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Entropy, Volume 15, Issue 10 (October 2013), Pages 3983-4552

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Open AccessArticle On a Generalized Entropy Measure Leading to the Pathway Model with a Preliminary Application to Solar Neutrino Data
Entropy 2013, 15(10), 4011-4025; doi:10.3390/e15104011
Received: 4 September 2013 / Accepted: 22 September 2013 / Published: 25 September 2013
Cited by 9 | PDF Full-text (257 KB) | HTML Full-text | XML Full-text
Abstract
An entropy for the scalar variable case, parallel to Havrda-Charvat entropy, was introduced by the first author, and the properties and its connection to Tsallis non-extensive statistical mechanics and the Mathai pathway model were examined by the authors in previous papers. In the
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An entropy for the scalar variable case, parallel to Havrda-Charvat entropy, was introduced by the first author, and the properties and its connection to Tsallis non-extensive statistical mechanics and the Mathai pathway model were examined by the authors in previous papers. In the current paper, we extend the entropy to cover the scalar case, multivariable case, and matrix variate case. Then, this measure is optimized under different types of restrictions, and a number of models in the multivariable case and matrix variable case are obtained. Connections of these models to problems in statistical and physical sciences are pointed out. An application of the simplest case of the pathway model to the interpretation of solar neutrino data by applying standard deviation analysis and diffusion entropy analysis is provided. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Evaluating the Spectrum of Unlocked Injection Frequency Dividers in Pulling Mode
Entropy 2013, 15(10), 4026-4041; doi:10.3390/e15104026
Received: 12 August 2013 / Revised: 17 September 2013 / Accepted: 18 September 2013 / Published: 25 September 2013
Cited by 1 | PDF Full-text (290 KB) | HTML Full-text | XML Full-text
Abstract
We study the phenomenon of periodic pulling which occurs in certain integrated microcircuits of relevant interest in applications, namely the injection-locked frequency dividers (ILFDs). They are modelled as second-order driven oscillators working in the subharmonic (secondary) resonance regime, i.e., when the self-oscillating
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We study the phenomenon of periodic pulling which occurs in certain integrated microcircuits of relevant interest in applications, namely the injection-locked frequency dividers (ILFDs). They are modelled as second-order driven oscillators working in the subharmonic (secondary) resonance regime, i.e., when the self-oscillating frequency is close (resonant) to an integer submultiple n of the driving frequency. Under the assumption of weak injection, we find the spectrum of the system’s oscillatory response in the unlocked mode through closed-form expressions, showing that such spectrum is double-sided and asymmetric, unlike the single-sided spectrum of systems with primary resonance (n=1). An analytical expression for the amplitude modulation of the oscillatory response is also presented. Numerical results are presented to support theoretical relations derived. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Examples of the Application of Nonparametric Information Geometry to Statistical Physics
Entropy 2013, 15(10), 4042-4065; doi:10.3390/e15104042
Received: 15 August 2013 / Revised: 13 September 2013 / Accepted: 16 September 2013 / Published: 25 September 2013
Cited by 3 | PDF Full-text (278 KB) | HTML Full-text | XML Full-text
Abstract
We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is
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We review a nonparametric version of Amari’s information geometry in which the set of positive probability densities on a given sample space is endowed with an atlas of charts to form a differentiable manifold modeled on Orlicz Banach spaces. This nonparametric setting is used to discuss the setting of typical problems in machine learning and statistical physics, such as black-box optimization, Kullback-Leibler divergence, Boltzmann-Gibbs entropy and the Boltzmann equation. Full article
(This article belongs to the collection Advances in Applied Statistical Mechanics)
Open AccessArticle Navigating the Chemical Space of HCN Polymerization and Hydrolysis: Guiding Graph Grammars by Mass Spectrometry Data
Entropy 2013, 15(10), 4066-4083; doi:10.3390/e15104066
Received: 22 February 2013 / Revised: 10 September 2013 / Accepted: 11 September 2013 / Published: 25 September 2013
Cited by 12 | PDF Full-text (869 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
Polymers of hydrogen cyanide and their hydrolysis products constitute a plausible, but still poorly understood proposal for early prebiotic chemistry on Earth. HCN polymers are generated by the interplay of more than a dozen distinctive reaction mechanisms and form a highly complex mixture.
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Polymers of hydrogen cyanide and their hydrolysis products constitute a plausible, but still poorly understood proposal for early prebiotic chemistry on Earth. HCN polymers are generated by the interplay of more than a dozen distinctive reaction mechanisms and form a highly complex mixture. Here we use a computational model based on graph grammars as a means of exploring the chemical spaces of HCN polymerization and hydrolysis. A fundamental issue is to understand the combinatorial explosion inherent in large, complex chemical systems. We demonstrate that experimental data, here obtained by mass spectrometry, and computationally predicted free energies together can be used to guide the exploration of the chemical space and makes it feasible to investigate likely pathways and chemical motifs even in potentially open-ended chemical systems. Full article
(This article belongs to the Special Issue Equilibrium and Non-Equilibrium Entropy in the Origin of Life)
Open AccessArticle Concurrent Phenomena at the Reaction Path of the SN2 Reaction CH3Cl + F. Information Planes and Statistical Complexity Analysis
Entropy 2013, 15(10), 4084-4104; doi:10.3390/e15104084
Received: 7 June 2013 / Revised: 22 August 2013 / Accepted: 27 August 2013 / Published: 26 September 2013
Cited by 2 | PDF Full-text (896 KB) | HTML Full-text | XML Full-text
Abstract
An information-theoretical complexity analysis of the SN2 exchange reaction for CH3Cl + F is performed in both position and momentum spaces by means of the following composite functionals of the one-particle density: D-L and I-J planes and Fisher-Shannon’s
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An information-theoretical complexity analysis of the SN2 exchange reaction for CH3Cl + F is performed in both position and momentum spaces by means of the following composite functionals of the one-particle density: D-L and I-J planes and Fisher-Shannon’s (FS) and López-Ruiz-Mancini-Calbet (LMC) shape complexities. It was found that all the chemical concepts traditionally assigned to elementary reactions such as the breaking/forming regions (B-B/F), the charge transfer/reorganization and the charge repulsion can be unraveled from the phenomenological analysis performed in this study through aspects of localizability, uniformity and disorder associated with the information-theoretical functionals. In contrast, no energy-based functionals can reveal the above mentioned chemical concepts. In addition, it is found that the TS critical point for this reaction does not show any chemical meaning (other than the barrier height) as compared with the concurrent processes revealed by the information-theoretical analysis. Instead, it is apparent from this study that a maximum delocalized state could be identified in the transition region which is associated to the charge transfer process as a new concurrent phenomenon associated with the charge transfer region (CT) for the ion-complex is identified. Finally it is discussed why most of the chemical features of interest (e.g., CT, B-B/F) are only revealed when some information-theoretic properties are taken into account, such as localizability, uniformity and disorder. Full article
Open AccessArticle Time Eigenstates for Potential Functions without Extremal Points
Entropy 2013, 15(10), 4105-4121; doi:10.3390/e15104105
Received: 18 August 2013 / Revised: 21 September 2013 / Accepted: 22 September 2013 / Published: 26 September 2013
Cited by 2 | PDF Full-text (379 KB)
Abstract
In a previous paper, we introduced a way to generate a time coordinate system for classical and quantum systems when the potential function has extremal points. In this paper, we deal with the case in which the potential function has no extremal points
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In a previous paper, we introduced a way to generate a time coordinate system for classical and quantum systems when the potential function has extremal points. In this paper, we deal with the case in which the potential function has no extremal points at all, and we illustrate the method with the harmonic and linear potentials. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Figures

Open AccessArticle Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion
Entropy 2013, 15(10), 4122-4133; doi:10.3390/e15104122
Received: 27 August 2013 / Revised: 22 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
Cited by 10 | PDF Full-text (230 KB) | HTML Full-text | XML Full-text
Abstract
The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R)  and a matrix (R <  r < ∞)  being in perfect thermal contact at r = R  is considered. The heat conduction in each region
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The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R)  and a matrix (R <  r < ∞)  being in perfect thermal contact at r = R  is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional order 0 < a ≤ 2 and 0 < β ≤ 2,  respectively. The Laplace transform with respect to time is used. The approximate solution valid for small values of time is obtained in terms of the Mittag-Leffler, Wright, and Mainardi functions. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Spectral Entropy, Empirical Entropy and Empirical Exergy for Deterministic Boundary-Layer Structures
Entropy 2013, 15(10), 4134-4158; doi:10.3390/e15104134
Received: 15 May 2013 / Revised: 14 August 2013 / Accepted: 18 September 2013 / Published: 27 September 2013
Cited by 4 | PDF Full-text (701 KB) | HTML Full-text | XML Full-text
Abstract
A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow in a methane fired combustion channel flow environment. The objective of this study is to explore the applicability of a set of
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A modified form of the Townsend equations for the fluctuating velocity wave vectors is applied to a laminar three-dimensional boundary-layer flow in a methane fired combustion channel flow environment. The objective of this study is to explore the applicability of a set of low dimensional, coupled, nonlinear differential equations for the prediction of possible deterministic ordered structures within a specific boundary-layer environment. Four increasing channel pressures are considered. The equations are cast into a Lorenz-type system of equations, which yields the low-dimensional set of equations. The solutions indicate the presence of several organized flow structures. Singular value decomposition of the nonlinear time series solutions indicate that nearly ninety-eight percent of the fluctuating directed kinetic energy is contained within the first four empirical modes of the decomposition. The empirical entropy computed from these results indicates that these four lowest modes are largely coherent structures with lower entropy rates. Four regions are observed: low-entropy structures over the first four modes; steep increase in entropy over three modes; steady, high entropy over seven modes; and an increase to maximum entropy over the last two modes. A measure, called the empirical exergy, characterizes the extent of directed kinetic energy produced in the nonlinear solution of the deterministic equations used to model the flow environment. The effect of increasing pressure is to produce more distinct ordered structures within the nonlinear time series solutions. Full article
Open AccessArticle Learning Entropy: Multiscale Measure for Incremental Learning
Entropy 2013, 15(10), 4159-4187; doi:10.3390/e15104159
Received: 26 July 2013 / Revised: 17 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
Cited by 6 | PDF Full-text (729 KB) | HTML Full-text | XML Full-text
Abstract
First, this paper recalls a recently introduced method of adaptive monitoring of dynamical systems and presents the most recent extension with a multiscale-enhanced approach. Then, it is shown that this concept of real-time data monitoring establishes a novel non-Shannon and non-probabilistic concept of
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First, this paper recalls a recently introduced method of adaptive monitoring of dynamical systems and presents the most recent extension with a multiscale-enhanced approach. Then, it is shown that this concept of real-time data monitoring establishes a novel non-Shannon and non-probabilistic concept of novelty quantification, i.e., Entropy of Learning, or in short the Learning Entropy. This novel cognitive measure can be used for evaluation of each newly measured sample of data, or even of whole intervals. The Learning Entropy is quantified in respect to the inconsistency of data to the temporary governing law of system behavior that is incrementally learned by adaptive models such as linear or polynomial adaptive filters or neural networks. The paper presents this novel concept on the example of gradient descent learning technique with normalized learning rate. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle The Fractional Differential Polynomial Neural Network for Approximation of Functions
Entropy 2013, 15(10), 4188-4198; doi:10.3390/e15104188
Received: 26 August 2013 / Revised: 5 September 2013 / Accepted: 24 September 2013 / Published: 30 September 2013
Cited by 7 | PDF Full-text (296 KB) | HTML Full-text | XML Full-text
Abstract
In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization
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In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe modelling of complex systems. Furthermore, the total information is calculated by using the fractional Poisson process. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Analogue Realization of Fractional-Order Dynamical Systems
Entropy 2013, 15(10), 4199-4214; doi:10.3390/e15104199
Received: 27 August 2013 / Revised: 24 September 2013 / Accepted: 25 September 2013 / Published: 7 October 2013
Cited by 12 | PDF Full-text (844 KB) | HTML Full-text | XML Full-text
Abstract
As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of
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As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description, circuit diagrams and design procedure, simulation and measured results are also presented. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Modeling Dynamics of Diffusion Across Heterogeneous Social Networks: News Diffusion in Social Media
Entropy 2013, 15(10), 4215-4242; doi:10.3390/e15104215
Received: 29 August 2013 / Revised: 13 September 2013 / Accepted: 17 September 2013 / Published: 8 October 2013
Cited by 7 | PDF Full-text (525 KB) | HTML Full-text | XML Full-text
Abstract
Diverse online social networks are becoming increasingly interconnected by sharing information. Accordingly, emergent macro-level phenomena have been observed, such as the synchronous spread of information across different types of social media. Attempting to analyze the emergent global behavior is impossible from the examination
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Diverse online social networks are becoming increasingly interconnected by sharing information. Accordingly, emergent macro-level phenomena have been observed, such as the synchronous spread of information across different types of social media. Attempting to analyze the emergent global behavior is impossible from the examination of a single social platform, and dynamic influences between different social networks are not negligible. Furthermore, the underlying structural property of networks is important, as it drives the diffusion process in a stochastic way. In this paper, we propose a macro-level diffusion model with a probabilistic approach by combining both the heterogeneity and structural connectivity of social networks. As real-world phenomena, we explore instances of news diffusion across different social media platforms from a dataset that contains over 386 million web documents covering a one-month period in early 2011. We find that influence between different media types is varied by the context of information. News media are the most influential in the arts and economy categories, while social networking sites (SNS) and blog media are in the politics and culture categories, respectively. Furthermore, controversial topics, such as political protests and multiculturalism failure, tend to spread concurrently across social media, while entertainment topics, such as film releases and celebrities, are more likely driven by interactions within single social platforms. We expect that the proposed model applies to a wider class of diffusion phenomena in diverse fields and that it provides a way of interpreting the dynamics of diffusion in terms of the strength and directionality of influences among populations. Full article
(This article belongs to the Special Issue Social Networks and Information Diffusion)
Open AccessArticle Conditioning, Correlation and Entropy Generation in Maxwell’s Demon
Entropy 2013, 15(10), 4243-4265; doi:10.3390/e15104243
Received: 29 July 2013 / Revised: 22 September 2013 / Accepted: 24 September 2013 / Published: 9 October 2013
PDF Full-text (300 KB) | HTML Full-text | XML Full-text
Abstract
Maxwell’s Demon conspires to use information about the state of a confined molecule in a Szilard engine (randomly frozen into a state subspace by his own actions) to derive work from a single-temperature heat bath. It is widely accepted that, if the Demon
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Maxwell’s Demon conspires to use information about the state of a confined molecule in a Szilard engine (randomly frozen into a state subspace by his own actions) to derive work from a single-temperature heat bath. It is widely accepted that, if the Demon can achieve this at all, he can do so without violating the Second Law only because of a counterbalancing price that must be paid to erase information when the Demon’s memory is reset at the end of his operating cycle. In this paper, Maxwell’s Demon is analyzed within a “referential” approach to physical information that defines and quantifies the Demon’s information via correlations between the joint physical state of the confined molecule and that of the Demon’s memory. On this view, which received early emphasis in Fahn’s 1996 classical analysis of Maxwell’s Demon, information is erased not during the memory reset step of the Demon’s cycle, but rather during the expansion step, when these correlations are destroyed. Dissipation and work extraction are analyzed here for a Demon that operates a generalized quantum mechanical Szilard engine embedded in a globally closed composite, which also includes a work reservoir, a heat bath and the remainder of the Demon’s environment. Memory-engine correlations lost during the expansion step, which enable extraction of work from the Demon via operations conditioned on the memory contents, are shown to be dissipative when this decorrelation is achieved unconditionally so no work can be extracted. Fahn’s essential conclusions are upheld in generalized form, and his quantitative results supported via appropriate specialization to the Demon of his classical analysis, all without external appeal to classical thermodynamics, the Second Law, phase space conservation arguments or Landauer’s Principle. Full article
(This article belongs to the Special Issue Maxwell’s Demon 2013)
Open AccessArticle Multivariate Conditional Granger Causality Analysis for Lagged Response of Soil Respiration in a Temperate Forest
Entropy 2013, 15(10), 4266-4284; doi:10.3390/e15104266
Received: 21 June 2013 / Revised: 10 September 2013 / Accepted: 25 September 2013 / Published: 9 October 2013
Cited by 6 | PDF Full-text (1435 KB) | HTML Full-text | XML Full-text
Abstract
Ecological multivariate systems offer a suitable data set on which to apply recent advances in information theory and causality detection. These systems are driven by the interplay of various environmental factors: meteorological and hydrological forcing, which are often correlated with each other at
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Ecological multivariate systems offer a suitable data set on which to apply recent advances in information theory and causality detection. These systems are driven by the interplay of various environmental factors: meteorological and hydrological forcing, which are often correlated with each other at different time lags; and biological factors, primary producers and decomposers with both autonomous and coupled dynamics. Here, using conditional spectral Granger causality, we quantify directional causalities in a complex atmosphere-plant-soil system involving the carbon cycle. Granger causality is a statistical approach, originating in econometrics, used to identify the presence of linear causal interactions between time series of data, based on prediction theory. We first test to see if there was a significant difference in the causal structure among two treatments where carbon allocation to roots was interrupted by girdling. We then expanded the analysis, introducing radiation and soil moisture. The results showed a complex pattern of multilevel interactions, with some of these interactions depending upon the number of variables in the system. However, no significant differences emerged in the causal structure of above and below ground carbon cycle among the two treatments. Full article
Open AccessArticle Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains
Entropy 2013, 15(10), 4285-4299; doi:10.3390/e15104285
Received: 29 August 2013 / Revised: 30 September 2013 / Accepted: 30 September 2013 / Published: 10 October 2013
Cited by 2 | PDF Full-text (452 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we present a methodological framework for conceptual modeling of assembly supply chain (ASC) networks. Models of such ASC networks are divided into classes on the basis of the numbers of initial suppliers. We provide a brief overview of select literature
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In this paper, we present a methodological framework for conceptual modeling of assembly supply chain (ASC) networks. Models of such ASC networks are divided into classes on the basis of the numbers of initial suppliers. We provide a brief overview of select literature on the topic of structural complexity in assembly systems. Subsequently, the so called Vertex degree index for measuring a structural complexity of ASC networks is applied. This measure, which is based on the Shannon entropy, is well suited for the given purpose. Finally, we outline a generic model of quantitative complexity scale for ASC Networks. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Elastic Properties of CaSiO3 Perovskite from ab initio Molecular Dynamics
Entropy 2013, 15(10), 4300-4309; doi:10.3390/e15104300
Received: 26 June 2013 / Accepted: 6 October 2013 / Published: 10 October 2013
Cited by 1 | PDF Full-text (275 KB) | HTML Full-text | XML Full-text
Abstract
Ab initio molecular dynamics simulations were performed to investigate the elasticity of cubic CaSiO3 perovskite at high pressure and temperature. All three independent elastic constants for cubic CaSiO3 perovskite, C11, C12, and C44, were calculated
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Ab initio molecular dynamics simulations were performed to investigate the elasticity of cubic CaSiO3 perovskite at high pressure and temperature. All three independent elastic constants for cubic CaSiO3 perovskite, C11, C12, and C44, were calculated from the computation of stress generated by small strains. The elastic constants were used to estimate the moduli and seismic wave velocities at the high pressure and high temperature characteristic of the Earth’s interior. The dependence of temperature for sound wave velocities decreased as the pressure increased. There was little difference between the estimated compressional sound wave velocity (VP) in cubic CaSiO3 perovskite and that in the Earth’s mantle, determined by seismological data. By contrast, a significant difference between the estimated shear sound wave velocity (VS) and that in the Earth’s mantle was confirmed. The elastic properties of cubic CaSiO3 perovskite cannot explain the properties of the Earth’s lower mantle, indicating that the cubic CaSiO3 perovskite phase is a minor mineral in the Earth’s lower mantle. Full article
(This article belongs to the Special Issue Molecular Dynamics Simulation) Print Edition available
Open AccessArticle Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation
Entropy 2013, 15(10), 4310-4318; doi:10.3390/e15104310
Received: 14 August 2013 / Revised: 25 September 2013 / Accepted: 1 October 2013 / Published: 14 October 2013
Cited by 3 | PDF Full-text (133 KB) | HTML Full-text | XML Full-text
Abstract
Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We
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Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent -1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle The Phase Space Elementary Cell in Classical and Generalized Statistics
Entropy 2013, 15(10), 4319-4333; doi:10.3390/e15104319
Received: 5 September 2013 / Revised: 25 September 2013 / Accepted: 27 September 2013 / Published: 15 October 2013
Cited by 3 | PDF Full-text (213 KB) | HTML Full-text | XML Full-text
Abstract
In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the
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In the past, the phase-space elementary cell of a non-quantized system was set equal to the third power of the Planck constant; in fact, it is not a necessary assumption. We discuss how the phase space volume, the number of states and the elementary-cell volume of a system of non-interacting N particles, changes when an interaction is switched on and the system becomes or evolves to a system of correlated non-Boltzmann particles and derives the appropriate expressions. Even if we assume that nowadays the volume of the elementary cell is equal to the cube of the Planck constant, h3, at least for quantum systems, we show that there is a correspondence between different values of h in the past, with important and, in principle, measurable cosmological and astrophysical consequences, and systems with an effective smaller (or even larger) phase-space volume described by non-extensive generalized statistics. Full article
(This article belongs to the collection Advances in Applied Statistical Mechanics)
Open AccessArticle Synchronization of a Novel Hyperchaotic Complex-Variable System Based on Finite-Time Stability Theory
Entropy 2013, 15(10), 4334-4344; doi:10.3390/e15104334
Received: 31 August 2013 / Revised: 6 October 2013 / Accepted: 11 October 2013 / Published: 16 October 2013
Cited by 4 | PDF Full-text (278 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we investigate the finite-time synchronization problem of a novel hyperchaotic complex-variable system which generates 2-, 3- and 4-scroll attractors. Based on the finite-time stability theory, two control strategies are proposed to realize synchronization of the novel hyperchaotic complex-variable system in
[...] Read more.
In this paper, we investigate the finite-time synchronization problem of a novel hyperchaotic complex-variable system which generates 2-, 3- and 4-scroll attractors. Based on the finite-time stability theory, two control strategies are proposed to realize synchronization of the novel hyperchaotic complex-variable system in finite time. Finally, two numerical examples have been provided to illustrate the effectiveness of the theoretical analysis. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Symmetry-Based Balance Equation for Local Entropy Density in a Dissipative Multibaker Chain System
Entropy 2013, 15(10), 4345-4375; doi:10.3390/e15104345
Received: 5 August 2013 / Revised: 24 September 2013 / Accepted: 9 October 2013 / Published: 16 October 2013
Cited by 1 | PDF Full-text (6991 KB) | HTML Full-text | XML Full-text
Abstract
In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the time-evolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion
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In this study, the balance equation for local entropy density defined on each partition is obtained by the decomposition of the time-evolution operator for local entropy density, on the level of the master equation, by using symmetric and antisymmetric properties for the inversion of partition, density pairs and a given drift velocity. The resultant equation includes the following terms: convection, diffusion, entropy flow due to a thermostat and entropy production. The averaging of the four terms recover the corresponding terms in a balance equation for the macroscopic entropy density of irreversible thermodynamics for a thermostated system. Moreover, an empirical law of order estimation is introduced to explain the limiting behavior of the averaged quantities in the macroscopic limit for the bulk system. The law makes it possible to separate some minor contributions from the major four terms and, for example, to explain the positive entropy production rate in a nonequilibrium state for volume-preserving systems, even if the state is far from steady state. They are numerically confirmed on an invertible, dissipative multibaker chain system, named a circuit model. These properties are independent of partitioning. Full article
(This article belongs to the Special Issue Advances in Methods and Foundations of Non-Equilibrium Thermodynamics)
Open AccessArticle Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process?
Entropy 2013, 15(10), 4376-4391; doi:10.3390/e15104376
Received: 30 August 2013 / Revised: 23 September 2013 / Accepted: 10 October 2013 / Published: 16 October 2013
PDF Full-text (428 KB) | HTML Full-text | XML Full-text
Abstract
Solute transport through homogeneous media has long been assumed to be scale-independent and can be quantified by the second-order advection-dispersion equation (ADE). This study, however, observed the opposite in the laboratory, where transport of CuSO4 through relatively homogeneous silica-sand columns exhibits sub-diffusion growing
[...] Read more.
Solute transport through homogeneous media has long been assumed to be scale-independent and can be quantified by the second-order advection-dispersion equation (ADE). This study, however, observed the opposite in the laboratory, where transport of CuSO4 through relatively homogeneous silica-sand columns exhibits sub-diffusion growing with the spatial scale. Only at a very small travel distance (approximately 10 cm) and a relatively short temporal scale can the transport be approximated by normal diffusion. This is also the only spatiotemporal scale where the fundamental concept of the “representative element volume” (which defines the scale of homogeneous cells used by the ADE-based hydrologic models) is valid. The failure of the standard ADE motivated us to apply a tempered-stable, fractional advection-dispersion equation (TS-FADE) to capture the transient anomalous dispersion with exponentially truncated power-law late-time tails in CuSO4 breakthrough curves. Results show that the tempering parameter in the TS-FADE model generally decreases with an increase of the column length (probably due to the higher probability of long retention processes), while the time index (which is a non-local parameter) remains stable for the uniformly packed columns. Transport in sand columns filled with relatively homogeneous silica sand, therefore, is scale-dependent, and the resultant transient sub-diffusion can be quantified by the TS-FADE model. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Efficiently Measuring Complexity on the Basis of Real-World Data
Entropy 2013, 15(10), 4392-4415; doi:10.3390/e15104392
Received: 23 August 2013 / Accepted: 9 October 2013 / Published: 16 October 2013
Cited by 9 | PDF Full-text (616 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
Permutation entropy, introduced by Bandt and Pompe, is a conceptually simple and well-interpretable measure of time series complexity. In this paper, we propose efficient methods for computing it and related ordinal-patterns-based characteristics. The methods are based on precomputing values of successive ordinal patterns
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Permutation entropy, introduced by Bandt and Pompe, is a conceptually simple and well-interpretable measure of time series complexity. In this paper, we propose efficient methods for computing it and related ordinal-patterns-based characteristics. The methods are based on precomputing values of successive ordinal patterns of order d, considering the fact that they are “overlapped” in d points, and on precomputing successive values of the permutation entropy related to “overlapping” successive time-windows. The proposed methods allow for measurement of the complexity of very large datasets in real-time. Full article
Open AccessArticle Bayesian Testing of a Point Null Hypothesis Based on the Latent Information Prior
Entropy 2013, 15(10), 4416-4431; doi:10.3390/e15104416
Received: 9 August 2013 / Revised: 16 September 2013 / Accepted: 10 October 2013 / Published: 17 October 2013
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Abstract
Bayesian testing of a point null hypothesis is considered. The null hypothesisis that an observation, x, is distributed according to the normal distribution with a mean ofzero and known variance q2. The alternative hypothesis is that x is distributed accordingto
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Bayesian testing of a point null hypothesis is considered. The null hypothesisis that an observation, x, is distributed according to the normal distribution with a mean ofzero and known variance q2. The alternative hypothesis is that x is distributed accordingto a normal distribution with an unknown nonzero mean, μ, and variance q2. The testingproblem is formulated as a prediction problem. Bayesian testing based on priors constructedby using conditional mutual information is investigated. Full article
Open AccessArticle All Shook Up: Fluctuations, Maxwell’s Demon and the Thermodynamics of Computation
Entropy 2013, 15(10), 4432-4483; doi:10.3390/e15104432
Received: 3 July 2013 / Revised: 17 September 2013 / Accepted: 22 September 2013 / Published: 17 October 2013
Cited by 12 | PDF Full-text (758 KB) | HTML Full-text | XML Full-text
Abstract
The most successful exorcism of Maxwell’s demon is Smoluchowski’s 1912 observation that thermal fluctuations would likely disrupt the operation of any molecular-scale demonic machine. A later tradition sought to exorcise Maxwell’s demon by assessing the entropic cost of the demon’s processing of information.
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The most successful exorcism of Maxwell’s demon is Smoluchowski’s 1912 observation that thermal fluctuations would likely disrupt the operation of any molecular-scale demonic machine. A later tradition sought to exorcise Maxwell’s demon by assessing the entropic cost of the demon’s processing of information. This later tradition fails since these same thermal fluctuations invalidate the molecular-scale manipulations upon which the thermodynamics of computation is based. A new argument concerning conservation of phase space volume shows that all Maxwell’s demons must fail. Full article
(This article belongs to the Special Issue Maxwell’s Demon 2013)
Open AccessArticle Energy, Entropy and Exergy in Communication Networks
Entropy 2013, 15(10), 4484-4503; doi:10.3390/e15104484
Received: 3 June 2013 / Revised: 2 October 2013 / Accepted: 11 October 2013 / Published: 18 October 2013
Cited by 7 | PDF Full-text (1508 KB) | HTML Full-text | XML Full-text
Abstract
The information and communication technology (ICT) sector is continuously growing, mainly due to the fast penetration of ICT into many areas of business and society. Growth is particularly high in the area of technologies and applications for communication networks, which can be used,
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The information and communication technology (ICT) sector is continuously growing, mainly due to the fast penetration of ICT into many areas of business and society. Growth is particularly high in the area of technologies and applications for communication networks, which can be used, among others, to optimize systems and processes. The ubiquitous application of ICT opens new perspectives and emphasizes the importance of understanding the complex interactions between ICT and other sectors. Complex and interacting heterogeneous systems can only properly be addressed by a holistic framework. Thermodynamic theory, and, in particular, the second law of thermodynamics, is a universally applicable tool to analyze flows of energy. Communication systems and their processes can be seen, similar to many other natural processes and systems, as dissipative transformations that level differences in energy density between participating subsystems and their surroundings. This paper shows how to apply thermodynamics to analyze energy flows through communication networks. Application of the second law of thermodynamics in the context of the Carnot heat engine is emphasized. The use of exergy-based lifecycle analysis to assess the sustainability of ICT systems is shown on an example of a radio access network. Full article
(This article belongs to the Special Issue Entropy and the Second Law of Thermodynamics)
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Open AccessArticle Searching for Next Single-Phase High-Entropy Alloy Compositions
Entropy 2013, 15(10), 4504-4519; doi:10.3390/e15104504
Received: 22 September 2013 / Revised: 11 October 2013 / Accepted: 16 October 2013 / Published: 18 October 2013
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Abstract
There has been considerable technological interest in high-entropy alloys (HEAs) since the initial publications on the topic appeared in 2004. However, only several of the alloys investigated are truly single-phase solid solution compositions. These include the FCC alloys CoCrFeNi and CoCrFeMnNi based on
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There has been considerable technological interest in high-entropy alloys (HEAs) since the initial publications on the topic appeared in 2004. However, only several of the alloys investigated are truly single-phase solid solution compositions. These include the FCC alloys CoCrFeNi and CoCrFeMnNi based on 3d transition metals elements and BCC alloys NbMoTaW, NbMoTaVW, and HfNbTaTiZr based on refractory metals. The search for new single-phase HEAs compositions has been hindered by a lack of an effective scientific strategy for alloy design. This report shows that the chemical interactions and atomic diffusivities predicted from ab initio molecular dynamics simulations which are closely related to primary crystallization during solidification can be used to assist in identifying single phase high-entropy solid solution compositions. Further, combining these simulations with phase diagram calculations via the CALPHAD method and inspection of existing phase diagrams is an effective strategy to accelerate the discovery of new single-phase HEAs. This methodology was used to predict new single-phase HEA compositions. These are FCC alloys comprised of CoFeMnNi, CuNiPdPt and CuNiPdPtRh, and HCP alloys of CoOsReRu. Full article
(This article belongs to the Special Issue High Entropy Alloys)
Open AccessArticle 3D Reconstruction of Coronal Loops by the Principal Component Analysis
Entropy 2013, 15(10), 4520-4539; doi:10.3390/e15104520
Received: 24 August 2013 / Revised: 27 September 2013 / Accepted: 10 October 2013 / Published: 22 October 2013
Cited by 3 | PDF Full-text (7780 KB) | HTML Full-text | XML Full-text
Abstract
Knowing the three dimensional structure of plasma filaments in the uppermost part of the solar atmosphere, known as coronal loops, and especially their length, is an important parameter in the wave-based diagnostics of this part of the Sun. The combination of observations of
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Knowing the three dimensional structure of plasma filaments in the uppermost part of the solar atmosphere, known as coronal loops, and especially their length, is an important parameter in the wave-based diagnostics of this part of the Sun. The combination of observations of the Sun from different points of observations in space, thanks to the most recent missions, including the Solar Dynamics Observatory (SDO) and the Solar TErrestrial RElations Observatory (STEREO), allows us to infer information about the geometrical shape of coronal loops in 3D space. Here, we propose a new method to reconstruct the loop shape starting from stereoscopically determined 3D points, which sample the loop length, by principal component analysis. This method is shown to retrieve in an easy way the main parameters that define the loop, e.g., the minor and major axes, the loop plane, the azimuthal and inclination angles, for the special case of a coplanar loop. Full article
(This article belongs to the Special Issue Advanced Signal Processing in Heliospheric Physics)
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Open AccessArticle A Kernel-Based Calculation of Information on a Metric Space
Entropy 2013, 15(10), 4540-4552; doi:10.3390/e15104540
Received: 24 July 2013 / Revised: 14 October 2013 / Accepted: 14 October 2013 / Published: 22 October 2013
Cited by 1 | PDF Full-text (242 KB) | HTML Full-text | XML Full-text
Abstract
Kernel density estimation is a technique for approximating probability distributions. Here, it is applied to the calculation of mutual information on a metric space. This is motivated by the problem in neuroscience of calculating the mutual information between stimuli and spiking responses; the
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Kernel density estimation is a technique for approximating probability distributions. Here, it is applied to the calculation of mutual information on a metric space. This is motivated by the problem in neuroscience of calculating the mutual information between stimuli and spiking responses; the space of these responses is a metric space. It is shown that kernel density estimation on a metric space resembles the k-nearest-neighbor approach. This approach is applied to a toy dataset designed to mimic electrophysiological data. Full article

Review

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Open AccessReview Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions
Entropy 2013, 15(10), 3983-4010; doi:10.3390/e15103983
Received: 29 July 2013 / Revised: 17 September 2013 / Accepted: 17 September 2013 / Published: 25 September 2013
Cited by 27 | PDF Full-text (303 KB) | HTML Full-text | XML Full-text
Abstract
We present the main features of the mathematical theory generated by the κ-deformed exponential function expk(x)=(1+k2x2+kx)1k, with 0 κ < 1, developed
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We present the main features of the mathematical theory generated by the κ-deformed exponential function exp k (x) = ( 1 + k 2 x 2 + kx) 1 k , with 0 κ < 1, developed in the last twelve years, which turns out to be a continuous one parameter deformation of the ordinary mathematics generated by the Euler exponential function. The κ-mathematics has its roots in special relativity and furnishes the theoretical foundations of the κ-statistical mechanics predicting power law tailed statistical distributions, which have been observed experimentally in many physical, natural and artificial systems. After introducing the κ-algebra, we present the associated κ-differential and κ-integral calculus. Then, we obtain the corresponding κ-exponential and κ-logarithm functions and give the κ-version of the main functions of the ordinary mathematics. Full article
(This article belongs to the collection Advances in Applied Statistical Mechanics)

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