Entropy

15 pages, 621 KiB

Open AccessEditor’s ChoiceArticle

Modified Distribution Entropy as a Complexity Measure of Heart Rate Variability (HRV) Signal

by Radhagayathri Udhayakumar, Chandan Karmakar, Peng Li, Xinpei Wang and Marimuthu Palaniswami

Entropy 2020, 22(10), 1077; https://doi.org/10.3390/e22101077 - 24 Sep 2020

Cited by 6 | Viewed by 3779

The complexity of a heart rate variability (HRV) signal is considered an important nonlinear feature to detect cardiac abnormalities. This work aims at explaining the physiological meaning of a recently developed complexity measurement method, namely, distribution entropy ( [...] Read more.

The complexity of a heart rate variability (HRV) signal is considered an important nonlinear feature to detect cardiac abnormalities. This work aims at explaining the physiological meaning of a recently developed complexity measurement method, namely, distribution entropy (

D i s t E n

), in the context of HRV signal analysis. We thereby propose modified distribution entropy (

m D i s t E n

) to remove the physiological discrepancy involved in the computation of

D i s t E n

. The proposed method generates a distance matrix that is devoid of over-exerted multi-lag signal changes. Restricted element selection in the distance matrix makes “

m D i s t E n

” a computationally inexpensive and physiologically more relevant complexity measure in comparison to

D i s t E n

. Full article

(This article belongs to the Special Issue Entropy in Data Analysis)

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14 pages, 1935 KiB

Open AccessEditor’s ChoiceArticle

The Quantum Friction and Optimal Finite-Time Performance of the Quantum Otto Cycle

by Andrea R. Insinga

Entropy 2020, 22(9), 1060; https://doi.org/10.3390/e22091060 - 22 Sep 2020

Cited by 24 | Viewed by 3263

Abstract

In this work we considered the quantum Otto cycle within an optimization framework. The goal was maximizing the power for a heat engine or maximizing the cooling power for a refrigerator. In the field of finite-time quantum thermodynamics it is common to consider [...] Read more.

In this work we considered the quantum Otto cycle within an optimization framework. The goal was maximizing the power for a heat engine or maximizing the cooling power for a refrigerator. In the field of finite-time quantum thermodynamics it is common to consider frictionless trajectories since these have been shown to maximize the work extraction during the adiabatic processes. Furthermore, for frictionless cycles, the energy of the system decouples from the other degrees of freedom, thereby simplifying the mathematical treatment. Instead, we considered general limit cycles and we used analytical techniques to compute the derivative of the work production over the whole cycle with respect to the time allocated for each of the adiabatic processes. By doing so, we were able to directly show that the frictionless cycle maximizes the work production, implying that the optimal power production must necessarily allow for some friction generation so that the duration of the cycle is reduced. Full article

(This article belongs to the Special Issue Finite-Time Thermodynamics)

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10 pages, 2686 KiB

Open AccessEditor’s ChoiceArticle

Surface-Codes-Based Quantum Communication Networks

by Ivan B. Djordjevic

Entropy 2020, 22(9), 1059; https://doi.org/10.3390/e22091059 - 22 Sep 2020

Cited by 5 | Viewed by 4356

Abstract

In this paper, we propose the surface codes (SCs)-based multipartite quantum communication networks (QCNs). We describe an approach that enables us to simultaneously entangle multiple nodes in an arbitrary network topology based on the SCs. We also describe how to extend the transmission [...] Read more.

In this paper, we propose the surface codes (SCs)-based multipartite quantum communication networks (QCNs). We describe an approach that enables us to simultaneously entangle multiple nodes in an arbitrary network topology based on the SCs. We also describe how to extend the transmission distance between arbitrary two nodes by using the SCs. The numerical results indicate that transmission distance between nodes can be extended to beyond 1000 km by employing simple syndrome decoding. Finally, we describe how to operate the proposed QCN by employing the software-defined networking (SDN) concept. Full article

(This article belongs to the Special Issue Physical-Layer Security, Quantum Key Distribution and Post-quantum Cryptography)

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22 pages, 3947 KiB

Open AccessEditor’s ChoiceArticle

TMEA: A Thermodynamically Motivated Framework for Functional Characterization of Biological Responses to System Acclimation

by Kevin Schneider, Benedikt Venn and Timo Mühlhaus

Entropy 2020, 22(9), 1030; https://doi.org/10.3390/e22091030 - 15 Sep 2020

Cited by 5 | Viewed by 4095

Abstract

The objective of gene set enrichment analysis (GSEA) in modern biological studies is to identify functional profiles in huge sets of biomolecules generated by high-throughput measurements of genes, transcripts, metabolites, and proteins. GSEA is based on a two-stage process using classical statistical analysis [...] Read more.

The objective of gene set enrichment analysis (GSEA) in modern biological studies is to identify functional profiles in huge sets of biomolecules generated by high-throughput measurements of genes, transcripts, metabolites, and proteins. GSEA is based on a two-stage process using classical statistical analysis to score the input data and subsequent testing for overrepresentation of the enrichment score within a given functional coherent set. However, enrichment scores computed by different methods are merely statistically motivated and often elusive to direct biological interpretation. Here, we propose a novel approach, called Thermodynamically Motivated Enrichment Analysis (TMEA), to account for the energy investment in biological relevant processes. Therefore, TMEA is based on surprisal analysis, which offers a thermodynamic-free energy-based representation of the biological steady state and of the biological change. The contribution of each biomolecule underlying the changes in free energy is used in a Monte Carlo resampling procedure resulting in a functional characterization directly coupled to the thermodynamic characterization of biological responses to system perturbations. To illustrate the utility of our method on real experimental data, we benchmark our approach on plant acclimation to high light and compare the performance of TMEA with the most frequently used method for GSEA. Full article

(This article belongs to the Special Issue Thermodynamics and Information Theory of Living Systems)

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Graphical abstract

27 pages, 1225 KiB

Open AccessEditor’s ChoiceArticle

Life’s Energy and Information: Contrasting Evolution of Volume- versus Surface-Specific Rates of Energy Consumption

by Anastassia M. Makarieva, Andrei V. Nefiodov and Bai-Lian Li

Entropy 2020, 22(9), 1025; https://doi.org/10.3390/e22091025 - 13 Sep 2020

Cited by 11 | Viewed by 5651

Abstract

As humanity struggles to find a path to resilience amidst global change vagaries, understanding organizing principles of living systems as the pillar for human existence is rapidly growing in importance. However, finding quantitative definitions for order, complexity, information and functionality of living systems [...] Read more.

As humanity struggles to find a path to resilience amidst global change vagaries, understanding organizing principles of living systems as the pillar for human existence is rapidly growing in importance. However, finding quantitative definitions for order, complexity, information and functionality of living systems remains a challenge. Here, we review and develop insights into this problem from the concept of the biotic regulation of the environment developed by Victor Gorshkov (1935–2019). Life’s extraordinary persistence—despite being a strongly non-equilibrium process—requires a quantum-classical duality: the program of life is written in molecules and thus can be copied without information loss, while life’s interaction with its non-equilibrium environment is performed by macroscopic classical objects (living individuals) that age. Life’s key energetic parameter, the volume-specific rate of energy consumption, is maintained within universal limits by most life forms. Contrary to previous suggestions, it cannot serve as a proxy for “evolutionary progress”. In contrast, ecosystem-level surface-specific energy consumption declines with growing animal body size in stable ecosystems. High consumption by big animals is associated with instability. We suggest that the evolutionary increase in body size may represent a spontaneous loss of information about environmental regulation, a manifestation of life’s algorithm ageing as a whole. Full article

(This article belongs to the Special Issue Evolution and Thermodynamics)

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25 pages, 10245 KiB

Open AccessEditor’s ChoiceArticle

Investigation of Forced Convection Enhancement and Entropy Generation of Nanofluid Flow through a Corrugated Minichannel Filled with a Porous Media

by Ehsan Aminian, Hesam Moghadasi, Hamid Saffari and Amir Mirza Gheitaghy

Entropy 2020, 22(9), 1008; https://doi.org/10.3390/e22091008 - 9 Sep 2020

Cited by 26 | Viewed by 3619

Abstract

Corrugating channel wall is considered to be an efficient procedure for achieving improved heat transfer. Further enhancement can be obtained through the utilization of nanofluids and porous media with high thermal conductivity. This paper presents the effect of geometrical parameters for the determination [...] Read more.

Corrugating channel wall is considered to be an efficient procedure for achieving improved heat transfer. Further enhancement can be obtained through the utilization of nanofluids and porous media with high thermal conductivity. This paper presents the effect of geometrical parameters for the determination of an appropriate configuration. Furthermore, the optimization of forced convective heat transfer and fluid/nanofluid flow through a sinusoidal wavy-channel inside a porous medium is performed through the optimization of entropy generation. The fluid flow in porous media is considered to be laminar and Darcy–Brinkman–Forchheimer model has been utilized. The obtained results were compared with the corresponding numerical data in order to ensure the accuracy and reliability of the numerical procedure. As a result, increasing the Darcy number leads to the increased portion of thermal entropy generation as well as the decreased portion of frictional entropy generation in all configurations. Moreover, configuration with wavelength of 10 mm, amplitude of 0.5 mm and phase shift of 60° was selected as an optimum geometry for further investigations on the addition of nanoparticles. Additionally, increasing trend of average Nusselt number and friction factor, besides the decreasing trend of performance evaluation criteria (PEC) index, were inferred by increasing the volume fraction of the nanofluid (Al₂O₃ and CuO). Full article

(This article belongs to the Special Issue Thermal Radiation and Entropy Analysis)

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20 pages, 1025 KiB

Open AccessEditor’s ChoiceArticle

Using Matrix-Product States for Open Quantum Many-Body Systems: Efficient Algorithms for Markovian and Non-Markovian Time-Evolution

by Regina Finsterhölzl, Manuel Katzer, Andreas Knorr and Alexander Carmele

Entropy 2020, 22(9), 984; https://doi.org/10.3390/e22090984 - 4 Sep 2020

Cited by 12 | Viewed by 6851

Abstract

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are [...] Read more.

This paper presents an efficient algorithm for the time evolution of open quantum many-body systems using matrix-product states (MPS) proposing a convenient structure of the MPS-architecture, which exploits the initial state of system and reservoir. By doing so, numerically expensive re-ordering protocols are circumvented. It is applicable to systems with a Markovian type of interaction, where only the present state of the reservoir needs to be taken into account. Its adaption to a non-Markovian type of interaction between the many-body system and the reservoir is demonstrated, where the information backflow from the reservoir needs to be included in the computation. Also, the derivation of the basis in the quantum stochastic Schrödinger picture is shown. As a paradigmatic model, the Heisenberg spin chain with nearest-neighbor interaction is used. It is demonstrated that the algorithm allows for the access of large systems sizes. As an example for a non-Markovian type of interaction, the generation of highly unusual steady states in the many-body system with coherent feedback control is demonstrated for a chain length of

N = 30

. Full article

(This article belongs to the Special Issue Open Quantum Systems (OQS) for Quantum Technologies)

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10 pages, 365 KiB

Open AccessEditor’s ChoiceArticle

Strong Coupling and Nonextensive Thermodynamics

by Rodrigo de Miguel and J. Miguel Rubí

Entropy 2020, 22(9), 975; https://doi.org/10.3390/e22090975 - 1 Sep 2020

Cited by 7 | Viewed by 3269

Abstract

We propose a Hamiltonian-based approach to the nonextensive thermodynamics of small systems, where small is a relative term comparing the size of the system to the size of the effective interaction region around it. We show that the effective Hamiltonian approach gives easy [...] Read more.

We propose a Hamiltonian-based approach to the nonextensive thermodynamics of small systems, where small is a relative term comparing the size of the system to the size of the effective interaction region around it. We show that the effective Hamiltonian approach gives easy accessibility to the thermodynamic properties of systems strongly coupled to their surroundings. The theory does not rely on the classical concept of dividing surface to characterize the system’s interaction with the environment. Instead, it defines an effective interaction region over which a system exchanges extensive quantities with its surroundings, easily producing laws recently shown to be valid at the nanoscale. Full article

(This article belongs to the Section Thermodynamics)

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18 pages, 331 KiB

Open AccessEditor’s ChoiceArticle

Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

by Rohisha Tuladhar, Fidel Santamaria and Ivanka Stamova

Entropy 2020, 22(9), 970; https://doi.org/10.3390/e22090970 - 31 Aug 2020

Cited by 11 | Viewed by 3077

Abstract

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect [...] Read more.

We present a biological fractional n-species delayed cooperation model of Lotka-Volterra type. The considered fractional derivatives are in the Caputo sense. Impulsive control strategies are applied for several stability properties of the states, namely Mittag-Leffler stability, practical stability and stability with respect to sets. The proposed results extend the existing stability results for integer-order

n -

species delayed Lotka-Volterra cooperation models to the fractional-order case under impulsive control. Full article

(This article belongs to the Special Issue Dynamics in Complex Neural Networks)

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36 pages, 2612 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On Products of Random Matrices

by Natalia Amburg, Aleksander Orlov and Dmitry Vasiliev

Entropy 2020, 22(9), 972; https://doi.org/10.3390/e22090972 - 31 Aug 2020

Cited by 9 | Viewed by 3187

Abstract

We introduce a family of models, which we name matrix models associated with children’s drawings—the so-called dessin d’enfant. Dessins d’enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are [...] Read more.

We introduce a family of models, which we name matrix models associated with children’s drawings—the so-called dessin d’enfant. Dessins d’enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the “spectrum of stars”. The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory. Full article

(This article belongs to the Special Issue Random Matrix Approaches in Classical and Quantum Information Theory)

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14 pages, 1854 KiB

Open AccessEditor’s ChoiceArticle

A New Look on Financial Markets Co-Movement through Cooperative Dynamics in Many-Body Physics

by María Nieves López-García, Miguel Angel Sánchez-Granero, Juan Evangelista Trinidad-Segovia, Antonio Manuel Puertas and Francisco Javier De las Nieves

Entropy 2020, 22(9), 954; https://doi.org/10.3390/e22090954 - 29 Aug 2020

Cited by 9 | Viewed by 3137

Abstract

One of the main contributions of the Capital Assets Pricing Model (CAPM) to portfolio theory was to explain the correlation between assets through its relationship with the market index. According to this approach, the market index is expected to explain the co-movement between [...] Read more.

One of the main contributions of the Capital Assets Pricing Model (CAPM) to portfolio theory was to explain the correlation between assets through its relationship with the market index. According to this approach, the market index is expected to explain the co-movement between two different stocks to a great extent. In this paper, we try to verify this hypothesis using a sample of 3.000 stocks of the USA market (attending to liquidity, capitalization, and free float criteria) by using some functions inspired by cooperative dynamics in physical particle systems. We will show that all of the co-movement among the stocks is completely explained by the market, even without considering the market beta of the stocks. Full article

(This article belongs to the Special Issue Information Theory and Economic Network)

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14 pages, 1365 KiB

Open AccessEditor’s ChoiceArticle

Effect of Machine Entropy Production on the Optimal Performance of a Refrigerator

by Michel Feidt and Monica Costea

Entropy 2020, 22(9), 913; https://doi.org/10.3390/e22090913 - 20 Aug 2020

Cited by 15 | Viewed by 2975

Abstract

The need for cooling is more and more important in current applications, as environmental constraints become more and more restrictive. Therefore, the optimization of reverse cycle machines is currently required. This optimization could be split in two parts, namely, (1) the design optimization, [...] Read more.

The need for cooling is more and more important in current applications, as environmental constraints become more and more restrictive. Therefore, the optimization of reverse cycle machines is currently required. This optimization could be split in two parts, namely, (1) the design optimization, leading to an optimal dimensioning to fulfill the specific demand (static or nominal steady state optimization); and (2) the dynamic optimization, where the demand fluctuates, and the system must be continuously adapted. Thus, the variability of the system load (with or without storage) implies its careful control-command. The topic of this paper is concerned with part (1) and proposes a novel and more complete modeling of an irreversible Carnot refrigerator that involves the coupling between sink (source) and machine through a heat transfer constraint. Moreover, it induces the choice of a reference heat transfer entropy, which is the heat transfer entropy at the source of a Carnot irreversible refrigerator. The thermodynamic optimization of the refrigerator provides new results regarding the optimal allocation of heat transfer conductances and minimum energy consumption with associated coefficient of performance (COP) when various forms of entropy production owing to internal irreversibility are considered. The reported results and their consequences represent a new fundamental step forward regarding the performance upper bound of Carnot irreversible refrigerator. Full article

(This article belongs to the Special Issue Thermodynamics of Heat Pump and Refrigeration Cycles)

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35 pages, 4137 KiB

Open AccessEditor’s ChoiceArticle

Rate of Entropy Production in Evolving Interfaces and Membranes under Astigmatic Kinematics: Shape Evolution in Geometric-Dissipation Landscapes

by Ziheng Wang, Phillip Servio and Alejandro D. Rey

Entropy 2020, 22(9), 909; https://doi.org/10.3390/e22090909 - 19 Aug 2020

Cited by 10 | Viewed by 3924

Abstract

This paper presents theory and simulation of viscous dissipation in evolving interfaces and membranes under kinematic conditions, known as astigmatic flow, ubiquitous during growth processes in nature. The essential aim is to characterize and explain the underlying connections between curvedness and shape evolution [...] Read more.

This paper presents theory and simulation of viscous dissipation in evolving interfaces and membranes under kinematic conditions, known as astigmatic flow, ubiquitous during growth processes in nature. The essential aim is to characterize and explain the underlying connections between curvedness and shape evolution and the rate of entropy production due to viscous bending and torsion rates. The membrane dissipation model used here is known as the Boussinesq-Scriven fluid model. Since the standard approaches in morphological evolution are based on the average, Gaussian and deviatoric curvatures, which comingle shape with curvedness, this paper introduces a novel decoupled approach whereby shape is independent of curvedness. In this curvedness-shape landscape, the entropy production surface under constant homogeneous normal velocity decays with growth but oscillates with shape changes. Saddles and spheres are minima while cylindrical patches are maxima. The astigmatic flow trajectories on the entropy production surface, show that only cylinders and spheres grow under the constant shape. Small deviations from cylindrical shapes evolve towards spheres or saddles depending on the initial condition, where dissipation rates decrease. Taken together the results and analysis provide novel and significant relations between shape evolution and viscous dissipation in deforming viscous membrane and surfaces. Full article

(This article belongs to the Special Issue Statistical Physics of Soft Matter and Complex Systems)

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20 pages, 315 KiB

Open AccessEditor’s ChoiceArticle

Finite-Time Thermodynamics in Economics

by Anatoly Tsirlin and Larisa Gagarina

Entropy 2020, 22(8), 891; https://doi.org/10.3390/e22080891 - 13 Aug 2020

Cited by 19 | Viewed by 3449

Abstract

In this paper, we consider optimal trading processes in economic systems. The analysis is based on accounting for irreversibility factors using the wealth function concept. The existence of the welfare function is proved, the concept of capital dissipation is introduced as a measure [...] Read more.

In this paper, we consider optimal trading processes in economic systems. The analysis is based on accounting for irreversibility factors using the wealth function concept. The existence of the welfare function is proved, the concept of capital dissipation is introduced as a measure of the irreversibility of processes in the microeconomic system, and the economic balances are recorded, including capital dissipation. Problems in the form of kinetic equations leading to given conditions of minimal dissipation are considered. Full article

(This article belongs to the Special Issue Finite-Time Thermodynamics)

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17 pages, 5758 KiB

Open AccessEditor’s ChoiceArticle

Optimization of a New Design of Molten Salt-to-CO₂ Heat Exchanger Using Exergy Destruction Minimization

by María José Montes, José Ignacio Linares, Rubén Barbero and Beatriz Yolanda Moratilla

Entropy 2020, 22(8), 883; https://doi.org/10.3390/e22080883 - 12 Aug 2020

Cited by 9 | Viewed by 5184

Abstract

One of the ways to make cost-competitive electricity, from concentrated solar thermal energy, is increasing the thermoelectric conversion efficiency. To achieve this objective, the most promising scheme is a molten salt central receiver, coupled to a supercritical carbon dioxide cycle. A key element [...] Read more.

One of the ways to make cost-competitive electricity, from concentrated solar thermal energy, is increasing the thermoelectric conversion efficiency. To achieve this objective, the most promising scheme is a molten salt central receiver, coupled to a supercritical carbon dioxide cycle. A key element to be developed in this scheme is the molten salt-to-CO₂ heat exchanger. This paper presents a heat exchanger design that avoids the molten salt plugging and the mechanical stress due to the high pressure of the CO₂, while improving the heat transfer of the supercritical phase, due to its compactness with a high heat transfer area. This design is based on a honeycomb-like configuration, in which a thermal unit consists of a circular channel for the molten salt surrounded by six smaller trapezoidal ducts for the CO₂. Further, an optimization based on the exergy destruction minimization has been accomplished, obtained the best working conditions of this heat exchanger: a temperature approach of 50 °C between both streams and a CO₂ pressure drop of 2.7 bar. Full article

(This article belongs to the Special Issue Thermodynamic Optimization of Complex Energy Systems)

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15 pages, 1578 KiB

Open AccessEditor’s ChoiceReview

Role of Entropy in Colloidal Self-Assembly

by Brunno C. Rocha, Sanjib Paul and Harish Vashisth

Entropy 2020, 22(8), 877; https://doi.org/10.3390/e22080877 - 10 Aug 2020

Cited by 26 | Viewed by 7551

Abstract

Entropy plays a key role in the self-assembly of colloidal particles. Specifically, in the case of hard particles, which do not interact or overlap with each other during the process of self-assembly, the free energy is minimized due to an increase in the [...] Read more.

Entropy plays a key role in the self-assembly of colloidal particles. Specifically, in the case of hard particles, which do not interact or overlap with each other during the process of self-assembly, the free energy is minimized due to an increase in the entropy of the system. Understanding the contribution of entropy and engineering it is increasingly becoming central to modern colloidal self-assembly research, because the entropy serves as a guide to design a wide variety of self-assembled structures for many technological and biomedical applications. In this work, we highlight the importance of entropy in different theoretical and experimental self-assembly studies. We discuss the role of shape entropy and depletion interactions in colloidal self-assembly. We also highlight the effect of entropy in the formation of open and closed crystalline structures, as well as describe recent advances in engineering entropy to achieve targeted self-assembled structures. Full article

(This article belongs to the Special Issue Thermodynamics and Entropy for Self-Assembly and Self-Organization)

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52 pages, 6023 KiB

Open AccessEditor’s ChoiceReview

Phase-Coherent Dynamics of Quantum Devices with Local Interactions

by Michele Filippone, Arthur Marguerite, Karyn Le Hur, Gwendal Fève and Christophe Mora

Entropy 2020, 22(8), 847; https://doi.org/10.3390/e22080847 - 31 Jul 2020

Cited by 12 | Viewed by 4826

Abstract

This review illustrates how Local Fermi Liquid (LFL) theories describe the strongly correlated and coherent low-energy dynamics of quantum dot devices. This approach consists in an effective elastic scattering theory, accounting exactly for strong correlations. Here, we focus on the mesoscopic capacitor and [...] Read more.

This review illustrates how Local Fermi Liquid (LFL) theories describe the strongly correlated and coherent low-energy dynamics of quantum dot devices. This approach consists in an effective elastic scattering theory, accounting exactly for strong correlations. Here, we focus on the mesoscopic capacitor and recent experiments achieving a Coulomb-induced quantum state transfer. Extending to out-of-equilibrium regimes, aimed at triggered single electron emission, we illustrate how inelastic effects become crucial, requiring approaches beyond LFLs, shedding new light on past experimental data by showing clear interaction effects in the dynamics of mesoscopic capacitors. Full article

(This article belongs to the Special Issue Quantum Transport in Mesoscopic Systems)

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28 pages, 2477 KiB

Open AccessEditor’s ChoiceArticle

Direct and Indirect Effects—An Information Theoretic Perspective

by Gabriel Schamberg, William Chapman, Shang-Ping Xie and Todd P. Coleman

Entropy 2020, 22(8), 854; https://doi.org/10.3390/e22080854 - 31 Jul 2020

Cited by 6 | Viewed by 5637

Abstract

Information theoretic (IT) approaches to quantifying causal influences have experienced some popularity in the literature, in both theoretical and applied (e.g., neuroscience and climate science) domains. While these causal measures are desirable in that they are model agnostic and can capture non-linear interactions, [...] Read more.

Information theoretic (IT) approaches to quantifying causal influences have experienced some popularity in the literature, in both theoretical and applied (e.g., neuroscience and climate science) domains. While these causal measures are desirable in that they are model agnostic and can capture non-linear interactions, they are fundamentally different from common statistical notions of causal influence in that they (1) compare distributions over the effect rather than values of the effect and (2) are defined with respect to random variables representing a cause rather than specific values of a cause. We here present IT measures of direct, indirect, and total causal effects. The proposed measures are unlike existing IT techniques in that they enable measuring causal effects that are defined with respect to specific values of a cause while still offering the flexibility and general applicability of IT techniques. We provide an identifiability result and demonstrate application of the proposed measures in estimating the causal effect of the El Niño–Southern Oscillation on temperature anomalies in the North American Pacific Northwest. Full article

(This article belongs to the Special Issue Information Theoretic Measures and Their Applications)

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11 pages, 636 KiB

Open AccessEditor’s ChoiceArticle

Forecasting Bitcoin Trends Using Algorithmic Learning Systems

by Gil Cohen

Entropy 2020, 22(8), 838; https://doi.org/10.3390/e22080838 - 30 Jul 2020

Cited by 24 | Viewed by 6007

Abstract

This research has examined the ability of two forecasting methods to forecast Bitcoin’s price trends. The research is based on Bitcoin—USA dollar prices from the beginning of 2012 until the end of March 2020. Such a long period of time that includes volatile [...] Read more.

This research has examined the ability of two forecasting methods to forecast Bitcoin’s price trends. The research is based on Bitcoin—USA dollar prices from the beginning of 2012 until the end of March 2020. Such a long period of time that includes volatile periods with strong up and downtrends introduces challenges to any forecasting system. We use particle swarm optimization to find the best forecasting combinations of setups. Results show that Bitcoin’s price changes do not follow the “Random Walk” efficient market hypothesis and that both Darvas Box and Linear Regression techniques can help traders to predict the bitcoin’s price trends. We also find that both methodologies work better predicting an uptrend than a downtrend. The best setup for the Darvas Box strategy is six days of formation. A Darvas box uptrend signal was found efficient predicting four sequential daily returns while a downtrend signal faded after two days on average. The best setup for the Linear Regression model is 42 days with 1 standard deviation. Full article

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22 pages, 2072 KiB

Open AccessEditor’s ChoiceArticle

Deep Learning for Stock Market Prediction

by M. Nabipour, P. Nayyeri, H. Jabani, A. Mosavi, E. Salwana and Shahab S.

Entropy 2020, 22(8), 840; https://doi.org/10.3390/e22080840 - 30 Jul 2020

Cited by 280 | Viewed by 38625

Abstract

The prediction of stock groups values has always been attractive and challenging for shareholders due to its inherent dynamics, non-linearity, and complex nature. This paper concentrates on the future prediction of stock market groups. Four groups named diversified financials, petroleum, non-metallic minerals, and [...] Read more.

The prediction of stock groups values has always been attractive and challenging for shareholders due to its inherent dynamics, non-linearity, and complex nature. This paper concentrates on the future prediction of stock market groups. Four groups named diversified financials, petroleum, non-metallic minerals, and basic metals from Tehran stock exchange were chosen for experimental evaluations. Data were collected for the groups based on 10 years of historical records. The value predictions are created for 1, 2, 5, 10, 15, 20, and 30 days in advance. Various machine learning algorithms were utilized for prediction of future values of stock market groups. We employed decision tree, bagging, random forest, adaptive boosting (Adaboost), gradient boosting, and eXtreme gradient boosting (XGBoost), and artificial neural networks (ANN), recurrent neural network (RNN) and long short-term memory (LSTM). Ten technical indicators were selected as the inputs into each of the prediction models. Finally, the results of the predictions were presented for each technique based on four metrics. Among all algorithms used in this paper, LSTM shows more accurate results with the highest model fitting ability. In addition, for tree-based models, there is often an intense competition between Adaboost, Gradient Boosting, and XGBoost. Full article

(This article belongs to the Special Issue Information Transfer in Multilayer/Deep Architectures)

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12 pages, 966 KiB

Open AccessEditor’s ChoiceArticle

Hybrid Quantum-Classical Neural Network for Calculating Ground State Energies of Molecules

by Rongxin Xia and Sabre Kais

Entropy 2020, 22(8), 828; https://doi.org/10.3390/e22080828 - 29 Jul 2020

Cited by 29 | Viewed by 8460

Abstract

We present a hybrid quantum-classical neural network that can be trained to perform electronic structure calculation and generate potential energy curves of simple molecules. The method is based on the combination of parameterized quantum circuits and measurements. With unsupervised training, the neural network [...] Read more.

We present a hybrid quantum-classical neural network that can be trained to perform electronic structure calculation and generate potential energy curves of simple molecules. The method is based on the combination of parameterized quantum circuits and measurements. With unsupervised training, the neural network can generate electronic potential energy curves based on training at certain bond lengths. To demonstrate the power of the proposed new method, we present the results of using the quantum-classical hybrid neural network to calculate ground state potential energy curves of simple molecules such as H

_{2}

, LiH, and BeH

_{2}

. The results are very accurate and the approach could potentially be used to generate complex molecular potential energy surfaces. Full article

(This article belongs to the Special Issue Noisy Intermediate-Scale Quantum Technologies (NISQ))

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16 pages, 708 KiB

Open AccessEditor’s ChoiceArticle

An Efficient Method Based on Framelets for Solving Fractional Volterra Integral Equations

by Mutaz Mohammad, Alexander Trounev and Carlo Cattani

Entropy 2020, 22(8), 824; https://doi.org/10.3390/e22080824 - 28 Jul 2020

Cited by 13 | Viewed by 3208

Abstract

This paper is devoted to shedding some light on the advantages of using tight frame systems for solving some types of fractional Volterra integral equations (FVIEs) involved by the Caputo fractional order derivative. A tight frame or simply framelet, is a generalization of [...] Read more.

This paper is devoted to shedding some light on the advantages of using tight frame systems for solving some types of fractional Volterra integral equations (FVIEs) involved by the Caputo fractional order derivative. A tight frame or simply framelet, is a generalization of an orthonormal basis. A lot of applications are modeled by non-negative functions; taking this into account in this paper, we consider framelet systems generated using some refinable non-negative functions, namely, B-splines. The FVIEs we considered were reduced to a set of linear system of equations and were solved numerically based on a collocation discretization technique. We present many important examples of FVIEs for which accurate and efficient numerical solutions have been accomplished and the numerical results converge very rapidly to the exact ones. Full article

(This article belongs to the Special Issue Entropy and Its Applications across Disciplines II)

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54 pages, 5393 KiB

Open AccessEditor’s ChoiceReview

Thermodynamics in Ecology—An Introductory Review

by Søren Nors Nielsen, Felix Müller, Joao Carlos Marques, Simone Bastianoni and Sven Erik Jørgensen

Entropy 2020, 22(8), 820; https://doi.org/10.3390/e22080820 - 27 Jul 2020

Cited by 62 | Viewed by 17793

Abstract

How to predict the evolution of ecosystems is one of the numerous questions asked of ecologists by managers and politicians. To answer this we will need to give a scientific definition to concepts like sustainability, integrity, resilience and ecosystem health. This is not [...] Read more.

How to predict the evolution of ecosystems is one of the numerous questions asked of ecologists by managers and politicians. To answer this we will need to give a scientific definition to concepts like sustainability, integrity, resilience and ecosystem health. This is not an easy task, as modern ecosystem theory exemplifies. Ecosystems show a high degree of complexity, based upon a high number of compartments, interactions and regulations. The last two decades have offered proposals for interpretation of ecosystems within a framework of thermodynamics. The entrance point of such an understanding of ecosystems was delivered more than 50 years ago through Schrödinger’s and Prigogine’s interpretations of living systems as “negentropy feeders” and “dissipative structures”, respectively. Combining these views from the far from equilibrium thermodynamics to traditional classical thermodynamics, and ecology is obviously not going to happen without problems. There seems little reason to doubt that far from equilibrium systems, such as organisms or ecosystems, also have to obey fundamental physical principles such as mass conservation, first and second law of thermodynamics. Both have been applied in ecology since the 1950s and lately the concepts of exergy and entropy have been introduced. Exergy has recently been proposed, from several directions, as a useful indicator of the state, structure and function of the ecosystem. The proposals take two main directions, one concerned with the exergy stored in the ecosystem, the other with the exergy degraded and entropy formation. The implementation of exergy in ecology has often been explained as a translation of the Darwinian principle of “survival of the fittest” into thermodynamics. The fittest ecosystem, being the one able to use and store fluxes of energy and materials in the most efficient manner. The major problem in the transfer to ecology is that thermodynamic properties can only be calculated and not measured. Most of the supportive evidence comes from aquatic ecosystems. Results show that natural and culturally induced changes in the ecosystems, are accompanied by a variations in exergy. In brief, ecological succession is followed by an increase of exergy. This paper aims to describe the state-of-the-art in implementation of thermodynamics into ecology. This includes a brief outline of the history and the derivation of the thermodynamic functions used today. Examples of applications and results achieved up to now are given, and the importance to management laid out. Some suggestions for essential future research agendas of issues that needs resolution are given. Full article

(This article belongs to the Special Issue Evolution and Thermodynamics)

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18 pages, 5556 KiB

Open AccessEditor’s ChoiceArticle

Thermodynamic Analysis of Working Fluids for a New “Heat from Cold” Cycle

by Ilya Girnik, Mikhail Tokarev and Yuri Aristov

Entropy 2020, 22(8), 808; https://doi.org/10.3390/e22080808 - 23 Jul 2020

Cited by 8 | Viewed by 2740

Abstract

Adsorptive Heat Transformation systems are at the interface between thermal and chemical engineering. Their study and development need a thorough thermodynamic analysis aimed at the smart choice of adsorbent-adsorptive pair and its fitting with a particular heat transformation cycle. This paper addresses such [...] Read more.

Adsorptive Heat Transformation systems are at the interface between thermal and chemical engineering. Their study and development need a thorough thermodynamic analysis aimed at the smart choice of adsorbent-adsorptive pair and its fitting with a particular heat transformation cycle. This paper addresses such an analysis for a new “Heat from Cold” cycle proposed for amplification of the ambient heat in cold countries. A comparison of four working fluids is made in terms of the useful heat per cycle and the temperature lift. The useful heat increases in the row water > ammonia ≥ methanol > hydrofluorocarbon R32. A threshold mass of exchanged adsorbate, below which the useful heat equals zero, raises in the same sequence. The most promising adsorbents for this cycle are activated carbons Maxsorb III and SRD 1352/2. For all the adsorptives studied, a linear relationship F = A·ΔT is found between the Dubinin adsorption potential and the driving temperature difference ΔT between the two natural thermal baths. It allows the maximum temperature lift during the heat generation stage to be assessed. Thus, a larger ΔT-value promotes the removal of the more strongly bound adsorbate. Full article

(This article belongs to the Special Issue Thermodynamic Approaches in Modern Engineering Systems)

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11 pages, 2028 KiB

Open AccessEditor’s ChoiceArticle

Relative Entropy as a Measure of Difference between Hermitian and Non-Hermitian Systems

by Kabgyun Jeong, Kyu-Won Park and Jaewan Kim

Entropy 2020, 22(8), 809; https://doi.org/10.3390/e22080809 - 23 Jul 2020

Cited by 3 | Viewed by 4151

Abstract

We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is [...] Read more.

We employ the relative entropy as a measure to quantify the difference of eigenmodes between Hermitian and non-Hermitian systems in elliptic optical microcavities. We have found that the average value of the relative entropy in the range of the collective Lamb shift is large, while that in the range of self-energy is small. Furthermore, the weak and strong interactions in the non-Hermitian system exhibit rather different behaviors in terms of the relative entropy, and thus it displays an obvious exchange of eigenmodes in the elliptic microcavity. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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35 pages, 3124 KiB

Open AccessEditor’s ChoiceArticle

Interacting Particle Solutions of Fokker–Planck Equations Through Gradient–Log–Density Estimation

by Dimitra Maoutsa, Sebastian Reich and Manfred Opper

Entropy 2020, 22(8), 802; https://doi.org/10.3390/e22080802 - 22 Jul 2020

Cited by 40 | Viewed by 11514

Abstract

Fokker–Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort [...] Read more.

Fokker–Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker–Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker–Planck equations in low and moderate dimensions. The proposed gradient–log–density estimator is also of independent interest, for example, in the context of optimal control. Full article

(This article belongs to the Special Issue Entropy Methods for Stochastic Dynamical Systems and Evolution Equations)

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40 pages, 2031 KiB

Open AccessEditor’s ChoiceArticle

Power Conversion and Its Efficiency in Thermoelectric Materials

by Armin Feldhoff

Entropy 2020, 22(8), 803; https://doi.org/10.3390/e22080803 - 22 Jul 2020

Cited by 33 | Viewed by 4520

Abstract

The basic principles of thermoelectrics rely on the coupling of entropy and electric charge. However, the long-standing dispute of energetics versus entropy has long paralysed the field. Herein, it is shown that treating entropy and electric charge in a symmetric manner enables a [...] Read more.

The basic principles of thermoelectrics rely on the coupling of entropy and electric charge. However, the long-standing dispute of energetics versus entropy has long paralysed the field. Herein, it is shown that treating entropy and electric charge in a symmetric manner enables a simple transport equation to be obtained and the power conversion and its efficiency to be deduced for a single thermoelectric material apart from a device. The material’s performance in both generator mode (thermo-electric) and entropy pump mode (electro-thermal) are discussed on a single voltage-electrical current curve, which is presented in a generalized manner by relating it to the electrically open-circuit voltage and the electrically closed-circuited electrical current. The electrical and thermal power in entropy pump mode are related to the maximum electrical power in generator mode, which depends on the material’s power factor. Particular working points on the material’s voltage-electrical current curve are deduced, namely, the electrical open circuit, electrical short circuit, maximum electrical power, maximum power conversion efficiency, and entropy conductivity inversion. Optimizing a thermoelectric material for different working points is discussed with respect to its figure-of-merit

z T

and power factor. The importance of the results to state-of-the-art and emerging materials is emphasized. Full article

(This article belongs to the Special Issue Simulation with Entropy Thermodynamics)

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16 pages, 435 KiB

Open AccessEditor’s ChoiceArticle

Evolution Equations for Quantum Semi-Markov Dynamics

by Nina Megier, Andrea Smirne and Bassano Vacchini

Entropy 2020, 22(7), 796; https://doi.org/10.3390/e22070796 - 21 Jul 2020

Cited by 10 | Viewed by 3318

Abstract

Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the [...] Read more.

Using a newly introduced connection between the local and non-local description of open quantum system dynamics, we investigate the relationship between these two characterisations in the case of quantum semi-Markov processes. This class of quantum evolutions, which is a direct generalisation of the corresponding classical concept, guarantees mathematically well-defined master equations, while accounting for a wide range of phenomena, possibly in the non-Markovian regime. In particular, we analyse the emergence of a dephasing term when moving from one type of master equation to the other, by means of several examples. We also investigate the corresponding Redfield-like approximated dynamics, which are obtained after a coarse graining in time. Relying on general properties of the associated classical random process, we conclude that such an approximation always leads to a Markovian evolution for the considered class of dynamics. Full article

(This article belongs to the Special Issue Open Quantum Systems (OQS) for Quantum Technologies)

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13 pages, 893 KiB

Open AccessEditor’s ChoiceArticle

Blind Witnesses Quench Quantum Interference without Transfer of Which-Path Information

by Craig S. Lent

Entropy 2020, 22(7), 776; https://doi.org/10.3390/e22070776 - 16 Jul 2020

Viewed by 2975

Abstract

Quantum computation is often limited by environmentally-induced decoherence. We examine the loss of coherence for a two-branch quantum interference device in the presence of multiple witnesses, representing an idealized environment. Interference oscillations are visible in the output as the magnetic flux through the [...] Read more.

Quantum computation is often limited by environmentally-induced decoherence. We examine the loss of coherence for a two-branch quantum interference device in the presence of multiple witnesses, representing an idealized environment. Interference oscillations are visible in the output as the magnetic flux through the branches is varied. Quantum double-dot witnesses are field-coupled and symmetrically attached to each branch. The global system—device and witnesses—undergoes unitary time evolution with no increase in entropy. Witness states entangle with the device state, but for these blind witnesses, which-path information is not able to be transferred to the quantum state of witnesses—they cannot “see” or make a record of which branch is traversed. The system which-path information leaves no imprint on the environment. Yet, the presence of a multiplicity of witnesses rapidly quenches quantum interference. Full article

(This article belongs to the Special Issue Physical Information and the Physical Foundations of Computation)

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22 pages, 812 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Quantum-Gravity Stochastic Effects on the de Sitter Event Horizon

by Claudio Cremaschini and Massimo Tessarotto

Entropy 2020, 22(6), 696; https://doi.org/10.3390/e22060696 - 22 Jun 2020

Cited by 4 | Viewed by 3418

Abstract

The stochastic character of the cosmological constant arising from the non-linear quantum-vacuum Bohm interaction in the framework of the manifestly-covariant theory of quantum gravity (CQG theory) is pointed out. This feature is shown to be consistent with the axiomatic formulation of quantum gravity [...] Read more.

The stochastic character of the cosmological constant arising from the non-linear quantum-vacuum Bohm interaction in the framework of the manifestly-covariant theory of quantum gravity (CQG theory) is pointed out. This feature is shown to be consistent with the axiomatic formulation of quantum gravity based on the hydrodynamic representation of the same CQG theory developed recently. The conclusion follows by investigating the indeterminacy properties of the probability density function and its representation associated with the quantum gravity state, which corresponds to a hydrodynamic continuity equation that satisfies the unitarity principle. As a result, the corresponding form of stochastic quantum-modified Einstein field equations is obtained and shown to admit a stochastic cosmological de Sitter solution for the space-time metric tensor. The analytical calculation of the stochastic averages of relevant physical observables is obtained. These include in particular the radius of the de Sitter sphere fixing the location of the event horizon and the expression of the Hawking temperature associated with the related particle tunneling effect. Theoretical implications for cosmology and field theories are pointed out. Full article

(This article belongs to the Special Issue Quantum Mechanics and Its Foundations)

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53 pages, 574 KiB

Open AccessEditor’s ChoiceArticle

Mathematical Models of Consciousness

by Johannes Kleiner

Entropy 2020, 22(6), 609; https://doi.org/10.3390/e22060609 - 30 May 2020

Cited by 27 | Viewed by 11623

Abstract

In recent years, promising mathematical models have been proposed that aim to describe conscious experience and its relation to the physical domain. Whereas the axioms and metaphysical ideas of these theories have been carefully motivated, their mathematical formalism has not. In this article, [...] Read more.

In recent years, promising mathematical models have been proposed that aim to describe conscious experience and its relation to the physical domain. Whereas the axioms and metaphysical ideas of these theories have been carefully motivated, their mathematical formalism has not. In this article, we aim to remedy this situation. We give an account of what warrants mathematical representation of phenomenal experience, derive a general mathematical framework that takes into account consciousness’ epistemic context, and study which mathematical structures some of the key characteristics of conscious experience imply, showing precisely where mathematical approaches allow to go beyond what the standard methodology can do. The result is a general mathematical framework for models of consciousness that can be employed in the theory-building process. Full article

(This article belongs to the Special Issue Models of Consciousness)

44 pages, 628 KiB

Open AccessEditor’s ChoiceReview

What Is So Special about Quantum Clicks?

by Karl Svozil

Entropy 2020, 22(6), 602; https://doi.org/10.3390/e22060602 - 28 May 2020

Cited by 24 | Viewed by 5410

Abstract

This is an elaboration of the “extra” advantage of the performance of quantized physical systems over classical ones, both in terms of single outcomes as well as probabilistic predictions. From a formal point of view, it is based on entities related to (dual) [...] Read more.

This is an elaboration of the “extra” advantage of the performance of quantized physical systems over classical ones, both in terms of single outcomes as well as probabilistic predictions. From a formal point of view, it is based on entities related to (dual) vectors in (dual) Hilbert spaces, as compared to the Boolean algebra of subsets of a set and the additive measures they support. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness II)

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25 pages, 494 KiB

Open AccessEditor’s ChoiceArticle

Differential Parametric Formalism for the Evolution of Gaussian States: Nonunitary Evolution and Invariant States

by Julio A. López-Saldívar, Margarita A. Man’ko and Vladimir I. Man’ko

Entropy 2020, 22(5), 586; https://doi.org/10.3390/e22050586 - 23 May 2020

Cited by 13 | Viewed by 3521

Abstract

In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in [...] Read more.

In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian

\hat{H}

. We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness II)

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14 pages, 1842 KiB

Open AccessEditor’s ChoiceArticle

A Method to Present and Analyze Ensembles of Information Sources

by Nicholas M. Timme, David Linsenbardt and Christopher C. Lapish

Entropy 2020, 22(5), 580; https://doi.org/10.3390/e22050580 - 21 May 2020

Viewed by 4320

Abstract

Information theory is a powerful tool for analyzing complex systems. In many areas of neuroscience, it is now possible to gather data from large ensembles of neural variables (e.g., data from many neurons, genes, or voxels). The individual variables can be analyzed with [...] Read more.

Information theory is a powerful tool for analyzing complex systems. In many areas of neuroscience, it is now possible to gather data from large ensembles of neural variables (e.g., data from many neurons, genes, or voxels). The individual variables can be analyzed with information theory to provide estimates of information shared between variables (forming a network between variables), or between neural variables and other variables (e.g., behavior or sensory stimuli). However, it can be difficult to (1) evaluate if the ensemble is significantly different from what would be expected in a purely noisy system and (2) determine if two ensembles are different. Herein, we introduce relatively simple methods to address these problems by analyzing ensembles of information sources. We demonstrate how an ensemble built of mutual information connections can be compared to null surrogate data to determine if the ensemble is significantly different from noise. Next, we show how two ensembles can be compared using a randomization process to determine if the sources in one contain more information than the other. All code necessary to carry out these analyses and demonstrations are provided. Full article

(This article belongs to the Special Issue Information Theory in Computational Neuroscience)

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26 pages, 7851 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Goal-Directed Planning for Habituated Agents by Active Inference Using a Variational Recurrent Neural Network

by Takazumi Matsumoto and Jun Tani

Entropy 2020, 22(5), 564; https://doi.org/10.3390/e22050564 - 18 May 2020

Cited by 28 | Viewed by 7187

Abstract

It is crucial to ask how agents can achieve goals by generating action plans using only partial models of the world acquired through habituated sensory-motor experiences. Although many existing robotics studies use a forward model framework, there are generalization issues with high degrees [...] Read more.

It is crucial to ask how agents can achieve goals by generating action plans using only partial models of the world acquired through habituated sensory-motor experiences. Although many existing robotics studies use a forward model framework, there are generalization issues with high degrees of freedom. The current study shows that the predictive coding (PC) and active inference (AIF) frameworks, which employ a generative model, can develop better generalization by learning a prior distribution in a low dimensional latent state space representing probabilistic structures extracted from well habituated sensory-motor trajectories. In our proposed model, learning is carried out by inferring optimal latent variables as well as synaptic weights for maximizing the evidence lower bound, while goal-directed planning is accomplished by inferring latent variables for maximizing the estimated lower bound. Our proposed model was evaluated with both simple and complex robotic tasks in simulation, which demonstrated sufficient generalization in learning with limited training data by setting an intermediate value for a regularization coefficient. Furthermore, comparative simulation results show that the proposed model outperforms a conventional forward model in goal-directed planning, due to the learned prior confining the search of motor plans within the range of habituated trajectories. Full article

(This article belongs to the Special Issue Probabilistic Inference in Goal-Directed Human and Animal Decision-Making)

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35 pages, 1693 KiB

Open AccessEditor’s ChoiceArticle

Re-Thinking the World with Neutral Monism:Removing the Boundaries Between Mind, Matter, and Spacetime

by Michael Silberstein and William Stuckey

Entropy 2020, 22(5), 551; https://doi.org/10.3390/e22050551 - 14 May 2020

Cited by 3 | Viewed by 11805

Abstract

Herein we are not interested in merely using dynamical systems theory, graph theory, information theory, etc., to model the relationship between brain dynamics and networks, and various states and degrees of conscious processes. We are interested in the question of how phenomenal conscious [...] Read more.

Herein we are not interested in merely using dynamical systems theory, graph theory, information theory, etc., to model the relationship between brain dynamics and networks, and various states and degrees of conscious processes. We are interested in the question of how phenomenal conscious experience and fundamental physics are most deeply related. Any attempt to mathematically and formally model conscious experience and its relationship to physics must begin with some metaphysical assumption in mind about the nature of conscious experience, the nature of matter and the nature of the relationship between them. These days the most prominent metaphysical fixed points are strong emergence or some variant of panpsychism. In this paper we will detail another distinct metaphysical starting point known as neutral monism. In particular, we will focus on a variant of the neutral monism of William James and Bertrand Russell. Rather than starting with physics as fundamental, as both strong emergence and panpsychism do in their own way, our goal is to suggest how one might derive fundamental physics from neutral monism. Thus, starting with two axioms grounded in our characterization of neutral monism, we will sketch out a derivation of and explanation for some key features of relativity and quantum mechanics that suggest a unity between those two theories that is generally unappreciated. Our mode of explanation throughout will be of the principle as opposed to constructive variety in something like Einstein’s sense of those terms. We will argue throughout that a bias towards property dualism and a bias toward reductive dynamical and constructive explanation lead to the hard problem and the explanatory gap in consciousness studies, and lead to serious unresolved problems in fundamental physics, such as the measurement problem and the mystery of entanglement in quantum mechanics and lack of progress in producing an empirically well-grounded theory of quantum gravity. We hope to show that given our take on neutral monism and all that follows from it, the aforementioned problems can be satisfactorily resolved leaving us with a far more intuitive and commonsense model of the relationship between conscious experience and physics. Full article

(This article belongs to the Special Issue Models of Consciousness)

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18 pages, 865 KiB

Open AccessEditor’s ChoiceArticle

Prediction and Variable Selection in High-Dimensional Misspecified Binary Classification

by Konrad Furmańczyk and Wojciech Rejchel

Entropy 2020, 22(5), 543; https://doi.org/10.3390/e22050543 - 13 May 2020

Cited by 5 | Viewed by 4227

Abstract

In this paper, we consider prediction and variable selection in the misspecified binary classification models under the high-dimensional scenario. We focus on two approaches to classification, which are computationally efficient, but lead to model misspecification. The first one is to apply penalized logistic [...] Read more.

In this paper, we consider prediction and variable selection in the misspecified binary classification models under the high-dimensional scenario. We focus on two approaches to classification, which are computationally efficient, but lead to model misspecification. The first one is to apply penalized logistic regression to the classification data, which possibly do not follow the logistic model. The second method is even more radical: we just treat class labels of objects as they were numbers and apply penalized linear regression. In this paper, we investigate thoroughly these two approaches and provide conditions, which guarantee that they are successful in prediction and variable selection. Our results hold even if the number of predictors is much larger than the sample size. The paper is completed by the experimental results. Full article

(This article belongs to the Special Issue Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods)

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20 pages, 2082 KiB

Open AccessEditor’s ChoiceArticle

Inferring What to Do (And What Not to)

by Thomas Parr

Entropy 2020, 22(5), 536; https://doi.org/10.3390/e22050536 - 11 May 2020

Cited by 8 | Viewed by 4758

Abstract

In recent years, the “planning as inference” paradigm has become central to the study of behaviour. The advance offered by this is the formalisation of motivation as a prior belief about “how I am going to act”. This paper provides an overview of [...] Read more.

In recent years, the “planning as inference” paradigm has become central to the study of behaviour. The advance offered by this is the formalisation of motivation as a prior belief about “how I am going to act”. This paper provides an overview of the factors that contribute to this prior. These are rooted in optimal experimental design, information theory, and statistical decision making. We unpack how these factors imply a functional architecture for motivated behaviour. This raises an important question: how can we put this architecture to work in the service of understanding observed neurobiological structure? To answer this question, we draw from established techniques in experimental studies of behaviour. Typically, these examine the influence of perturbations of the nervous system—which include pathological insults or optogenetic manipulations—to see their influence on behaviour. Here, we argue that the message passing that emerges from inferring what to do can be similarly perturbed. If a given perturbation elicits the same behaviours as a focal brain lesion, this provides a functional interpretation of empirical findings and an anatomical grounding for theoretical results. We highlight examples of this approach that influence different sorts of goal-directed behaviour, active learning, and decision making. Finally, we summarise their implications for the neuroanatomy of inferring what to do (and what not to). Full article

(This article belongs to the Special Issue Probabilistic Inference in Goal-Directed Human and Animal Decision-Making)

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21 pages, 1102 KiB

Open AccessEditor’s ChoiceArticle

Non-Hermitian Hamiltonians and Quantum Transport in Multi-Terminal Conductors

by Nikolay M. Shubin, Alexander A. Gorbatsevich and Gennadiy Ya. Krasnikov

Entropy 2020, 22(4), 459; https://doi.org/10.3390/e22040459 - 17 Apr 2020

Cited by 3 | Viewed by 4201

Abstract

We study the transport properties of multi-terminal Hermitian structures within the non-equilibrium Green’s function formalism in a tight-binding approximation. We show that non-Hermitian Hamiltonians naturally appear in the description of coherent tunneling and are indispensable for the derivation of a general compact expression [...] Read more.

We study the transport properties of multi-terminal Hermitian structures within the non-equilibrium Green’s function formalism in a tight-binding approximation. We show that non-Hermitian Hamiltonians naturally appear in the description of coherent tunneling and are indispensable for the derivation of a general compact expression for the lead-to-lead transmission coefficients of an arbitrary multi-terminal system. This expression can be easily analyzed, and a robust set of conditions for finding zero and unity transmissions (even in the presence of extra electrodes) can be formulated. Using the proposed formalism, a detailed comparison between three- and two-terminal systems is performed, and it is shown, in particular, that transmission at bound states in the continuum does not change with the third electrode insertion. The main conclusions are illustratively exemplified by some three-terminal toy models. For instance, the influence of the tunneling coupling to the gate electrode is discussed for a model of quantum interference transistor. The results of this paper will be of high interest, in particular, within the field of quantum design of molecular electronic devices. Full article

(This article belongs to the Special Issue Quantum Dynamics with Non-Hermitian Hamiltonians)

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36 pages, 531 KiB

Open AccessEditor’s ChoiceArticle

Towards a Unified Theory of Learning and Information

by Ibrahim Alabdulmohsin

Entropy 2020, 22(4), 438; https://doi.org/10.3390/e22040438 - 13 Apr 2020

Cited by 7 | Viewed by 5119

Abstract

In this paper, we introduce the notion of “learning capacity” for algorithms that learn from data, which is analogous to the Shannon channel capacity for communication systems. We show how “learning capacity” bridges the gap between statistical learning theory and information theory, and [...] Read more.

In this paper, we introduce the notion of “learning capacity” for algorithms that learn from data, which is analogous to the Shannon channel capacity for communication systems. We show how “learning capacity” bridges the gap between statistical learning theory and information theory, and we will use it to derive generalization bounds for finite hypothesis spaces, differential privacy, and countable domains, among others. Moreover, we prove that under the Axiom of Choice, the existence of an empirical risk minimization (ERM) rule that has a vanishing learning capacity is equivalent to the assertion that the hypothesis space has a finite Vapnik–Chervonenkis (VC) dimension, thus establishing an equivalence relation between two of the most fundamental concepts in statistical learning theory and information theory. In addition, we show how the learning capacity of an algorithm provides important qualitative results, such as on the relation between generalization and algorithmic stability, information leakage, and data processing. Finally, we conclude by listing some open problems and suggesting future directions of research. Full article

(This article belongs to the Special Issue Nonparametric Statistical Inference with an Emphasis on Information-Theoretic Methods)

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10 pages, 618 KiB

Open AccessEditor’s ChoiceArticle

Measuring the Tangle of Three-Qubit States

by Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto and José I. Latorre

Entropy 2020, 22(4), 436; https://doi.org/10.3390/e22040436 - 11 Apr 2020

Cited by 15 | Viewed by 5854

Abstract

We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters [...] Read more.

We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method. Full article

(This article belongs to the Section Quantum Information)

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21 pages, 1900 KiB

Open AccessEditor’s ChoiceArticle

Sensitivity Analysis of Selected Parameters in the Order Picking Process Simulation Model, with Randomly Generated Orders

by Mariusz Kostrzewski

Entropy 2020, 22(4), 423; https://doi.org/10.3390/e22040423 - 9 Apr 2020

Cited by 31 | Viewed by 5024

Abstract

Sensitivity analysis of selected parameters in simulation models of logistics facilities is one of the key aspects in functioning of self-conscious and efficient management. In order to develop simulation models adequate of real logistics facilities’ processes, it is important to input actual data [...] Read more.

Sensitivity analysis of selected parameters in simulation models of logistics facilities is one of the key aspects in functioning of self-conscious and efficient management. In order to develop simulation models adequate of real logistics facilities’ processes, it is important to input actual data connected to material flows on entry to models, whereas most models assume unified load units as default. To provide such data, pseudorandom number generators (PRNGs) are used. The original generator described in the paper was employed in order to generate picking lists for order picking process (OPP). This ensures building a hypothetical, yet close to reality process in terms of unpredictable customers’ orders. Models with applied PRNGs ensure more detailed and more understandable representation of OPPs in comparison to analytical models. Therefore, the author’s motivation was to present the original model as a tool for enterprises’ managers who might control OPP, devices and means of transport employed therein. The outcomes and implications of the contribution are connected to presentation of selected possibilities in OPP analyses, which might be developed and solved within the model. The presented model has some limitations. One of them is assumption that one mean of transport per one aisle is taken into consideration. Another limitation is the indirectly randomization of certain model’s parameters. Full article

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22 pages, 3546 KiB

Open AccessEditor’s ChoiceArticle

On Geometry of Information Flow for Causal Inference

by Sudam Surasinghe and Erik M. Bollt

Entropy 2020, 22(4), 396; https://doi.org/10.3390/e22040396 - 30 Mar 2020

Cited by 5 | Viewed by 6512

Abstract

Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. [...] Read more.

Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspective of information flow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information flow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information flow. Therefore, contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write

G e o C_{y \to x}

. This avoids some of the boundedness issues that we show exist for the transfer entropy,

T_{y \to x}

. We will highlight our discussions with data developed from synthetic models of successively more complex nature: these include the Hénon map example, and finally a real physiological example relating breathing and heart rate function. Full article

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29 pages, 4801 KiB

Open AccessEditor’s ChoiceArticle

Spectral Structure and Many-Body Dynamics of Ultracold Bosons in a Double-Well

by Frank Schäfer, Miguel A. Bastarrachea-Magnani, Axel U. J. Lode, Laurent de Forges de Parny and Andreas Buchleitner

Entropy 2020, 22(4), 382; https://doi.org/10.3390/e22040382 - 26 Mar 2020

Cited by 6 | Viewed by 5013

Abstract

We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior [...] Read more.

We examine the spectral structure and many-body dynamics of two and three repulsively interacting bosons trapped in a one-dimensional double-well, for variable barrier height, inter-particle interaction strength, and initial conditions. By exact diagonalization of the many-particle Hamiltonian, we specifically explore the dynamical behavior of the particles launched either at the single-particle ground state or saddle-point energy, in a time-independent potential. We complement these results by a characterization of the cross-over from diabatic to quasi-adiabatic evolution under finite-time switching of the potential barrier, via the associated time evolution of a single particle’s von Neumann entropy. This is achieved with the help of the multiconfigurational time-dependent Hartree method for indistinguishable particles (MCTDH-X)—which also allows us to extrapolate our results for increasing particle numbers. Full article

(This article belongs to the Special Issue Quantum Entropies and Complexity)

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24 pages, 1285 KiB

Open AccessEditor’s ChoiceArticle

Endoreversible Modeling of a Hydraulic Recuperation System

by Robin Masser and Karl Heinz Hoffmann

Entropy 2020, 22(4), 383; https://doi.org/10.3390/e22040383 - 26 Mar 2020

Cited by 22 | Viewed by 3882

Abstract

Hybrid drive systems able to recover and reuse braking energy of the vehicle can reduce fuel consumption, air pollution and operating costs. Among them, hydraulic recuperation systems are particularly suitable for commercial vehicles, especially if they are already equipped with a hydraulic system. [...] Read more.

Hybrid drive systems able to recover and reuse braking energy of the vehicle can reduce fuel consumption, air pollution and operating costs. Among them, hydraulic recuperation systems are particularly suitable for commercial vehicles, especially if they are already equipped with a hydraulic system. Thus far, the investigation of such systems has been limited to individual components or optimizing their control. In this paper, we focus on thermodynamic effects and their impact on the overall systems energy saving potential using endoreversible thermodynamics as the ideal framework for modeling. The dynamical behavior of the hydraulic recuperation system as well as energy savings are estimated using real data of a vehicle suitable for application. Here, energy savings accelerating the vehicle around 10% and a reduction in energy transferred to the conventional disc brakes around 58% are predicted. We further vary certain design and loss parameters—such as accumulator volume, displacement of the hydraulic unit, heat transfer coefficients or pipe diameter—and discuss their influence on the energy saving potential of the system. It turns out that heat transfer coefficients and pipe diameter are of less importance than accumulator volume and displacement of the hydraulic unit. Full article

(This article belongs to the Section Thermodynamics)

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28 pages, 4982 KiB

Open AccessEditor’s ChoiceArticle

Theory, Analysis, and Applications of the Entropic Lattice Boltzmann Model for Compressible Flows

by Nicolò Frapolli, Shyam Chikatamarla and Ilya Karlin

Entropy 2020, 22(3), 370; https://doi.org/10.3390/e22030370 - 24 Mar 2020

Cited by 17 | Viewed by 6088

Abstract

The entropic lattice Boltzmann method for the simulation of compressible flows is studied in detail and new opportunities for extending operating range are explored. We address limitations on the maximum Mach number and temperature range allowed for a given lattice. Solutions to both [...] Read more.

The entropic lattice Boltzmann method for the simulation of compressible flows is studied in detail and new opportunities for extending operating range are explored. We address limitations on the maximum Mach number and temperature range allowed for a given lattice. Solutions to both these problems are presented by modifying the original lattices without increasing the number of discrete velocities and without altering the numerical algorithm. In order to increase the Mach number, we employ shifted lattices while the magnitude of lattice speeds is increased in order to extend the temperature range. Accuracy and efficiency of the shifted lattices are demonstrated with simulations of the supersonic flow field around a diamond-shaped and NACA0012 airfoil, the subsonic, transonic, and supersonic flow field around the Busemann biplane, and the interaction of vortices with a planar shock wave. For the lattices with extended temperature range, the model is validated with the simulation of the Richtmyer–Meshkov instability. We also discuss some key ideas of how to reduce the number of discrete speeds in three-dimensional simulations by pruning of the higher-order lattices, and introduce a new construction of the corresponding guided equilibrium by entropy minimization. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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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 6698

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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14 pages, 2541 KiB

Open AccessEditor’s ChoiceReview

New Invariant Expressions in Chemical Kinetics

by Gregory S. Yablonsky, Daniel Branco, Guy B. Marin and Denis Constales

Entropy 2020, 22(3), 373; https://doi.org/10.3390/e22030373 - 24 Mar 2020

Cited by 6 | Viewed by 3632

Abstract

This paper presents a review of our original results obtained during the last decade. These results have been found theoretically for classical mass-action-law models of chemical kinetics and justified experimentally. In contrast with the traditional invariances, they relate to a special battery of [...] Read more.

This paper presents a review of our original results obtained during the last decade. These results have been found theoretically for classical mass-action-law models of chemical kinetics and justified experimentally. In contrast with the traditional invariances, they relate to a special battery of kinetic experiments, not a single experiment. Two types of invariances are distinguished and described in detail: thermodynamic invariants, i.e., special combinations of kinetic dependences that yield the equilibrium constants, or simple functions of the equilibrium constants; and “mixed” kinetico-thermodynamic invariances, functions both of equilibrium constants and non-thermodynamic ratios of kinetic coefficients. Full article

(This article belongs to the Special Issue Entropies: Between Information Geometry and Kinetics)

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18 pages, 3218 KiB

Open AccessEditor’s ChoiceArticle

BiEntropy, TriEntropy and Primality

by Grenville J. Croll

Entropy 2020, 22(3), 311; https://doi.org/10.3390/e22030311 - 10 Mar 2020

Cited by 4 | Viewed by 8090

Abstract

The order and disorder of binary representations of the natural numbers < 2⁸ is measured using the BiEntropy function. Significant differences are detected between the primes and the non-primes. The BiEntropic prime density is shown to be quadratic with a very small [...] Read more.

The order and disorder of binary representations of the natural numbers < 2⁸ is measured using the BiEntropy function. Significant differences are detected between the primes and the non-primes. The BiEntropic prime density is shown to be quadratic with a very small Gaussian distributed error. The work is repeated in binary using a Monte Carlo simulation of a sample of natural numbers < 2³² and in trinary for all natural numbers < 3⁹ with similar but cubic results. We found a significant relationship between BiEntropy and TriEntropy such that we can discriminate between the primes and numbers divisible by six. We discuss the theoretical basis of these results and show how they generalise to give a tight bound on the variance of Pi(x)–Li(x) for all x. This bound is much tighter than the bound given by Von Koch in 1901 as an equivalence for proof of the Riemann Hypothesis. Since the primes are Gaussian due to a simple induction on the binary derivative, this implies that the twin primes conjecture is true. We also provide absolutely convergent asymptotes for the numbers of Fermat and Mersenne primes in the appendices. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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17 pages, 538 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Two Faced Janus of Quantum Nonlocality

by Andrei Khrennikov

Entropy 2020, 22(3), 303; https://doi.org/10.3390/e22030303 - 6 Mar 2020

Cited by 32 | Viewed by 4329

Abstract

This paper is a new step towards understanding why “quantum nonlocality” is a misleading concept. Metaphorically speaking, “quantum nonlocality” is Janus faced. One face is an apparent nonlocality of the Lüders projection and another face is Bell nonlocality (a wrong conclusion that the [...] Read more.

This paper is a new step towards understanding why “quantum nonlocality” is a misleading concept. Metaphorically speaking, “quantum nonlocality” is Janus faced. One face is an apparent nonlocality of the Lüders projection and another face is Bell nonlocality (a wrong conclusion that the violation of Bell type inequalities implies the existence of mysterious instantaneous influences between distant physical systems). According to the Lüders projection postulate, a quantum measurement performed on one of the two distant entangled physical systems modifies their compound quantum state instantaneously. Therefore, if the quantum state is considered to be an attribute of the individual physical system and if one assumes that experimental outcomes are produced in a perfectly random way, one quickly arrives at the contradiction. It is a primary source of speculations about a spooky action at a distance. Bell nonlocality as defined above was explained and rejected by several authors; thus, we concentrate in this paper on the apparent nonlocality of the Lüders projection. As already pointed out by Einstein, the quantum paradoxes disappear if one adopts the purely statistical interpretation of quantum mechanics (QM). In the statistical interpretation of QM, if probabilities are considered to be objective properties of random experiments we show that the Lüders projection corresponds to the passage from joint probabilities describing all set of data to some marginal conditional probabilities describing some particular subsets of data. If one adopts a subjective interpretation of probabilities, such as QBism, then the Lüders projection corresponds to standard Bayesian updating of the probabilities. The latter represents degrees of beliefs of local agents about outcomes of individual measurements which are placed or which will be placed at distant locations. In both approaches, probability-transformation does not happen in the physical space, but only in the information space. Thus, all speculations about spooky interactions or spooky predictions at a distance are simply misleading. Coming back to Bell nonlocality, we recall that in a recent paper we demonstrated, using exclusively the quantum formalism, that CHSH inequalities may be violated for some quantum states only because of the incompatibility of quantum observables and Bohr’s complementarity. Finally, we explain that our criticism of quantum nonlocality is in the spirit of Hertz-Boltzmann methodology of scientific theories. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness II)

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