Entropy

13 pages, 305 KiB

Open AccessArticle

On the Exact Variance of Tsallis Entanglement Entropy in a Random Pure State

by Lu Wei

Entropy 2019, 21(5), 539; https://doi.org/10.3390/e21050539 - 27 May 2019

Cited by 16 | Viewed by 3408

The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an [...] Read more.

The Tsallis entropy is a useful one-parameter generalization to the standard von Neumann entropy in quantum information theory. In this work, we study the variance of the Tsallis entropy of bipartite quantum systems in a random pure state. The main result is an exact variance formula of the Tsallis entropy that involves finite sums of some terminating hypergeometric functions. In the special cases of quadratic entropy and small subsystem dimensions, the main result is further simplified to explicit variance expressions. As a byproduct, we find an independent proof of the recently proven variance formula of the von Neumann entropy based on the derived moment relation to the Tsallis entropy. Full article

(This article belongs to the Special Issue Entropy in Foundations of Quantum Physics)

3 pages, 188 KiB

Open AccessEditorial

Nonadditive Entropies and Complex Systems

by Andrea Rapisarda, Stefan Thurner and Constantino Tsallis

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

Cited by 4 | Viewed by 4323

Abstract

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

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

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

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

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

[...] Full article

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

26 pages, 970 KiB

Open AccessArticle

Discriminatory Target Learning: Mining Significant Dependence Relationships from Labeled and Unlabeled Data

by Zhi-Yi Duan, Li-Min Wang, Musa Mammadov, Hua Lou and Ming-Hui Sun

Entropy 2019, 21(5), 537; https://doi.org/10.3390/e21050537 - 26 May 2019

Cited by 1 | Viewed by 3515

Abstract

Machine learning techniques have shown superior predictive power, among which Bayesian network classifiers (BNCs) have remained of great interest due to its capacity to demonstrate complex dependence relationships. Most traditional BNCs tend to build only one model to fit training instances by analyzing [...] Read more.

Machine learning techniques have shown superior predictive power, among which Bayesian network classifiers (BNCs) have remained of great interest due to its capacity to demonstrate complex dependence relationships. Most traditional BNCs tend to build only one model to fit training instances by analyzing independence between attributes using conditional mutual information. However, for different class labels, the conditional dependence relationships may be different rather than invariant when attributes take different values, which may result in classification bias. To address this issue, we propose a novel framework, called discriminatory target learning, which can be regarded as a tradeoff between probabilistic model learned from unlabeled instance at the uncertain end and that learned from labeled training data at the certain end. The final model can discriminately represent the dependence relationships hidden in unlabeled instance with respect to different possible class labels. Taking k-dependence Bayesian classifier as an example, experimental comparison on 42 publicly available datasets indicated that the final model achieved competitive classification performance compared to state-of-the-art learners such as Random forest and averaged one-dependence estimators. Full article

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

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29 pages, 1660 KiB

Open AccessArticle

Entropy and Mixing Entropy for Weakly Nonlinear Mechanical Vibrating Systems

by Zahra Sotoudeh

Entropy 2019, 21(5), 536; https://doi.org/10.3390/e21050536 - 26 May 2019

Cited by 4 | Viewed by 3943

Abstract

In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems [...] Read more.

In this paper, we examine Khinchin’s entropy for two weakly nonlinear systems of oscillators. We study a system of coupled Duffing oscillators and a set of Henon–Heiles oscillators. It is shown that the general method of deriving the Khinchin’s entropy for linear systems can be modified to account for weak nonlinearities. Nonlinearities are modeled as nonlinear springs. To calculate the Khinchin’s entropy, one needs to obtain an analytical expression of the system’s phase volume. We use a perturbation method to do so, and verify the results against the numerical calculation of the phase volume. It is shown that such an approach is valid for weakly nonlinear systems. In an extension of the author’s previous work for linear systems, a mixing entropy is defined for these two oscillators. The mixing entropy is the result of the generation of entropy when two systems are combined to create a complex system. It is illustrated that mixing entropy is always non-negative. The mixing entropy provides insight into the energy behavior of each system. The limitation of statistical energy analysis motivates this study. Using the thermodynamic relationship of temperature and entropy, and Khinchin’s entropy, one can define a vibrational temperature. Vibrational temperature can be used to derive the power flow proportionality, which is the backbone of the statistical energy analysis. Although this paper is motivated by statistical energy analysis application, it is not devoted to the statistical energy analysis of nonlinear systems. Full article

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14 pages, 6818 KiB

Open AccessArticle

A Giga-Stable Oscillator with Hidden and Self-Excited Attractors: A Megastable Oscillator Forced by His Twin

by Thoai Phu Vo, Yeganeh Shaverdi, Abdul Jalil M. Khalaf, Fawaz E. Alsaadi, Tasawar Hayat and Viet-Thanh Pham

Entropy 2019, 21(5), 535; https://doi.org/10.3390/e21050535 - 25 May 2019

Cited by 10 | Viewed by 4035

Abstract

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while [...] Read more.

In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors’ images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)

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9 pages, 247 KiB

Open AccessArticle

Summoning, No-Signalling and Relativistic Bit Commitments

by Adrian Kent

Entropy 2019, 21(5), 534; https://doi.org/10.3390/e21050534 - 25 May 2019

Viewed by 3120

Abstract

Summoning is a task between two parties, Alice and Bob, with distributed networks of agents in space-time. Bob gives Alice a random quantum state, known to him but not her, at some point. She is required to return the state at some later [...] Read more.

Summoning is a task between two parties, Alice and Bob, with distributed networks of agents in space-time. Bob gives Alice a random quantum state, known to him but not her, at some point. She is required to return the state at some later point, belonging to a subset defined by communications received from Bob at other points. Many results about summoning, including the impossibility of unrestricted summoning tasks and the necessary conditions for specific types of summoning tasks to be possible, follow directly from the quantum no-cloning theorem and the relativistic no-superluminal-signalling principle. The impossibility of cloning devices can be derived from the impossibility of superluminal signalling and the projection postulate, together with assumptions about the devices’ location-independent functioning. In this qualified sense, known summoning results follow from the causal structure of space-time and the properties of quantum measurements. Bounds on the fidelity of approximate cloning can be similarly derived. Bit commitment protocols and other cryptographic protocols based on the no-summoning theorem can thus be proven secure against some classes of post-quantum but non-signalling adversaries. Full article

(This article belongs to the Special Issue Relativistic Quantum Information)

15 pages, 916 KiB

Open AccessArticle

A SOM-Based Membrane Optimization Algorithm for Community Detection

by Chuang Liu, Yingkui Du and Jiahao Lei

Entropy 2019, 21(5), 533; https://doi.org/10.3390/e21050533 - 25 May 2019

Cited by 7 | Viewed by 3707

Abstract

The real world is full of rich and valuable complex networks. Community structure is an important feature in complex networks, which makes possible the discovery of some structure or hidden related information for an in-depth study of complex network structures and functional characteristics. [...] Read more.

The real world is full of rich and valuable complex networks. Community structure is an important feature in complex networks, which makes possible the discovery of some structure or hidden related information for an in-depth study of complex network structures and functional characteristics. Aimed at community detection in complex networks, this paper proposed a membrane algorithm based on a self-organizing map (SOM) network. Firstly, community detection was transformed as discrete optimization problems by selecting the optimization function. Secondly, three elements of the membrane algorithm, objects, reaction rules, and membrane structure were designed to analyze the properties and characteristics of the community structure. Thirdly, a SOM was employed to determine the number of membranes by learning and mining the structure of the current objects in the decision space, which is beneficial to guiding the local and global search of the proposed algorithm by constructing the neighborhood relationship. Finally, the simulation experiment was carried out on both synthetic benchmark networks and four real-world networks. The experiment proved that the proposed algorithm had higher accuracy, stability, and execution efficiency, compared with the results of other experimental algorithms. Full article

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

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25 pages, 2964 KiB

Open AccessArticle

Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning

by Jorge Useche and Rafael Hurtado

Entropy 2019, 21(5), 532; https://doi.org/10.3390/e21050532 - 25 May 2019

Cited by 7 | Viewed by 5990

Abstract

One of the most relevant features of musical pieces is the selection and utilization of musical elements by composers. For connecting the musical properties of a melodic line as a whole with those of its constituent elements, we propose a representation for musical [...] Read more.

One of the most relevant features of musical pieces is the selection and utilization of musical elements by composers. For connecting the musical properties of a melodic line as a whole with those of its constituent elements, we propose a representation for musical intervals based on physical quantities and a statistical model based on the minimization of relative entropy. The representation contains information about the size, location in the register, and level of tonal consonance of musical intervals. The statistical model involves expected values of relevant physical quantities that can be adopted as macroscopic constraints with musical meaning. We studied the occurrences of musical intervals in 20 melodic lines from seven masterpieces of Western tonal music. We found that all melodic lines are strictly ordered in terms of the physical quantities of the representation and that the formalism is suitable for approximately reproducing the final selection of musical intervals made by the composers, as well as for describing musical features as the asymmetry in the use of ascending and descending intervals, transposition processes, and the mean dissonance of a melodic line. Full article

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

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Graphical abstract

11 pages, 841 KiB

Open AccessArticle

A Matrix Information-Geometric Method for Change-Point Detection of Rigid Body Motion

by Xiaomin Duan, Huafei Sun and Xinyu Zhao

Entropy 2019, 21(5), 531; https://doi.org/10.3390/e21050531 - 25 May 2019

Cited by 2 | Viewed by 3267

Abstract

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group

S E (3)

, so the Riemannian mean based on the Lie group [...] Read more.

A matrix information-geometric method was developed to detect the change-points of rigid body motions. Note that the set of all rigid body motions is the special Euclidean group

S E (3)

, so the Riemannian mean based on the Lie group structures of

S E (3)

reflects the characteristics of change-points. Once a change-point occurs, the distance between the current point and the Riemannian mean of its neighbor points should be a local maximum. A gradient descent algorithm is proposed to calculate the Riemannian mean. Using the Baker–Campbell–Hausdorff formula, the first-order approximation of the Riemannian mean is taken as the initial value of the iterative procedure. The performance of our method was evaluated by numerical examples and manipulator experiments. Full article

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

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10 pages, 1213 KiB

Open AccessArticle

Nonlinear Stochastic Equation within an Itô Prescription for Modelling of Financial Market

by Leonardo S. Lima

Entropy 2019, 21(5), 530; https://doi.org/10.3390/e21050530 - 25 May 2019

Cited by 5 | Viewed by 3417

Abstract

The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of [...] Read more.

The stochastic nonlinear model based on Itô diffusion is proposed as a mathematical model for price dynamics of financial markets. We study this model with relation to concrete stylised facts about financial markets. We investigate the behavior of the long tail distribution of the volatilities and verify the inverse power law behavior which is obeyed for some financial markets. Furthermore, we obtain the behavior of the long range memory and obtain that it follows to a distinct behavior of other stochastic models that are used as models for the finances. Furthermore, we have made an analysis by using Fokker–Planck equation independent on time with the aim of obtaining the cumulative probability distribution of volatilities

P (g)

, however, the probability density found does not exhibit the cubic inverse law. Full article

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15 pages, 4421 KiB

Open AccessArticle

Entropy Generation via Ohmic Heating and Hall Current in Peristaltically-Flowing Carreau Fluid

by Saima Noreen, Asif Abbas and Abid Hussanan

Entropy 2019, 21(5), 529; https://doi.org/10.3390/e21050529 - 24 May 2019

Cited by 5 | Viewed by 4195

Abstract

The core objective of the present study is to examine entropy generation minimization via Hall current and Ohmic heating. Carreau fluid considerations interpret the unavailability of systems’ thermal energy (for mechanical work). The magneto hydrodynamic flow is in the channel, which is not [...] Read more.

The core objective of the present study is to examine entropy generation minimization via Hall current and Ohmic heating. Carreau fluid considerations interpret the unavailability of systems’ thermal energy (for mechanical work). The magneto hydrodynamic flow is in the channel, which is not symmetric. We have solved analytically the resulting nonlinear mathematical model. Moreover, physical exploration of important parameters on total entropy generation, temperature, and Bejan number is plotted and discussed. We observed that the generation of entropy takes place throughout the confined flow field y = W₁ and y = W₂ because of the viscous dissipation effect. In addition, reducing the operating temperature minimizes the entropy. Full article

(This article belongs to the Special Issue Challenges and Progresses in the Modelling of Entropy Generation in Fluid Mechanics, Heat Transfer and Porous Media)

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20 pages, 4127 KiB

Open AccessArticle

Dual-Space Information Modeling of Socio-Economic Systems under Information Asymmetry

by Lei Bao and Joseph Fritchman

Entropy 2019, 21(5), 528; https://doi.org/10.3390/e21050528 - 24 May 2019

Viewed by 3528

Abstract

Information definitions across many disciplines commonly treat information as a physical world entity. Information measures are used along with other physical variables undistinguished for modeling physical systems. Building on previous work, this research explicitly defines information as a unique category of entity that [...] Read more.

Information definitions across many disciplines commonly treat information as a physical world entity. Information measures are used along with other physical variables undistinguished for modeling physical systems. Building on previous work, this research explicitly defines information as a unique category of entity that is created by intelligent agents to represent aspects of the physical world but that is not part of the physical world. This leads to the formation of the dual-space information modeling (DSIM) framework, which clearly distinguishes an information space from the physically based material space. The separation of information and material spaces allows new insight and flexibility into modeling complex systems. In this research, DSIM based agent models are applied to study the impact of information asymmetry to marketing behaviors. This paper demonstrates the effectiveness of the DSIM framework in the modeling process and how emergent behavior from these systems is encapsulated in the dual-space. Full article

(This article belongs to the Special Issue Information Theory in Complex Systems)

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15 pages, 11189 KiB

Open AccessFeature PaperArticle

Current Correlations in a Quantum Dot Ring: A Role of Quantum Interference

by Bogdan R. Bułka and Jakub Łuczak

Entropy 2019, 21(5), 527; https://doi.org/10.3390/e21050527 - 24 May 2019

Cited by 6 | Viewed by 3855

Abstract

We present studies of the electron transport and circular currents induced by the bias voltage and the magnetic flux threading a ring of three quantum dots coupled with two electrodes. Quantum interference of electron waves passing through the states with opposite chirality plays [...] Read more.

We present studies of the electron transport and circular currents induced by the bias voltage and the magnetic flux threading a ring of three quantum dots coupled with two electrodes. Quantum interference of electron waves passing through the states with opposite chirality plays a relevant role in transport, where one can observe Fano resonance with destructive interference. The quantum interference effect is quantitatively described by local bond currents and their correlation functions. Fluctuations of the transport current are characterized by the Lesovik formula for the shot noise, which is a composition of the bond current correlation functions. In the presence of circular currents, the cross-correlation of the bond currents can be very large, but it is negative and compensates for the large positive auto-correlation functions. Full article

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

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16 pages, 1362 KiB

Open AccessArticle

Multiscale Information Decomposition Dissects Control Mechanisms of Heart Rate Variability at Rest and During Physiological Stress

by Jana Krohova, Luca Faes, Barbora Czippelova, Zuzana Turianikova, Nikoleta Mazgutova, Riccardo Pernice, Alessandro Busacca, Daniele Marinazzo, Sebastiano Stramaglia and Michal Javorka

Entropy 2019, 21(5), 526; https://doi.org/10.3390/e21050526 - 24 May 2019

Cited by 28 | Viewed by 5055

Abstract

Heart rate variability (HRV; variability of the RR interval of the electrocardiogram) results from the activity of several coexisting control mechanisms, which involve the influence of respiration (RESP) and systolic blood pressure (SBP) oscillations operating across multiple temporal scales and changing in different [...] Read more.

Heart rate variability (HRV; variability of the RR interval of the electrocardiogram) results from the activity of several coexisting control mechanisms, which involve the influence of respiration (RESP) and systolic blood pressure (SBP) oscillations operating across multiple temporal scales and changing in different physiological states. In this study, multiscale information decomposition is used to dissect the physiological mechanisms related to the genesis of HRV in 78 young volunteers monitored at rest and during postural and mental stress evoked by head-up tilt (HUT) and mental arithmetics (MA). After representing RR, RESP and SBP at different time scales through a recently proposed method based on multivariate state space models, the joint information transfer

T_{RESP, SBP \to RR}

is decomposed into unique, redundant and synergistic components, describing the strength of baroreflex modulation independent of respiration (

U_{SBP \to RR}

), nonbaroreflex (

U_{RESP \to RR}

) and baroreflex-mediated (

R_{RESP, SBP \to RR}

) respiratory influences, and simultaneous presence of baroreflex and nonbaroreflex respiratory influences (

S_{RESP, SBP \to RR}

), respectively. We find that fast (short time scale) HRV oscillations—respiratory sinus arrhythmia—originate from the coexistence of baroreflex and nonbaroreflex (central) mechanisms at rest, with a stronger baroreflex involvement during HUT. Focusing on slower HRV oscillations, the baroreflex origin is dominant and MA leads to its higher involvement. Respiration influences independent on baroreflex are present at long time scales, and are enhanced during HUT. Full article

(This article belongs to the Special Issue Information Dynamics in Brain and Physiological Networks)

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23 pages, 2884 KiB

Open AccessArticle

Evaluating Approximations and Heuristic Measures of Integrated Information

by André Sevenius Nilsen, Bjørn Erik Juel and William Marshall

Entropy 2019, 21(5), 525; https://doi.org/10.3390/e21050525 - 24 May 2019

Cited by 14 | Viewed by 6060

Abstract

Integrated information theory (IIT) proposes a measure of integrated information, termed Phi (Φ), to capture the level of consciousness of a physical system in a given state. Unfortunately, calculating Φ itself is currently possible only for very small model systems and far from [...] Read more.

Integrated information theory (IIT) proposes a measure of integrated information, termed Phi (Φ), to capture the level of consciousness of a physical system in a given state. Unfortunately, calculating Φ itself is currently possible only for very small model systems and far from computable for the kinds of system typically associated with consciousness (brains). Here, we considered several proposed heuristic measures and computational approximations, some of which can be applied to larger systems, and tested if they correlate well with Φ. While these measures and approximations capture intuitions underlying IIT and some have had success in practical applications, it has not been shown that they actually quantify the type of integrated information specified by the latest version of IIT and, thus, whether they can be used to test the theory. In this study, we evaluated these approximations and heuristic measures considering how well they estimated the Φ values of model systems and not on the basis of practical or clinical considerations. To do this, we simulated networks consisting of 3–6 binary linear threshold nodes randomly connected with excitatory and inhibitory connections. For each system, we then constructed the system’s state transition probability matrix (TPM) and generated observed data over time from all possible initial conditions. We then calculated Φ, approximations to Φ, and measures based on state differentiation, coalition entropy, state uniqueness, and integrated information. Our findings suggest that Φ can be approximated closely in small binary systems by using one or more of the readily available approximations (r > 0.95) but without major reductions in computational demands. Furthermore, the maximum value of Φ across states (a state-independent quantity) correlated strongly with measures of signal complexity (LZ, r_s = 0.722), decoder-based integrated information (Φ*, r_s = 0.816), and state differentiation (D1, r_s = 0.827). These measures could allow for the efficient estimation of a system’s capacity for high Φ or function as accurate predictors of low- (but not high-)Φ systems. While it is uncertain whether the results extend to larger systems or systems with other dynamics, we stress the importance that measures aimed at being practical alternatives to Φ be, at a minimum, rigorously tested in an environment where the ground truth can be established. Full article

(This article belongs to the Special Issue Integrated Information Theory)

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15 pages, 1006 KiB

Open AccessArticle

The Evolution of Neuroplasticity and the Effect on Integrated Information

by Leigh Sheneman, Jory Schossau and Arend Hintze

Entropy 2019, 21(5), 524; https://doi.org/10.3390/e21050524 - 24 May 2019

Cited by 3 | Viewed by 4814

Abstract

Information integration theory has been developed to quantify consciousness. Since conscious thought requires the integration of information, the degree of this integration can be used as a neural correlate (

Φ

) with the intent to measure degree of consciousness. Previous research has [...] Read more.

Information integration theory has been developed to quantify consciousness. Since conscious thought requires the integration of information, the degree of this integration can be used as a neural correlate (

Φ

) with the intent to measure degree of consciousness. Previous research has shown that the ability to integrate information can be improved by Darwinian evolution. The value

Φ

can change over many generations, and complex tasks require systems with at least a minimum

Φ

. This work was done using simple animats that were able to remember previous sensory inputs, but were incapable of fundamental change during their lifetime: actions were predetermined or instinctual. Here, we are interested in changes to

Φ

due to lifetime learning (also known as neuroplasticity). During lifetime learning, the system adapts to perform a task and necessitates a functional change, which in turn could change

Φ

. One can find arguments to expect one of three possible outcomes:

Φ

might remain constant, increase, or decrease due to learning. To resolve this, we need to observe systems that learn, but also improve their ability to learn over the many generations that Darwinian evolution requires. Quantifying

Φ

over the course of evolution, and over the course of their lifetimes, allows us to investigate how the ability to integrate information changes. To measure

Φ

, the internal states of the system must be experimentally observable. However, these states are notoriously difficult to observe in a natural system. Therefore, we use a computational model that not only evolves virtual agents (animats), but evolves animats to learn during their lifetime. We use this approach to show that a system that improves its performance due to feedback learning increases its ability to integrate information. In addition, we show that a system’s ability to increase

Φ

correlates with its ability to increase in performance. This suggests that systems that are very plastic regarding

Φ

learn better than those that are not. Full article

(This article belongs to the Special Issue Integrated Information Theory)

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12 pages, 2896 KiB

Open AccessArticle

Image Entropy for the Identification of Chimera States of Spatiotemporal Divergence in Complex Coupled Maps of Matrices

by Rasa Smidtaite, Guangqing Lu and Minvydas Ragulskis

Entropy 2019, 21(5), 523; https://doi.org/10.3390/e21050523 - 24 May 2019

Cited by 5 | Viewed by 4019

Abstract

Complex networks of coupled maps of matrices (NCMM) are investigated in this paper. It is shown that a NCMM can evolve into two different steady states—the quiet state or the state of divergence. It appears that chimera states of spatiotemporal divergence do exist [...] Read more.

Complex networks of coupled maps of matrices (NCMM) are investigated in this paper. It is shown that a NCMM can evolve into two different steady states—the quiet state or the state of divergence. It appears that chimera states of spatiotemporal divergence do exist in the regions around the boundary lines separating these two steady states. It is demonstrated that digital image entropy can be used as an effective measure for the visualization of these regions of chimera states in different networks (regular, feed-forward, random, and small-world NCMM). Full article

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

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17 pages, 3151 KiB

Open AccessArticle

An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy

by Zhongfan Zhu, Jingshan Yu, Jie Dou and Dingzhi Peng

Entropy 2019, 21(5), 522; https://doi.org/10.3390/e21050522 - 23 May 2019

Cited by 6 | Viewed by 3364

Abstract

In the context of river dynamics, some experimental results have shown that particle velocity is different from fluid velocity along the stream-wise direction for uniform sediment-laden open-channel flows; this velocity difference has been termed velocity lag in the literature. In this study, an [...] Read more.

In the context of river dynamics, some experimental results have shown that particle velocity is different from fluid velocity along the stream-wise direction for uniform sediment-laden open-channel flows; this velocity difference has been termed velocity lag in the literature. In this study, an analytical expression for estimating the velocity lag in open-channel flows was derived based on the Tsallis entropy theory together with the principle of maximum entropy. The derived expression represents the velocity lag as a function of a non-dimensional entropy parameter depending on the average and maximum values of velocity lag from experimental measurements. The derived expression was tested against twenty-two experimental datasets collected from the literature with three deterministic models and the developed Shannon entropy-based model. The Tsallis entropy-based model agreed better with the experimental datasets than the deterministic models for eighteen out of the twenty-two total real cases, and the prediction accuracy for the eighteen experimental datasets was comparable to that of the developed Shannon entropy-based model (the Tsallis entropy-based expression agreed slightly better than the Shannon entropy-based model for twelve out of eighteen test cases, whereas for the other six test cases, the Shannon entropy-based model had a slightly higher prediction accuracy). Finally, the effects of the friction velocity of the flow, the particle diameter, and the particles’ specific gravity on the velocity lag were analyzed based on the Tsallis entropy-based model. This study shows the potential of the Tsallis entropy theory together with the principle of maximum entropy to predict the stream-wise velocity lag between a particle and the surrounding fluid in sediment-laden open-channel flows. Full article

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18 pages, 771 KiB

Open AccessEditor’s ChoiceArticle

Is Independence Necessary for a Discontinuous Phase Transition within the q-Voter Model?

by Angelika Abramiuk, Jakub Pawłowski and Katarzyna Sznajd-Weron

Entropy 2019, 21(5), 521; https://doi.org/10.3390/e21050521 - 23 May 2019

Cited by 18 | Viewed by 4498

Abstract

We ask a question about the possibility of a discontinuous phase transition and the related social hysteresis within the q-voter model with anticonformity. Previously, it was claimed that within the q-voter model the social hysteresis can emerge only because of an [...] Read more.

We ask a question about the possibility of a discontinuous phase transition and the related social hysteresis within the q-voter model with anticonformity. Previously, it was claimed that within the q-voter model the social hysteresis can emerge only because of an independent behavior, and for the model with anticonformity only continuous phase transitions are possible. However, this claim was derived from the model, in which the size of the influence group needed for the conformity was the same as the size of the group needed for the anticonformity. Here, we abandon this assumption on the equality of two types of social response and introduce the generalized model, in which the size of the influence group needed for the conformity

q_{c}

and the size of the influence group needed for the anticonformity

q_{a}

are independent variables and in general

q_{c} \neq q_{a}

. We investigate the model on the complete graph, similarly as it was done for the original q-voter model with anticonformity, and we show that such a generalized model displays both types of phase transitions depending on parameters

q_{c}

and

q_{a}

. Full article

(This article belongs to the Special Issue Computational and Statistical Physics Approaches for Complex Systems and Social Phenomena)

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48 pages, 1293 KiB

Open AccessArticle

Statistical Lyapunov Theory Based on Bifurcation Analysis of Energy Cascade in Isotropic Homogeneous Turbulence: A Physical–Mathematical Review

by Nicola de Divitiis

Entropy 2019, 21(5), 520; https://doi.org/10.3390/e21050520 - 23 May 2019

Cited by 3 | Viewed by 4010

Abstract

This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study [...] Read more.

This work presents a review of previous articles dealing with an original turbulence theory proposed by the author and provides new theoretical insights into some related issues. The new theoretical procedures and methodological approaches confirm and corroborate the previous results. These articles study the regime of homogeneous isotropic turbulence for incompressible fluids and propose theoretical approaches based on a specific Lyapunov theory for determining the closures of the von Kármán–Howarth and Corrsin equations and the statistics of velocity and temperature difference. While numerous works are present in the literature which concern the closures of the autocorrelation equations in the Fourier domain (i.e., Lin equation closure), few articles deal with the closures of the autocorrelation equations in the physical space. These latter, being based on the eddy–viscosity concept, describe diffusive closure models. On the other hand, the proposed Lyapunov theory leads to nondiffusive closures based on the property that, in turbulence, contiguous fluid particles trajectories continuously diverge. Therefore, the main motivation of this review is to present a theoretical formulation which does not adopt the eddy–viscosity paradigm and summarizes the results of the previous works. Next, this analysis assumes that the current fluid placements, together with velocity and temperature fields, are fluid state variables. This leads to the closures of the autocorrelation equations and helps to interpret the mechanism of energy cascade as due to the continuous divergence of the contiguous trajectories. Furthermore, novel theoretical issues are here presented among which we can mention the following ones. The bifurcation rate of the velocity gradient, calculated along fluid particles trajectories, is shown to be much larger than the corresponding maximal Lyapunov exponent. On that basis, an interpretation of the energy cascade phenomenon is given and the statistics of finite time Lyapunov exponent of the velocity gradient is shown to be represented by normal distribution functions. Next, the self–similarity produced by the proposed closures is analyzed and a proper bifurcation analysis of the closed von Kármán–Howarth equation is performed. This latter investigates the route from developed turbulence toward the non–chaotic regimes, leading to an estimate of the critical Taylor scale Reynolds number. A proper statistical decomposition based on extended distribution functions and on the Navier–Stokes equations is presented, which leads to the statistics of velocity and temperature difference. Full article

(This article belongs to the Special Issue Intermittency and Self-Organisation in Turbulence and Statistical Mechanics)

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16 pages, 2057 KiB

Open AccessArticle

Fault Diagnosis for Rolling Element Bearings Based on Feature Space Reconstruction and Multiscale Permutation Entropy

by Weibo Zhang and Jianzhong Zhou

Entropy 2019, 21(5), 519; https://doi.org/10.3390/e21050519 - 23 May 2019

Cited by 18 | Viewed by 3572

Abstract

Aimed at distinguishing different fault categories of severity of rolling bearings, a novel method based on feature space reconstruction and multiscale permutation entropy is proposed in the study. Firstly, the ensemble empirical mode decomposition algorithm (EEMD) was employed to adaptively decompose the vibration [...] Read more.

Aimed at distinguishing different fault categories of severity of rolling bearings, a novel method based on feature space reconstruction and multiscale permutation entropy is proposed in the study. Firstly, the ensemble empirical mode decomposition algorithm (EEMD) was employed to adaptively decompose the vibration signal into multiple intrinsic mode functions (IMFs), and the representative IMFs which contained rich fault information were selected to reconstruct a feature vector space. Secondly, the multiscale permutation entropy (MPE) was used to calculate the complexity of reconstructed feature space. Finally, the value of multiscale permutation entropy was presented to a support vector machine for fault classification. The proposed diagnostic algorithm was applied to three groups of rolling bearing experiments. The experimental results indicate that the proposed method has better classification performance and robustness than other traditional methods. Full article

(This article belongs to the Section Signal and Data Analysis)

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19 pages, 916 KiB

Open AccessArticle

Quantum Identity Authentication in the Counterfactual Quantum Key Distribution Protocol

by Bin Liu, Zhifeng Gao, Di Xiao, Wei Huang, Zhiqing Zhang and Bingjie Xu

Entropy 2019, 21(5), 518; https://doi.org/10.3390/e21050518 - 23 May 2019

Cited by 15 | Viewed by 4161

Abstract

In this paper, a quantum identity authentication protocol is presented based on the counterfactual quantum key distribution system. Utilizing the proposed protocol, two participants can verify each other’s identity through the counterfactual quantum communication system. The security of the protocol is proved against [...] Read more.

In this paper, a quantum identity authentication protocol is presented based on the counterfactual quantum key distribution system. Utilizing the proposed protocol, two participants can verify each other’s identity through the counterfactual quantum communication system. The security of the protocol is proved against individual attacks. Furthermore, according to the characteristics of the counterfactual quantum key distribution system, we propose an authenticated counterfactual quantum key distribution protocol based on a novel strategy of mixing the two types of quantum cryptographic protocols randomly. The authenticated quantum key distribution can also be used to update the extent of the authentication keys. Full article

(This article belongs to the Collection Quantum Information)

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11 pages, 741 KiB

Open AccessArticle

Recurrence Networks in Natural Languages

by Edgar Baeza-Blancas, Bibiana Obregón-Quintana, Candelario Hernández-Gómez, Domingo Gómez-Meléndez, Daniel Aguilar-Velázquez, Larry S. Liebovitch and Lev Guzmán-Vargas

Entropy 2019, 21(5), 517; https://doi.org/10.3390/e21050517 - 23 May 2019

Cited by 4 | Viewed by 4016

Abstract

We present a study of natural language using the recurrence network method. In our approach, the repetition of patterns of characters is evaluated without considering the word structure in written texts from different natural languages. Our dataset comprises 85 ebookseBooks written in 17 [...] Read more.

We present a study of natural language using the recurrence network method. In our approach, the repetition of patterns of characters is evaluated without considering the word structure in written texts from different natural languages. Our dataset comprises 85 ebookseBooks written in 17 different European languages. The similarity between patterns of length m is determined by the Hamming distance and a value r is considered to define a matching between two patterns, i.e., a repetition is defined if the Hamming distance is equal or less than the given threshold value r. In this way, we calculate the adjacency matrix, where a connection between two nodes exists when a matching occurs. Next, the recurrence network is constructed for the texts and some representative network metrics are calculated. Our results show that average values of network density, clustering, and assortativity are larger than their corresponding shuffled versions, while for metrics like such as closeness, both original and random sequences exhibit similar values. Moreover, our calculations show similar average values for density among languages which that belong to the same linguistic family. In addition, the application of a linear discriminant analysis leads to well-separated clusters of family languages based on based on the network-density properties. Finally, we discuss our results in the context of the general characteristics of written texts. Full article

(This article belongs to the Special Issue Entropy, Nonlinear Dynamics and Complexity)

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48 pages, 1245 KiB

Open AccessArticle

The Arbitrarily Varying Relay Channel

by Uzi Pereg and Yossef Steinberg

Entropy 2019, 21(5), 516; https://doi.org/10.3390/e21050516 - 22 May 2019

Cited by 2 | Viewed by 3792

Abstract

We study the arbitrarily varying relay channel, which models communication with relaying in the presence of an active adversary. We establish the cutset bound and partial decode-forward bound on the random code capacity. We further determine the random code capacity for special cases. [...] Read more.

We study the arbitrarily varying relay channel, which models communication with relaying in the presence of an active adversary. We establish the cutset bound and partial decode-forward bound on the random code capacity. We further determine the random code capacity for special cases. Then, we consider conditions under which the deterministic code capacity is determined as well. In addition, we consider the arbitrarily varying Gaussian relay channel with sender frequency division under input and state constraints. We determine the random code capacity, and establish lower and upper bounds on the deterministic code capacity. Furthermore, we show that as opposed to previous relay models, the primitive relay channel has a different behavior compared to the non-primitive relay channel in the arbitrarily varying scenario. Full article

(This article belongs to the Special Issue The 20th Anniversary of Entropy - Recent Advances in Entropy and Information-Theoretic Concepts and Their Applications)

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12 pages, 383 KiB

Open AccessArticle

Artificial Noise Injection and Its Power Loading Methods for Secure Space-Time Line Coded Systems

by Jingon Joung, Jihoon Choi, Bang Chul Jung and Sungwook Yu

Entropy 2019, 21(5), 515; https://doi.org/10.3390/e21050515 - 22 May 2019

Cited by 18 | Viewed by 3715

Abstract

In this paper, we consider a

2 \times 2

space-time line coded (STLC) system having two-transmit and two-receive antennas. To improve the secrecy rate of the STLC system, in which an illegitimate receiver eavesdrops the information delivered from the STLC transmitter to the [...] Read more.

In this paper, we consider a

2 \times 2

space-time line coded (STLC) system having two-transmit and two-receive antennas. To improve the secrecy rate of the STLC system, in which an illegitimate receiver eavesdrops the information delivered from the STLC transmitter to the STLC receiver, we propose an artificial noise (AN) injection method. By exploiting the STLC structure, a novel AN for the STLC is designed and its optimal power loading factor is derived. Numerical results verify that the proposed secure STLC systems with the designed AN injection and the power control method can significantly improve the secrecy rate compared to the conventional STLC systems. It is observed that the proposed method is more effective if there is a significant gap between the main-channel and the eavesdropper-channel gains. Full article

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

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21 pages, 3504 KiB

Open AccessArticle

Entropy Generation of Forced Convection during Melting of Ice Slurry

by Beata Niezgoda-Żelasko

Entropy 2019, 21(5), 514; https://doi.org/10.3390/e21050514 - 21 May 2019

Cited by 2 | Viewed by 3591

Abstract

This paper looks at entropy generation during ice slurry flow in straight pipes and typical heat exchanger structures used in refrigeration and air-conditioning technology. A dimensionless relationship was proposed to determine the interdependency between flow velocity and the volume fraction of ice, for [...] Read more.

This paper looks at entropy generation during ice slurry flow in straight pipes and typical heat exchanger structures used in refrigeration and air-conditioning technology. A dimensionless relationship was proposed to determine the interdependency between flow velocity and the volume fraction of ice, for which the entropy generation rates were at the minimum level in the case of non-adiabatic ice slurry flow. For pipe flow, the correlation between the minimum entropy generation rate and the overall enhancement efficiency was analyzed. As regards heat exchange processes in heat exchangers, the authors analyzed the relationship between the minimum entropy generation rate and the heat exchange surface area and exchanger efficiency. Full article

(This article belongs to the Special Issue Entropy Production in Turbulent Flow)

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19 pages, 945 KiB

Open AccessArticle

Mimicking Anti-Viruses with Machine Learning and Entropy Profiles

by Héctor D. Menéndez and José Luis Llorente

Entropy 2019, 21(5), 513; https://doi.org/10.3390/e21050513 - 21 May 2019

Cited by 12 | Viewed by 4588

Abstract

The quality of anti-virus software relies on simple patterns extracted from binary files. Although these patterns have proven to work on detecting the specifics of software, they are extremely sensitive to concealment strategies, such as polymorphism or metamorphism. These limitations also make anti-virus [...] Read more.

The quality of anti-virus software relies on simple patterns extracted from binary files. Although these patterns have proven to work on detecting the specifics of software, they are extremely sensitive to concealment strategies, such as polymorphism or metamorphism. These limitations also make anti-virus software predictable, creating a security breach. Any black hat with enough information about the anti-virus behaviour can make its own copy of the software, without any access to the original implementation or database. In this work, we show how this is indeed possible by combining entropy patterns with classification algorithms. Our results, applied to 57 different anti-virus engines, show that we can mimic their behaviour with an accuracy close to 98% in the best case and 75% in the worst, applied on Windows’ disk resident malware. Full article

(This article belongs to the Section Multidisciplinary Applications)

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16 pages, 7438 KiB

Open AccessArticle

Magnetic Otto Engine for an Electron in a Quantum Dot: Classical and Quantum Approach

by Francisco J. Peña, Oscar Negrete, Gabriel Alvarado Barrios, David Zambrano, Alejandro González, Alvaro S. Nunez, Pedro A. Orellana and Patricio Vargas

Entropy 2019, 21(5), 512; https://doi.org/10.3390/e21050512 - 20 May 2019

Cited by 19 | Viewed by 4827

Abstract

We studied the performance of classical and quantum magnetic Otto cycle with a working substance composed of a single quantum dot using the Fock–Darwin model with the inclusion of the Zeeman interaction. Modulating an external/perpendicular magnetic field, in the classical approach, we found [...] Read more.

We studied the performance of classical and quantum magnetic Otto cycle with a working substance composed of a single quantum dot using the Fock–Darwin model with the inclusion of the Zeeman interaction. Modulating an external/perpendicular magnetic field, in the classical approach, we found an oscillating behavior in the total work extracted that was not present in the quantum formulation.We found that, in the classical approach, the engine yielded a greater performance in terms of total work extracted and efficiency than when compared with the quantum approach. This is because, in the classical case, the working substance can be in thermal equilibrium at each point of the cycle, which maximizes the energy extracted in the adiabatic strokes. Full article

(This article belongs to the Special Issue Fluctuation Relations and Nonequilibrium Thermodynamics in Classical and Quantum Systems)

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19 pages, 3302 KiB

Open AccessArticle

Variational Characterizations of Local Entropy and Heat Regularization in Deep Learning

by Nicolas García Trillos, Zachary Kaplan and Daniel Sanz-Alonso

Entropy 2019, 21(5), 511; https://doi.org/10.3390/e21050511 - 20 May 2019

Cited by 1 | Viewed by 4000

Abstract

The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses, we introduce variational characterizations that naturally suggest a two-step scheme for [...] Read more.

The aim of this paper is to provide new theoretical and computational understanding on two loss regularizations employed in deep learning, known as local entropy and heat regularization. For both regularized losses, we introduce variational characterizations that naturally suggest a two-step scheme for their optimization, based on the iterative shift of a probability density and the calculation of a best Gaussian approximation in Kullback–Leibler divergence. Disregarding approximation error in these two steps, the variational characterizations allow us to show a simple monotonicity result for training error along optimization iterates. The two-step optimization schemes for local entropy and heat regularized loss differ only over which argument of the Kullback–Leibler divergence is used to find the best Gaussian approximation. Local entropy corresponds to minimizing over the second argument, and the solution is given by moment matching. This allows replacing traditional backpropagation calculation of gradients by sampling algorithms, opening an avenue for gradient-free, parallelizable training of neural networks. However, our presentation also acknowledges the potential increase in computational cost of naive optimization of regularized costs, thus giving a less optimistic view than existing works of the gains facilitated by loss regularization. Full article

(This article belongs to the Special Issue Information-Theoretic Approaches in Deep Learning)

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23 pages, 481 KiB

Open AccessArticle

The Exponentiated Lindley Geometric Distribution with Applications

by Bo Peng, Zhengqiu Xu and Min Wang

Entropy 2019, 21(5), 510; https://doi.org/10.3390/e21050510 - 20 May 2019

Cited by 5 | Viewed by 3782

Abstract

We introduce a new three-parameter lifetime distribution, the exponentiated Lindley geometric distribution, which exhibits increasing, decreasing, unimodal, and bathtub shaped hazard rates. We provide statistical properties of the new distribution, including shape of the probability density function, hazard rate function, quantile function, order [...] Read more.

We introduce a new three-parameter lifetime distribution, the exponentiated Lindley geometric distribution, which exhibits increasing, decreasing, unimodal, and bathtub shaped hazard rates. We provide statistical properties of the new distribution, including shape of the probability density function, hazard rate function, quantile function, order statistics, moments, residual life function, mean deviations, Bonferroni and Lorenz curves, and entropies. We use maximum likelihood estimation of the unknown parameters, and an Expectation-Maximization algorithm is also developed to find the maximum likelihood estimates. The Fisher information matrix is provided to construct the asymptotic confidence intervals. Finally, two real-data examples are analyzed for illustrative purposes. Full article

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

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Entropy, Volume 21, Issue 5 (May 2019) – 103 articles

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