Topical Collection "Advances in Applied Statistical Mechanics"

A topical collection in Entropy (ISSN 1099-4300). This collection belongs to the section "Statistical Physics".

Editor

Dr. Antonio M. Scarfone
E-Mail Website
Collection Editor
Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche (ISC-CNR), c/o DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
Interests: nonextensive statistical mechanics; nonlinear Fokker–Planck equations; geometry information; nonlinear Schroedinger equation; quantum groups and quantum algebras; complex systems
Special Issues, Collections and Topics in MDPI journals

Topical Collection Information

Dear Colleagues,

There is a diffuse belief that statistical properties of physical systems are well described by Boltzmann­-Gibbs statistical mechanics. However, a constantly increasing amount of situations are known to violate the predictions of orthodox statistical mechanics. Systems where these emerging features are observed seem do not fulfill the standard ergodic and mixing properties on which the Boltzmann-Gibbs formalism are founded. In general, these systems are governed by nonlinear dynamics which establishes a deep relation among the parts. As a consequence, they reach a dynamical equilibrium in which the equilibrium probability distribution can differ deeply from the exponential shape typical of the Gibbs distribution.

In the last decades, we assisted to an intense research activity that has modified our understanding of statistical physics, extending and renewing its applicability considerably. Important developments, relating equilibrium and nonequilibrium statistical physics, kinetic theory, information theory and others, have produced a new understanding of the properties of complex systems that requires, in many cases, the extension of the theory beyond the Boltzmann-Gibbs formalism.

The aim of this collection, is to collect papers in both the foundations and the applications of Statistical Mechanics going outside its traditional application. In particular, foundations regard classical and quantum aspects of statistical physics including generalized entropies, free-scale distributions, information theory, geometry information, nonextensive statistical mechanics, kinetic theory, long-range interactions and small systems. Applications are different and may include biophysics, seismology, econophysics, social systems, physics of networks, physics of risk, traffic flow, complex systems, fractal systems and others.

Specific topics of interest include (but are not limited to): * Generalized entropies * Boltzmann entropy * Renyi entropy * Non linear kinetic * Fokker-Planck equations * Quantum information * Geometry information * Fractal systems * Complex systems * Networks * Econophysics * Sociophysics * Biophysics

Dr. Antonio Maria Scarfone
Collection Editor

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The article processing charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs).

Keywords

  • non extensive systems
  • generalized entropies
  • boltzmann entropy
  • renyi entropy
  • non linear kinetic
  • fokker-planck equations
  • quantum information
  • geometry information
  • fractal systems
  • complex systems
  • networks
  • econophysics
  • sociophysics
  • biophysics

Published Papers (39 papers)

2022

Jump to: 2021, 2020, 2019, 2017, 2016, 2015, 2014, 2013

Article
Stability and Evolution of Synonyms and Homonyms in Signaling Game
Entropy 2022, 24(2), 194; https://0-doi-org.brum.beds.ac.uk/10.3390/e24020194 (registering DOI) - 27 Jan 2022
Viewed by 108
Abstract
Synonyms and homonyms appear in all natural languages. We analyze their evolution within the framework of the signaling game. Agents in our model use reinforcement learning, where probabilities of selection of a communicated word or of its interpretation depend on weights equal to [...] Read more.
Synonyms and homonyms appear in all natural languages. We analyze their evolution within the framework of the signaling game. Agents in our model use reinforcement learning, where probabilities of selection of a communicated word or of its interpretation depend on weights equal to the number of accumulated successful communications. When the probabilities increase linearly with weights, synonyms appear to be very stable and homonyms decline relatively fast. Such behavior seems to be at odds with linguistic observations. A better agreement is obtained when probabilities increase faster than linearly with weights. Our results may suggest that a certain positive feedback, the so-called Metcalfe’s Law, possibly drives some linguistic processes. Evolution of synonyms and homonyms in our model can be approximately described using a certain nonlinear urn model. Full article

2021

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Article
Gradient Profile Estimation Using Exponential Cubic Spline Smoothing in a Bayesian Framework
Entropy 2021, 23(6), 674; https://0-doi-org.brum.beds.ac.uk/10.3390/e23060674 - 27 May 2021
Viewed by 506
Abstract
Attaining reliable gradient profiles is of utmost relevance for many physical systems. In many situations, the estimation of the gradient is inaccurate due to noise. It is common practice to first estimate the underlying system and then compute the gradient profile by taking [...] Read more.
Attaining reliable gradient profiles is of utmost relevance for many physical systems. In many situations, the estimation of the gradient is inaccurate due to noise. It is common practice to first estimate the underlying system and then compute the gradient profile by taking the subsequent analytic derivative of the estimated system. The underlying system is often estimated by fitting or smoothing the data using other techniques. Taking the subsequent analytic derivative of an estimated function can be ill-posed. This becomes worse as the noise in the system increases. As a result, the uncertainty generated in the gradient estimate increases. In this paper, a theoretical framework for a method to estimate the gradient profile of discrete noisy data is presented. The method was developed within a Bayesian framework. Comprehensive numerical experiments were conducted on synthetic data at different levels of noise. The accuracy of the proposed method was quantified. Our findings suggest that the proposed gradient profile estimation method outperforms the state-of-the-art methods. Full article
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Article
Evolution towards Linguistic Coherence in Naming Game with Migrating Agents
Entropy 2021, 23(3), 299; https://0-doi-org.brum.beds.ac.uk/10.3390/e23030299 - 28 Feb 2021
Viewed by 766
Abstract
As an integral part of our culture and way of life, language is intricately related to the migrations of people. To understand whether and how migration shapes language formation processes, we examine the dynamics of the naming game with migrating agents. (i) When [...] Read more.
As an integral part of our culture and way of life, language is intricately related to the migrations of people. To understand whether and how migration shapes language formation processes, we examine the dynamics of the naming game with migrating agents. (i) When all agents may migrate, the dynamics generates effective surface tension that drives the coarsening. Such behaviour is very robust and appears for a wide range of densities of agents and their migration rates. (ii) However, when only multilingual agents are allowed to migrate, monolingual islands are typically formed. In such a case, when the migration rate is sufficiently large, the majority of agents acquire a common language that spontaneously emerges with no indication of surface-tension-driven coarsening. Relatively slow coarsening that takes place in a dense static population is very fragile, and an arbitrarily small migration rate can most likely divert the system towards the quick formation of monolingual islands. Our work shows that migration influences language formation processes, but additional details such as density or mobility of agents are needed to more precisely specify this influence. Full article
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2020

Jump to: 2022, 2021, 2019, 2017, 2016, 2015, 2014, 2013

Review
Statistical and Proactive Analysis of an Inter-Laboratory Comparison: The Radiocarbon Dating of the Shroud of Turin
Entropy 2020, 22(9), 926; https://0-doi-org.brum.beds.ac.uk/10.3390/e22090926 - 24 Aug 2020
Viewed by 3136
Abstract
We review the sampling and results of the radiocarbon dating of the archaeological cloth known as the Shroud of Turin, in the light of recent statistical analyses of both published and raw data. The statistical analyses highlight an inter-laboratory heterogeneity of the means [...] Read more.
We review the sampling and results of the radiocarbon dating of the archaeological cloth known as the Shroud of Turin, in the light of recent statistical analyses of both published and raw data. The statistical analyses highlight an inter-laboratory heterogeneity of the means and a monotone spatial variation of the ages of subsamples that suggest the presence of contaminants unevenly removed by the cleaning pretreatments. We consider the significance and overall impact of the statistical analyses on assessing the reliability of the dating results and the design of correct sampling. These analyses suggest that the 1988 radiocarbon dating does not match the current accuracy requirements. Should this be the case, it would be interesting to know the accurate age of the Shroud of Turin. Taking into account the whole body of scientific data, we discuss whether it makes sense to date the Shroud again. Full article
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Correction
Correction: Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy 2016, 18, 382
Entropy 2020, 22(8), 892; https://0-doi-org.brum.beds.ac.uk/10.3390/e22080892 - 14 Aug 2020
Viewed by 1272
Abstract
Section 3.3 of “Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy2016, 18, 382” contains errors. Therefore, this section is retracted. However, these [...] Read more.
Section 3.3 of “Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy2016, 18, 382” contains errors. Therefore, this section is retracted. However, these changes do not influence the conclusions and the other results of the paper. Full article

2019

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Article
European Option Based on Least-Squares Method under Non-Extensive Statistical Mechanics
Entropy 2019, 21(10), 933; https://0-doi-org.brum.beds.ac.uk/10.3390/e21100933 - 25 Sep 2019
Viewed by 776
Abstract
This paper is devoted to the study of the pricing of European options under a non-Gaussian model. This model follows a non-extensive statistical mechanics which can better describe the fractal characteristics of price movement in the financial market. Moreover, we present a simple [...] Read more.
This paper is devoted to the study of the pricing of European options under a non-Gaussian model. This model follows a non-extensive statistical mechanics which can better describe the fractal characteristics of price movement in the financial market. Moreover, we present a simple but precise least-square method for approximation and obtain a closed-form solution of the price of European options. The advantages of this technique are illustrated by numerical simulation, which shows that the least-squares method is better compared with Borland’s two methods in 2002 and 2004. Full article
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Article
Asymptotic Behavior of Memristive Circuits
Entropy 2019, 21(8), 789; https://0-doi-org.brum.beds.ac.uk/10.3390/e21080789 - 13 Aug 2019
Cited by 4 | Viewed by 2804
Abstract
The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence [...] Read more.
The interest in memristors has risen due to their possible application both as memory units and as computational devices in combination with CMOS. This is in part due to their nonlinear dynamics, and a strong dependence on the circuit topology. We provide evidence that also purely memristive circuits can be employed for computational purposes. In the present paper we show that a polynomial Lyapunov function in the memory parameters exists for the case of DC controlled memristors. Such a Lyapunov function can be asymptotically approximated with binary variables, and mapped to quadratic combinatorial optimization problems. This also shows a direct parallel between memristive circuits and the Hopfield-Little model. In the case of Erdos-Renyi random circuits, we show numerically that the distribution of the matrix elements of the projectors can be roughly approximated with a Gaussian distribution, and that it scales with the inverse square root of the number of elements. This provides an approximated but direct connection with the physics of disordered system and, in particular, of mean field spin glasses. Using this and the fact that the interaction is controlled by a projector operator on the loop space of the circuit. We estimate the number of stationary points of the approximate Lyapunov function and provide a scaling formula as an upper bound in terms of the circuit topology only. Full article
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Article
Effects of Advective-Diffusive Transport of Multiple Chemoattractants on Motility of Engineered Chemosensory Particles in Fluidic Environments
Entropy 2019, 21(5), 465; https://0-doi-org.brum.beds.ac.uk/10.3390/e21050465 - 04 May 2019
Cited by 1 | Viewed by 1694
Abstract
Motility behavior of an engineered chemosensory particle (ECP) in fluidic environments is driven by its responses to chemical stimuli. One of the challenges to understanding such behaviors lies in tracking changes in chemical signal gradients of chemoattractants and ECP-fluid dynamics as the fluid [...] Read more.
Motility behavior of an engineered chemosensory particle (ECP) in fluidic environments is driven by its responses to chemical stimuli. One of the challenges to understanding such behaviors lies in tracking changes in chemical signal gradients of chemoattractants and ECP-fluid dynamics as the fluid is continuously disturbed by ECP motion. To address this challenge, we introduce a new multiscale numerical model to simulate chemotactic swimming of an ECP in confined fluidic environments by accounting for motility-induced disturbances in spatiotemporal chemoattractant distributions. The model accommodates advective-diffusive transport of unmixed chemoattractants, ECP-fluid hydrodynamics at the ECP-fluid interface, and spatiotemporal disturbances in the chemoattractant concentrations due to particle motion. Demonstrative simulations are presented with an ECP, mimicking Escherichia coli (E. coli) chemotaxis, released into initially quiescent fluids with different source configurations of the chemoattractants N-methyl-L-aspartate and L-serine. Simulations demonstrate that initial distributions and temporal evolution of chemoattractants and their release modes (instantaneous vs. continuous, point source vs. distributed) dictate time histories of chemotactic motility of an ECP. Chemotactic motility is shown to be largely determined by spatiotemporal variation in chemoattractant concentration gradients due to transient disturbances imposed by ECP-fluid hydrodynamics, an observation not captured in previous numerical studies that relied on static chemoattractant concentration fields. Full article
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Article
An Urban Scaling Estimation Method in a Heterogeneity Variance Perspective
Entropy 2019, 21(4), 337; https://0-doi-org.brum.beds.ac.uk/10.3390/e21040337 - 28 Mar 2019
Cited by 1 | Viewed by 1285
Abstract
Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this [...] Read more.
Urban scaling laws describe powerful universalities of the scaling relationships between urban attributes and the city size across different countries and times. There are still challenges in precise statistical estimation of the scaling exponent; the properties of variance require further study. In this paper, a statistical regression method based on the maximum likelihood estimation considering the lower bound constraints and the heterogeneous variance of error structure, termed as CHVR, is proposed for urban scaling estimation. In the CHVR method, the heterogeneous properties of variance are explored and modeled in the form of a power-of-the-mean variance model. The maximum likelihood fitting method is supplemented to satisfy the lower bound constraints in empirical data. The CHVR method has been applied to estimating the scaling exponents of six urban attributes covering three scaling regimes in China and compared with two traditional methods. Method evaluations based on three different criteria validate that compared with both classical methods, the CHVR method is more effective and robust. Moreover, a statistical test and long-term variations of the parameter in the variance function demonstrate that the proposed heterogeneous variance function can not only describe the heterogeneity in empirical data adequately but also provide more meaningful urban information. Therefore, the CHVR method shows great potential to provide a valuable tool for effective urban scaling studies across the world and be applied to scaling law estimation in other complex systems in the future. Full article
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Article
The Impact of Financial and Macroeconomic Shocks on the Entropy of Financial Markets
Entropy 2019, 21(3), 316; https://0-doi-org.brum.beds.ac.uk/10.3390/e21030316 - 23 Mar 2019
Cited by 3 | Viewed by 1252
Abstract
We propose here a method to analyze whether financial and macroeconomic shocks influence the entropy of financial networks. We derive a measure of entropy using the correlation matrix of the stock market components of the DOW Jones Industrial Average (DJIA) index. Using VAR [...] Read more.
We propose here a method to analyze whether financial and macroeconomic shocks influence the entropy of financial networks. We derive a measure of entropy using the correlation matrix of the stock market components of the DOW Jones Industrial Average (DJIA) index. Using VAR models in different specifications, we show that shocks in production or the DJIA index lead to an increase in the entropy of the financial markets. Full article
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2017

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Article
On Entropy Dynamics for Active “Living” Particles
Entropy 2017, 19(10), 525; https://0-doi-org.brum.beds.ac.uk/10.3390/e19100525 - 02 Oct 2017
Cited by 1 | Viewed by 1657
Abstract
This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living—hence, complex—systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the [...] Read more.
This paper presents a modeling approach, followed by entropy calculations of the dynamics of large systems of interacting active particles viewed as living—hence, complex—systems. Active particles are partitioned into functional subsystems, while their state is modeled by a discrete scalar variable, while the state of the overall system is defined by a probability distribution function over the state of the particles. The aim of this paper consists of contributing to a further development of the mathematical kinetic theory of active particles. Full article
Article
Initial Results of Testing Some Statistical Properties of Hard Disks Workload in Personal Computers in Terms of Non-Extensive Entropy and Long-Range Dependencies
Entropy 2017, 19(7), 335; https://0-doi-org.brum.beds.ac.uk/10.3390/e19070335 - 05 Jul 2017
Viewed by 1610
Abstract
The aim of this paper is to present some preliminary results and non-extensive statistical properties of selected operating system counters related to hard drive behaviour. A number of experiments have been carried out in order to generate the workload and analyse the behaviour [...] Read more.
The aim of this paper is to present some preliminary results and non-extensive statistical properties of selected operating system counters related to hard drive behaviour. A number of experiments have been carried out in order to generate the workload and analyse the behaviour of computers during man–machine interaction. All analysed computers were personal ones, worked under Windows operating systems. The research was conducted to demonstrate how the concept of non-extensive statistical mechanics can be helpful in the description of computer systems behaviour, especially in the context of statistical properties with scaling phenomena, long-term dependencies and statistical self-similarity. The studies have been made on the basis of perfmon tool that allows the user to trace operating systems counters during processing. Full article
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Article
On the Simplification of Statistical Mechanics for Space Plasmas
Entropy 2017, 19(6), 285; https://0-doi-org.brum.beds.ac.uk/10.3390/e19060285 - 18 Jun 2017
Cited by 9 | Viewed by 2952
Abstract
Space plasmas are frequently described by kappa distributions. Non-extensive statistical mechanics involves the maximization of the Tsallis entropic form under the constraints of canonical ensemble, considering also a dyadic formalism between the ordinary and escort probability distributions. This paper addresses the statistical origin [...] Read more.
Space plasmas are frequently described by kappa distributions. Non-extensive statistical mechanics involves the maximization of the Tsallis entropic form under the constraints of canonical ensemble, considering also a dyadic formalism between the ordinary and escort probability distributions. This paper addresses the statistical origin of kappa distributions, and shows that they can be connected with non-extensive statistical mechanics without considering the dyadic formalism of ordinary/escort distributions. While this concept does significantly simplify the usage of the theory, it costs the definition of a dyadic entropic formulation, in order to preserve the consistency between statistical mechanics and thermodynamics. Therefore, the simplification of the theory by means of avoiding dyadic formalism is impossible within the framework of non-extensive statistical mechanics. Full article
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Article
Peierls–Bogolyubov’s Inequality for Deformed Exponentials
Entropy 2017, 19(6), 271; https://0-doi-org.brum.beds.ac.uk/10.3390/e19060271 - 12 Jun 2017
Cited by 1 | Viewed by 2029
Abstract
We study the convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and in this way obtain new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls–Bogolyubov’s inequality. We use these results to improve [...] Read more.
We study the convexity or concavity of certain trace functions for the deformed logarithmic and exponential functions, and in this way obtain new trace inequalities for deformed exponentials that may be considered as generalizations of Peierls–Bogolyubov’s inequality. We use these results to improve previously-known lower bounds for the Tsallis relative entropy. Full article
Article
Quadratic Mutual Information Feature Selection
Entropy 2017, 19(4), 157; https://0-doi-org.brum.beds.ac.uk/10.3390/e19040157 - 01 Apr 2017
Cited by 11 | Viewed by 3031
Abstract
We propose a novel feature selection method based on quadratic mutual information which has its roots in Cauchy–Schwarz divergence and Renyi entropy. The method uses the direct estimation of quadratic mutual information from data samples using Gaussian kernel functions, and can detect second [...] Read more.
We propose a novel feature selection method based on quadratic mutual information which has its roots in Cauchy–Schwarz divergence and Renyi entropy. The method uses the direct estimation of quadratic mutual information from data samples using Gaussian kernel functions, and can detect second order non-linear relations. Its main advantages are: (i) unified analysis of discrete and continuous data, excluding any discretization; and (ii) its parameter-free design. The effectiveness of the proposed method is demonstrated through an extensive comparison with mutual information feature selection (MIFS), minimum redundancy maximum relevance (MRMR), and joint mutual information (JMI) on classification and regression problem domains. The experiments show that proposed method performs comparably to the other methods when applied to classification problems, except it is considerably faster. In the case of regression, it compares favourably to the others, but is slower. Full article
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2016

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Article
Intra-Day Trading System Design Based on the Integrated Model of Wavelet De-Noise and Genetic Programming
Entropy 2016, 18(12), 435; https://0-doi-org.brum.beds.ac.uk/10.3390/e18120435 - 06 Dec 2016
Cited by 1 | Viewed by 3640
Abstract
Technical analysis has been proved to be capable of exploiting short-term fluctuations in financial markets. Recent results indicate that the market timing approach beats many traditional buy-and-hold approaches in most of the short-term trading periods. Genetic programming (GP) was used to generate short-term [...] Read more.
Technical analysis has been proved to be capable of exploiting short-term fluctuations in financial markets. Recent results indicate that the market timing approach beats many traditional buy-and-hold approaches in most of the short-term trading periods. Genetic programming (GP) was used to generate short-term trade rules on the stock markets during the last few decades. However, few of the related studies on the analysis of financial time series with genetic programming considered the non-stationary and noisy characteristics of the time series. In this paper, to de-noise the original financial time series and to search profitable trading rules, an integrated method is proposed based on the Wavelet Threshold (WT) method and GP. Since relevant information that affects the movement of the time series is assumed to be fully digested during the market closed periods, to avoid the jumping points of the daily or monthly data, in this paper, intra-day high-frequency time series are used to fully exploit the short-term forecasting advantage of technical analysis. To validate the proposed integrated approach, an empirical study is conducted based on the China Securities Index (CSI) 300 futures in the emerging China Financial Futures Exchange (CFFEX) market. The analysis outcomes show that the wavelet de-noise approach outperforms many comparative models. Full article
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Article
Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus)
Entropy 2016, 18(11), 382; https://0-doi-org.brum.beds.ac.uk/10.3390/e18110382 - 26 Oct 2016
Cited by 21 | Viewed by 3294 | Correction
Abstract
Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, [...] Read more.
Mixture models are in high demand for machine-learning analysis due to their computational tractability, and because they serve as a good approximation for continuous densities. Predominantly, entropy applications have been developed in the context of a mixture of normal densities. In this paper, we consider a novel class of skew-normal mixture models, whose components capture skewness due to their flexibility. We find upper and lower bounds for Shannon and Rényi entropies for this model. Using such a pair of bounds, a confidence interval for the approximate entropy value can be calculated. In addition, an asymptotic expression for Rényi entropy by Stirling’s approximation is given, and upper and lower bounds are reported using multinomial coefficients and some properties and inequalities of L p metric spaces. Simulation studies are then applied to a swordfish (Xiphias gladius Linnaeus) length dataset. Full article
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Article
Entropy for the Quantized Field in the Atom-Field Interaction: Initial Thermal Distribution
Entropy 2016, 18(10), 346; https://0-doi-org.brum.beds.ac.uk/10.3390/e18100346 - 23 Sep 2016
Cited by 1 | Viewed by 2333
Abstract
We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical [...] Read more.
We study the entropy of a quantized field in interaction with a two-level atom (in a pure state) when the field is initially in a mixture of two number states. We then generalise the result for a thermal state; i.e., an (infinite) statistical mixture of number states. We show that for some specific interaction times, the atom passes its purity to the field and therefore the field entropy decreases from its initial value. Full article
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Letter
Combinatorial Intricacies of Labeled Fano Planes
Entropy 2016, 18(9), 312; https://0-doi-org.brum.beds.ac.uk/10.3390/e18090312 - 23 Aug 2016
Viewed by 2647
Abstract
Given a seven-element set X={1,2,3,4,5,6,7}, there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that [...] Read more.
Given a seven-element set X = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , there are 30 ways to define a Fano plane on it. Let us call a line of such a Fano plane—that is to say an unordered triple from X—ordinary or defective, according to whether the sum of two smaller integers from the triple is or is not equal to the remaining one, respectively. A point of the labeled Fano plane is said to be of the order s, 0 s 3 , if there are s defective lines passing through it. With such structural refinement in mind, the 30 Fano planes are shown to fall into eight distinct types. Out of the total of 35 lines, nine ordinary lines are of five different kinds, whereas the remaining 26 defective lines yield as many as ten distinct types. It is shown that no labeled Fano plane can have all points of zero-th order, or feature just one point of order two. A connection with prominent configurations in Steiner triple systems is also pointed out. Full article
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Article
Common Probability Patterns Arise from Simple Invariances
Entropy 2016, 18(5), 192; https://0-doi-org.brum.beds.ac.uk/10.3390/e18050192 - 19 May 2016
Cited by 13 | Viewed by 2351
Abstract
Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of [...] Read more.
Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of shift, stretch and rotation, plus a few additional invariances. Prior work described the three fundamental invariances as a consequence of the equilibrium canonical ensemble of statistical mechanics or the Jaynesian maximization of information entropy. By contrast, I emphasize the primacy and sufficiency of invariance alone to explain the commonly observed patterns. Primary invariance naturally creates the array of commonly observed scaling relations and associated probability patterns, whereas the classical approaches derived from statistical mechanics or information theory require special assumptions to derive commonly observed scales. Full article

2015

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Article
Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems
Entropy 2015, 17(5), 2853-2861; https://0-doi-org.brum.beds.ac.uk/10.3390/e17052853 - 05 May 2015
Cited by 16 | Viewed by 2636
Abstract
It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many [...] Read more.
It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form \(S_{BG}=-k\sum_i p_i \ln p_i\); the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the \(S_q=k \frac{1-\sum_i p_i^q}{q-1}\) one (\(q \in {\cal R}; \,S_1=S_{BG}\)), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate \(S_q\). We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization. Full article
Article
Optimum Accelerated Degradation Tests for the Gamma Degradation Process Case under the Constraint of Total Cost
Entropy 2015, 17(5), 2556-2572; https://0-doi-org.brum.beds.ac.uk/10.3390/e17052556 - 23 Apr 2015
Cited by 16 | Viewed by 2696
Abstract
An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of [...] Read more.
An accelerated degradation test (ADT) is regarded as an effective alternative to an accelerated life test in the sense that an ADT can provide more accurate information on product reliability, even when few or no failures may be expected before the end of a practical test period. In this paper, statistical methods for optimal designing ADT plans are developed assuming that the degradation characteristic follows a gamma process (GP). The GP-based approach has an advantage that it can deal with more frequently encountered situations in which the degradation should always be nonnegative and strictly increasing over time. The optimal ADT plan is developed under the total experimental cost constraint by determining the optimal settings of variables such as the number of measurements, the measurement times, the test stress levels and the number of units allocated to each stress level such that the asymptotic variance of the maximum likelihood estimator of the q-th quantile of the lifetime distribution at the use condition is minimized. In addition, compromise plans are developed to provide means to check the adequacy of the assumed acceleration model. Finally, sensitivity analysis procedures for assessing the effects of the uncertainties in the pre-estimates of unknown parameters are illustrated with an example. Full article
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2014

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Article
Extreme Value Laws for Superstatistics
Entropy 2014, 16(10), 5523-5536; https://0-doi-org.brum.beds.ac.uk/10.3390/e16105523 - 20 Oct 2014
Cited by 5 | Viewed by 2606
Abstract
We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three [...] Read more.
We study the extreme value distribution of stochastic processes modeled by superstatistics. Classical extreme value theory asserts that (under mild asymptotic independence assumptions) only three possible limit distributions are possible, namely: Gumbel, Fréchet and Weibull distribution. On the other hand, superstatistics contains three important universality classes, namely χ2-superstatistics, inverse χ2 -superstatistics, and lognormal superstatistics, all maximizing different effective entropy measures. We investigate how the three classes of extreme value theory are related to the three classes of superstatistics. We show that for any superstatistical process whose local equilibrium distribution does not live on a finite support, the Weibull distribution cannot occur. Under the above mild asymptotic independence assumptions, we also show that χ2-superstatistics generally leads an extreme value statistics described by a Fréchet distribution, whereas inverse χ2 -superstatistics, as well as lognormal superstatistics, lead to an extreme value statistics associated with the Gumbel distribution. Full article
Article
Nonlinearities in Elliptic Curve Authentication
Entropy 2014, 16(9), 5144-5158; https://0-doi-org.brum.beds.ac.uk/10.3390/e16095144 - 25 Sep 2014
Cited by 9 | Viewed by 2543
Abstract
In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area [...] Read more.
In order to construct the border solutions for nonsupersingular elliptic curve equations, some common used models need to be adapted from linear treated cases for use in particular nonlinear cases. There are some approaches that conclude with these solutions. Optimization in this area means finding the majority of points on the elliptic curve and minimizing the time to compute the solution in contrast with the necessary time to compute the inverse solution. We can compute the positive solution of PDE (partial differential equation) like oscillations of f(s)/s around the principal eigenvalue λ1 of -Δ in H 0 1 (Ω).Translating mathematics into cryptographic applications will be relevant in everyday life, where in there are situations in which two parts that communicate need a third part to confirm this process. For example, if two persons want to agree on something they need an impartial person to confirm this agreement, like a notary. This third part does not influence in anyway the communication process. It is just a witness to the agreement. We present a system where the communicating parties do not authenticate one another. Each party authenticates itself to a third part who also sends the keys for the encryption/decryption process. Another advantage of such a system is that if someone (sender) wants to transmit messages to more than one person (receivers), he needs only one authentication, unlike the classic systems where he would need to authenticate himself to each receiver. We propose an authentication method based on zero-knowledge and elliptic curves. Full article
Article
Measures of Causality in Complex Datasets with Application to Financial Data
Entropy 2014, 16(4), 2309-2349; https://0-doi-org.brum.beds.ac.uk/10.3390/e16042309 - 24 Apr 2014
Cited by 23 | Viewed by 3864
Abstract
This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert–Schmidt norm of the cross-covariance operator) and transfer entropy, examining [...] Read more.
This article investigates the causality structure of financial time series. We concentrate on three main approaches to measuring causality: linear Granger causality, kernel generalisations of Granger causality (based on ridge regression and the Hilbert–Schmidt norm of the cross-covariance operator) and transfer entropy, examining each method and comparing their theoretical properties, with special attention given to the ability to capture nonlinear causality. We also present the theoretical benefits of applying non-symmetrical measures rather than symmetrical measures of dependence. We apply the measures to a range of simulated and real data. The simulated data sets were generated with linear and several types of nonlinear dependence, using bivariate, as well as multivariate settings. An application to real-world financial data highlights the practical difficulties, as well as the potential of the methods. We use two real data sets: (1) U.S. inflation and one-month Libor; (2) S&P data and exchange rates for the following currencies: AUDJPY, CADJPY, NZDJPY, AUDCHF, CADCHF, NZDCHF. Overall, we reach the conclusion that no single method can be recognised as the best in all circumstances, and each of the methods has its domain of best applicability. We also highlight areas for improvement and future research. Full article
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Article
From Random Motion of Hamiltonian Systems to Boltzmann’s H Theorem and Second Law of Thermodynamics: a Pathway by Path Probability
Entropy 2014, 16(2), 885-894; https://0-doi-org.brum.beds.ac.uk/10.3390/e16020885 - 13 Feb 2014
Cited by 3 | Viewed by 3341
Abstract
A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields [...] Read more.
A numerical experiment of ideal stochastic motion of a particle subject to conservative forces and Gaussian noise reveals that the path probability depends exponentially on action. This distribution implies a fundamental principle generalizing the least action principle of the Hamiltonian/Lagrangian mechanics and yields an extended formalism of mechanics for random dynamics. Within this theory, Liouville’s theorem of conservation of phase density distribution must be modified to allow time evolution of phase density and consequently the Boltzmann H theorem. We argue that the gap between the regular Newtonian dynamics and the random dynamics was not considered in the criticisms of the H theorem. Full article
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2013

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Review
Statistical Mechanics Ideas and Techniques Applied to Selected Problems in Ecology
Entropy 2013, 15(12), 5237-5276; https://0-doi-org.brum.beds.ac.uk/10.3390/e15125237 - 27 Nov 2013
Cited by 6 | Viewed by 3852
Abstract
Ecosystem dynamics provides an interesting arena for the application of a plethora concepts and techniques from statistical mechanics. Here I review three examples corresponding each one to an important problem in ecology. First, I start with an analytical derivation of clumpy patterns for [...] Read more.
Ecosystem dynamics provides an interesting arena for the application of a plethora concepts and techniques from statistical mechanics. Here I review three examples corresponding each one to an important problem in ecology. First, I start with an analytical derivation of clumpy patterns for species relative abundances (SRA) empirically observed in several ecological communities involving a high number n of species, a phenomenon which have puzzled ecologists for decades. An interesting point is that this derivation uses results obtained from a statistical mechanics model for ferromagnets. Second, going beyond the mean field approximation, I study the spatial version of a popular ecological model involving just one species representing vegetation. The goal is to address the phenomena of catastrophic shifts—gradual cumulative variations in some control parameter that suddenly lead to an abrupt change in the system—illustrating it by means of the process of desertification of arid lands. The focus is on the aggregation processes and the effects of diffusion that combined lead to the formation of non trivial spatial vegetation patterns. It is shown that different quantities—like the variance, the two-point correlation function and the patchiness—may serve as early warnings for the desertification of arid lands. Remarkably, in the onset of a desertification transition the distribution of vegetation patches exhibits scale invariance typical of many physical systems in the vicinity a phase transition. I comment on similarities of and differences between these catastrophic shifts and paradigmatic thermodynamic phase transitions like the liquid-vapor change of state for a fluid. Third, I analyze the case of many species interacting in space. I choose tropical forests, which are mega-diverse ecosystems that exhibit remarkable dynamics. Therefore these ecosystems represent a research paradigm both for studies of complex systems dynamics as well as to unveil the mechanisms responsible for the assembly of species-rich communities. The more classical equilibrium approaches are compared versus non-equilibrium ones and in particular I discuss a recently introduced cellular automaton model in which species compete both locally in physical space and along a niche axis. Full article
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Article
The κ-Generalizations of Stirling Approximation and Multinominal Coefficients
Entropy 2013, 15(12), 5144-5153; https://0-doi-org.brum.beds.ac.uk/10.3390/e15125144 - 26 Nov 2013
Cited by 5 | Viewed by 3191
Abstract
Stirling approximation of the factorials and multinominal coefficients are generalized based on the κ-generalized functions introduced by Kaniadakis. We have related the κ-generalized multinominal coefficients to the κ-entropy by introducing a new κ-product operation, which exists only when κ ≠ 0. Full article
Article
Dynamics of Instantaneous Condensation in the ZRP Conditioned on an Atypical Current
Entropy 2013, 15(11), 5065-5083; https://0-doi-org.brum.beds.ac.uk/10.3390/e15115065 - 19 Nov 2013
Cited by 27 | Viewed by 4632
Abstract
Using a generalized Doob’s h-transform we consider the zero-range process (ZRP) conditioned to carry an atypical current, with focus on the regime where the Gallavotti-Cohen symmetry loses its validity. For a single site we compute explicitly the boundary injection and absorption rates of [...] Read more.
Using a generalized Doob’s h-transform we consider the zero-range process (ZRP) conditioned to carry an atypical current, with focus on the regime where the Gallavotti-Cohen symmetry loses its validity. For a single site we compute explicitly the boundary injection and absorption rates of an effective process which maps to a biased random walk. Our approach provides a direct probabilistic confirmation of the theory of “instantaneous condensation” which was proposed some while ago to explain the dynamical origin of the the failure of the Gallavotti-Cohen symmetry for high currents in the ZRP. However, it turns out that for stochastic dynamics with infinite state space care needs to be taken in the application of the Doob’s transform—we discuss in detail the sense in which the effective dynamics can be interpreted as “typical” for different regimes of the current phase diagram. Full article
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Review
Statistical Mechanics and Information-Theoretic Perspectives on Complexity in the Earth System
Entropy 2013, 15(11), 4844-4888; https://0-doi-org.brum.beds.ac.uk/10.3390/e15114844 - 07 Nov 2013
Cited by 53 | Viewed by 5674
Abstract
This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of [...] Read more.
This review provides a summary of methods originated in (non-equilibrium) statistical mechanics and information theory, which have recently found successful applications to quantitatively studying complexity in various components of the complex system Earth. Specifically, we discuss two classes of methods: (i) entropies of different kinds (e.g., on the one hand classical Shannon and R´enyi entropies, as well as non-extensive Tsallis entropy based on symbolic dynamics techniques and, on the other hand, approximate entropy, sample entropy and fuzzy entropy); and (ii) measures of statistical interdependence and causality (e.g., mutual information and generalizations thereof, transfer entropy, momentary information transfer). We review a number of applications and case studies utilizing the above-mentioned methodological approaches for studying contemporary problems in some exemplary fields of the Earth sciences, highlighting the potentials of different techniques. Full article
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Article
Group Invariance of Information Geometry on q-Gaussian Distributions Induced by Beta-Divergence
Entropy 2013, 15(11), 4732-4747; https://0-doi-org.brum.beds.ac.uk/10.3390/e15114732 - 04 Nov 2013
Cited by 5 | Viewed by 3528
Abstract
We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this [...] Read more.
We demonstrate that the q-exponential family particularly admits natural geometrical structures among deformed exponential families. The property is the invariance of structures with respect to a general linear group, which transitively acts on the space of positive definite matrices. We prove this property via the correspondence between information geometry induced by a deformed potential on the space and the one induced by what we call β-divergence defined on the q-exponential family with q = β + 1. The results are fundamental in robust multivariate analysis using the q-Gaussian family. Full article
Article
Law of Multiplicative Error and Its Generalization to the Correlated Observations Represented by the q-Product
Entropy 2013, 15(11), 4634-4647; https://0-doi-org.brum.beds.ac.uk/10.3390/e15114634 - 28 Oct 2013
Cited by 2 | Viewed by 2686
Abstract
The law of multiplicative error is presented for independent observations and correlated observations represented by the q-product, respectively. We obtain the standard log-normal distribution in the former case and the log-q-normal distribution in the latter case. Queirós’ q-log normal distribution is also reconsidered [...] Read more.
The law of multiplicative error is presented for independent observations and correlated observations represented by the q-product, respectively. We obtain the standard log-normal distribution in the former case and the log-q-normal distribution in the latter case. Queirós’ q-log normal distribution is also reconsidered in the framework of the law of error. These results are presented with mathematical conditions to give rise to these distributions. Full article
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Article
The Phase Space Elementary Cell in Classical and Generalized Statistics
Entropy 2013, 15(10), 4319-4333; https://0-doi-org.brum.beds.ac.uk/10.3390/e15104319 - 15 Oct 2013
Cited by 7 | Viewed by 3002
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 [...] Read more.
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
Article
Examples of the Application of Nonparametric Information Geometry to Statistical Physics
Entropy 2013, 15(10), 4042-4065; https://0-doi-org.brum.beds.ac.uk/10.3390/e15104042 - 25 Sep 2013
Cited by 17 | Viewed by 3952
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 [...] Read more.
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
Review
Theoretical Foundations and Mathematical Formalism of the Power-Law Tailed Statistical Distributions
Entropy 2013, 15(10), 3983-4010; https://0-doi-org.brum.beds.ac.uk/10.3390/e15103983 - 25 Sep 2013
Cited by 61 | Viewed by 5071
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 [...] Read more.
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
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Article
Solutions of Some Nonlinear Diffusion Equations and Generalized Entropy Framework
Entropy 2013, 15(9), 3931-3940; https://0-doi-org.brum.beds.ac.uk/10.3390/e15093931 - 18 Sep 2013
Cited by 3 | Viewed by 3619
Abstract
We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the q-exponential [...] Read more.
We investigate solutions of a generalized diffusion equation that contains nonlinear terms in the presence of external forces and reaction terms. The solutions found here can have a compact or long tail behavior and can be expressed in terms of the q-exponential functions present in the Tsallis framework. In the case of the long-tailed behavior, in the asymptotic limit, these solutions can also be connected with the L´evy distributions. In addition, from the results presented here, a rich class of diffusive processes, including normal and anomalous ones, can be obtained. Full article
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Article
Deformed Exponentials and Applications to Finance
Entropy 2013, 15(9), 3471-3489; https://0-doi-org.brum.beds.ac.uk/10.3390/e15093471 - 02 Sep 2013
Cited by 33 | Viewed by 3632
Abstract
We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are [...] Read more.
We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are proposed, which generalized the well-known Black and Scholes model. Full article
Article
Kinetic Theory Microstructure Modeling in Concentrated Suspensions
Entropy 2013, 15(7), 2805-2832; https://0-doi-org.brum.beds.ac.uk/10.3390/e15072805 - 19 Jul 2013
Cited by 18 | Viewed by 4064
Abstract
When suspensions involving rigid rods become too concentrated, standard dilute theories fail to describe their behavior. Rich microstructures involving complex clusters are observed, and no model allows describing its kinematics and rheological effects. In previous works the authors propose a first attempt to [...] Read more.
When suspensions involving rigid rods become too concentrated, standard dilute theories fail to describe their behavior. Rich microstructures involving complex clusters are observed, and no model allows describing its kinematics and rheological effects. In previous works the authors propose a first attempt to describe such clusters from a micromechanical model, but neither its validity nor the rheological effects were addressed. Later, authors applied this model for fitting the rheological measurements in concentrated suspensions of carbon nanotubes (CNTs) by assuming a rheo-thinning behavior at the constitutive law level. However, three major issues were never addressed until now: (i) the validation of the micromechanical model by direct numerical simulation; (ii) the establishment of a general enough multi-scale kinetic theory description, taking into account interaction, diffusion and elastic effects; and (iii) proposing a numerical technique able to solve the kinetic theory description. This paper focuses on these three major issues, proving the validity of the micromechanical model, establishing a multi-scale kinetic theory description and, then, solving it by using an advanced and efficient separated representation of the cluster distribution function. These three aspects, never until now addressed in the past, constitute the main originality and the major contribution of the present paper. Full article
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Article
Statistical Properties of the Foreign Exchange Network at Different Time Scales: Evidence from Detrended Cross-Correlation Coefficient and Minimum Spanning Tree
Entropy 2013, 15(5), 1643-1662; https://0-doi-org.brum.beds.ac.uk/10.3390/e15051643 - 06 May 2013
Cited by 65 | Viewed by 6128
Abstract
We investigate the statistical properties of the foreign exchange (FX) network at different time scales by two approaches, namely the methods of detrended cross-correlation coefficient (DCCA coefficient) and minimum spanning tree (MST). The daily FX rates of 44 major currencies in the period [...] Read more.
We investigate the statistical properties of the foreign exchange (FX) network at different time scales by two approaches, namely the methods of detrended cross-correlation coefficient (DCCA coefficient) and minimum spanning tree (MST). The daily FX rates of 44 major currencies in the period of 2007–2012 are chosen as the empirical data. Based on the analysis of statistical properties of cross-correlation coefficients, we find that the cross-correlation coefficients of the FX market are fat-tailed. By examining three MSTs at three special time scales (i.e., the minimum, medium, and maximum scales), we come to some conclusions: USD and EUR are confirmed as the predominant world currencies; the Middle East cluster is very stable while the Asian cluster and the Latin America cluster are not stable in the MSTs; the Commonwealth cluster is also found in the MSTs. By studying four evaluation criteria, we find that the MSTs of the FX market present diverse topological and statistical properties at different time scales. The scale-free behavior is observed in the FX network at most of time scales. We also find that most of links in the FX network survive from one time scale to the next. Full article
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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Tentative Title: Entropy dynamics in the statistical dynamics of large systems of active particles
Authors: A. Elaiw, M. A. Alghamdi, and N. Bellomo

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