrative literature review of information entropy metrics for complex networks is conducted following the PRISMA guidelines. Existing entropy metrics are classified according to three different criteria: whether the metric provides a property of the graph or a graph component (such as the nodes), the chosen probability distribution, and the types of complex networks to which the metrics are applicable. Consequently, this work identifies the areas in need for further development aiming to guide future research efforts. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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17 pages, 5105 KiB

Open AccessReview

A Review of Fractional Order Entropies

by António M. Lopes and José A. Tenreiro Machado

Entropy 2020, 22(12), 1374; https://doi.org/10.3390/e22121374 - 5 Dec 2020

Cited by 24 | Viewed by 3167

Abstract

Fractional calculus (FC) is the area of calculus that generalizes the operations of differentiation and integration. FC operators are non-local and capture the history of dynamical effects present in many natural and artificial phenomena. Entropy is a measure of uncertainty, diversity and randomness [...] Read more.

Fractional calculus (FC) is the area of calculus that generalizes the operations of differentiation and integration. FC operators are non-local and capture the history of dynamical effects present in many natural and artificial phenomena. Entropy is a measure of uncertainty, diversity and randomness often adopted for characterizing complex dynamical systems. Stemming from the synergies between the two areas, this paper reviews the concept of entropy in the framework of FC. Several new entropy definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. However, FC is not yet well disseminated in the community of entropy. Therefore, new definitions based on FC can generalize both concepts in the theoretical and applied points of view. The time to come will prove to what extend the new formulations will be useful. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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26 pages, 425 KiB

Open AccessReview

Geometric Aspects of the Isentropic Liquid Dynamics and Vorticity Invariants

by Alexander A. Balinsky, Denis Blackmore, Radosław Kycia and Anatolij K. Prykarpatski

Entropy 2020, 22(11), 1241; https://doi.org/10.3390/e22111241 - 31 Oct 2020

Cited by 2 | Viewed by 2139

Abstract

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the [...] Read more.

We review a modern differential geometric description of fluid isentropic motion and features of it including diffeomorphism group structure, modelling the related dynamics, as well as its compatibility with the quasi-stationary thermodynamical constraints. We analyze the adiabatic liquid dynamics, within which, following the general approach, the nature of the related Poissonian structure on the fluid motion phase space as a semidirect Banach groups product, and a natural reduction of the canonical symplectic structure on its cotangent space to the classical Lie-Poisson bracket on the adjoint space to the corresponding semidirect Lie algebras product are explained in detail. We also present a modification of the Hamiltonian analysis in case of a flow governed by isothermal liquid dynamics. We study the differential-geometric structure of isentropic magneto-hydrodynamic superfluid phase space and its related motion within the Hamiltonian analysis and related invariant theory. In particular, we construct an infinite hierarchy of different kinds of integral magneto-hydrodynamic invariants, generalizing those previously constructed in the literature, and analyzing their differential-geometric origins. A charged liquid dynamics on the phase space invariant with respect to an abelian gauge group transformation is also investigated, and some generalizations of the canonical Lie-Poisson type bracket is presented. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

61 pages, 2469 KiB

Open AccessReview

An Elementary Introduction to Information Geometry

by Frank Nielsen

Entropy 2020, 22(10), 1100; https://doi.org/10.3390/e22101100 - 29 Sep 2020

Cited by 81 | Viewed by 43622

Abstract

In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential [...] Read more.

In this survey, we describe the fundamental differential-geometric structures of information manifolds, state the fundamental theorem of information geometry, and illustrate some use cases of these information manifolds in information sciences. The exposition is self-contained by concisely introducing the necessary concepts of differential geometry. Proofs are omitted for brevity. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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24 pages, 11131 KiB

Open AccessReview

Thermophoretic Micron-Scale Devices: Practical Approach and Review

by Namkyu Lee and Simone Wiegand

Entropy 2020, 22(9), 950; https://doi.org/10.3390/e22090950 - 28 Aug 2020

Cited by 22 | Viewed by 4401

Abstract

In recent years, there has been increasing interest in the development of micron-scale devices utilizing thermal gradients to manipulate molecules and colloids, and to measure their thermophoretic properties quantitatively. Various devices have been realized, such as on-chip implements, micro-thermogravitational columns and other micron-scale [...] Read more.

In recent years, there has been increasing interest in the development of micron-scale devices utilizing thermal gradients to manipulate molecules and colloids, and to measure their thermophoretic properties quantitatively. Various devices have been realized, such as on-chip implements, micro-thermogravitational columns and other micron-scale thermophoretic cells. The advantage of the miniaturized devices lies in the reduced sample volume. Often, a direct observation of particles using various microscopic techniques is possible. On the other hand, the small dimensions lead to some technical problems, such as a precise temperature measurement on small length scale with high spatial resolution. In this review, we will focus on the “state of the art” thermophoretic micron-scale devices, covering various aspects such as generating temperature gradients, temperature measurement, and the analysis of the current micron-scale devices. We want to give researchers an orientation for their development of thermophoretic micron-scale devices for biological, chemical, analytical, and medical applications. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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18 pages, 422 KiB

Open AccessReview

Stock Market Volatility and Return Analysis: A Systematic Literature Review

by Roni Bhowmik and Shouyang Wang

Entropy 2020, 22(5), 522; https://doi.org/10.3390/e22050522 - 4 May 2020

Cited by 61 | Viewed by 19780

Abstract

In the field of business research method, a literature review is more relevant than ever. Even though there has been lack of integrity and inflexibility in traditional literature reviews with questions being raised about the quality and trustworthiness of these types of reviews. [...] Read more.

In the field of business research method, a literature review is more relevant than ever. Even though there has been lack of integrity and inflexibility in traditional literature reviews with questions being raised about the quality and trustworthiness of these types of reviews. This research provides a literature review using a systematic database to examine and cross-reference snowballing. In this paper, previous studies featuring a generalized autoregressive conditional heteroskedastic (GARCH) family-based model stock market return and volatility have also been reviewed. The stock market plays a pivotal role in today’s world economic activities, named a “barometer” and “alarm” for economic and financial activities in a country or region. In order to prevent uncertainty and risk in the stock market, it is particularly important to measure effectively the volatility of stock index returns. However, the main purpose of this review is to examine effective GARCH models recommended for performing market returns and volatilities analysis. The secondary purpose of this review study is to conduct a content analysis of return and volatility literature reviews over a period of 12 years (2008–2019) and in 50 different papers. The study found that there has been a significant change in research work within the past 10 years and most of researchers have worked for developing stock markets. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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6 pages, 211 KiB

Open AccessReview

How Incomputable Is Kolmogorov Complexity?

by Paul M.B. Vitányi

Entropy 2020, 22(4), 408; https://doi.org/10.3390/e22040408 - 3 Apr 2020

Cited by 20 | Viewed by 4063

Abstract

Kolmogorov complexity is the length of the ultimately compressed version of a file (i.e., anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. We discuss the incomputability of [...] Read more.

Kolmogorov complexity is the length of the ultimately compressed version of a file (i.e., anything which can be put in a computer). Formally, it is the length of a shortest program from which the file can be reconstructed. We discuss the incomputability of Kolmogorov complexity, which formal loopholes this leaves us with, recent approaches to compute or approximate Kolmogorov complexity, which approaches are problematic, and which approaches are viable. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

32 pages, 5963 KiB

Open AccessEditor’s ChoiceReview

On the Evidence of Thermodynamic Self-Organization during Fatigue: A Review

by Mehdi Naderi

Entropy 2020, 22(3), 372; https://doi.org/10.3390/e22030372 - 24 Mar 2020

Cited by 7 | Viewed by 6020

Abstract

In this review paper, the evidence and application of thermodynamic self-organization are reviewed for metals typically with single crystals subjected to cyclic loading. The theory of self-organization in thermodynamic processes far from equilibrium is a cutting-edge theme for the development of a new [...] Read more.

In this review paper, the evidence and application of thermodynamic self-organization are reviewed for metals typically with single crystals subjected to cyclic loading. The theory of self-organization in thermodynamic processes far from equilibrium is a cutting-edge theme for the development of a new generation of materials. It could be interpreted as the formation of globally coherent patterns, configurations and orderliness through local interactivities by “cascade evolution of dissipative structures”. Non-equilibrium thermodynamics, entropy, and dissipative structures connected to self-organization phenomenon (patterning, orderliness) are briefly discussed. Some example evidences are reviewed in detail to show how thermodynamics self-organization can emerge from a non-equilibrium process; fatigue. Evidences including dislocation density evolution, stored energy, temperature, and acoustic signals can be considered as the signature of self-organization. Most of the attention is given to relate an analogy between persistent slip bands (PSBs) and self-organization in metals with single crystals. Some aspects of the stability of dislocations during fatigue of single crystals are discussed using the formulation of excess entropy generation. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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13 pages, 447 KiB

Open AccessOpinion

Closed Formula for Transport across Constrictions

by Paolo Malgaretti and Jens Harting

Entropy 2023, 25(3), 470; https://doi.org/10.3390/e25030470 - 8 Mar 2023

Cited by 4 | Viewed by 1561

Abstract

In the last decade, the Fick–Jacobs approximation has been exploited to capture transport across constrictions. Here, we review the derivation of the Fick–Jacobs equation with particular emphasis on its linear response regime. We show that, for fore-aft symmetric channels, the flux of noninteracting [...] Read more.

In the last decade, the Fick–Jacobs approximation has been exploited to capture transport across constrictions. Here, we review the derivation of the Fick–Jacobs equation with particular emphasis on its linear response regime. We show that, for fore-aft symmetric channels, the flux of noninteracting systems is fully captured by its linear response regime. For this case, we derive a very simple formula that captures the correct trends and can be exploited as a simple tool to design experiments or simulations. Lastly, we show that higher-order corrections in the flux may appear for nonsymmetric channels. Full article

(This article belongs to the Special Issue Review Papers for Entropy)

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