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Entropy, Volume 13, Issue 10 (October 2011), Pages 1746-1903

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Research

Jump to: Review

Open AccessArticle Projective Power Entropy and Maximum Tsallis Entropy Distributions
Entropy 2011, 13(10), 1746-1764; doi:10.3390/e13101746
Received: 26 July 2011 / Revised: 20 September 2011 / Accepted: 20 September 2011 / Published: 26 September 2011
Cited by 10 | PDF Full-text (2244 KB) | HTML Full-text | XML Full-text
Abstract
We discuss a one-parameter family of generalized cross entropy between two distributions with the power index, called the projective power entropy. The cross entropy is essentially reduced to the Tsallis entropy if two distributions are taken to be equal. Statistical and probabilistic [...] Read more.
We discuss a one-parameter family of generalized cross entropy between two distributions with the power index, called the projective power entropy. The cross entropy is essentially reduced to the Tsallis entropy if two distributions are taken to be equal. Statistical and probabilistic properties associated with the projective power entropy are extensively investigated including a characterization problem of which conditions uniquely determine the projective power entropy up to the power index. A close relation of the entropy with the Lebesgue space Lp and the dual Lq is explored, in which the escort distribution associates with an interesting property. When we consider maximum Tsallis entropy distributions under the constraints of the mean vector and variance matrix, the model becomes a multivariate q-Gaussian model with elliptical contours, including a Gaussian and t-distribution model. We discuss the statistical estimation by minimization of the empirical loss associated with the projective power entropy. It is shown that the minimum loss estimator for the mean vector and variance matrix under the maximum entropy model are the sample mean vector and the sample variance matrix. The escort distribution of the maximum entropy distribution plays the key role for the derivation. Full article
(This article belongs to the Special Issue Tsallis Entropy)
Open AccessArticle Tsallis Entropy for Geometry Simplification
Entropy 2011, 13(10), 1805-1828; doi:10.3390/e13101805
Received: 1 August 2011 / Revised: 20 September 2011 / Accepted: 27 September 2011 / Published: 29 September 2011
Cited by 1 | PDF Full-text (10102 KB)
Abstract
This paper presents a study and a comparison of the use of different information-theoretic measures for polygonal mesh simplification. Generalized measures from Information Theory such as Havrda–Charvát–Tsallis entropy and mutual information have been applied. These measures have been used in the error [...] Read more.
This paper presents a study and a comparison of the use of different information-theoretic measures for polygonal mesh simplification. Generalized measures from Information Theory such as Havrda–Charvát–Tsallis entropy and mutual information have been applied. These measures have been used in the error metric of a surfaces implification algorithm. We demonstrate that these measures are useful for simplifying three-dimensional polygonal meshes. We have also compared these metrics with the error metrics used in a geometry-based method and in an image-driven method. Quantitative results are presented in the comparison using the root-mean-square error (RMSE). Full article
(This article belongs to the Special Issue Tsallis Entropy)
Open AccessArticle Entropy Generation Analysis of Desalination Technologies
Entropy 2011, 13(10), 1829-1864; doi:10.3390/e13101829
Received: 5 September 2011 / Revised: 27 September 2011 / Accepted: 27 September 2011 / Published: 30 September 2011
Cited by 46 | PDF Full-text (297 KB)
Abstract
Increasing global demand for fresh water is driving the development and implementation of a wide variety of seawater desalination technologies. Entropy generation analysis, and specifically, Second Law efficiency, is an important tool for illustrating the influence of irreversibilities within a system on [...] Read more.
Increasing global demand for fresh water is driving the development and implementation of a wide variety of seawater desalination technologies. Entropy generation analysis, and specifically, Second Law efficiency, is an important tool for illustrating the influence of irreversibilities within a system on the required energy input. When defining Second Law efficiency, the useful exergy output of the system must be properly defined. For desalination systems, this is the minimum least work of separation required to extract a unit of water from a feed stream of a given salinity. In order to evaluate the Second Law efficiency, entropy generation mechanisms present in a wide range of desalination processes are analyzed. In particular, entropy generated in the run down to equilibrium of discharge streams must be considered. Physical models are applied to estimate the magnitude of entropy generation by component and individual processes. These formulations are applied to calculate the total entropy generation in several desalination systems including multiple effect distillation, multistage flash, membrane distillation, mechanical vapor compression, reverse osmosis, and humidification-dehumidification. Within each technology, the relative importance of each source of entropy generation is discussed in order to determine which should be the target of entropy generation minimization. As given here, the correct application of Second Law efficiency shows which systems operate closest to the reversible limit and helps to indicate which systems have the greatest potential for improvement. Full article
(This article belongs to the Special Issue Entropy Generation Minimization)
Figures

Open AccessArticle Quantifying Dynamical Complexity of Magnetic Storms and Solar Flares via Nonextensive Tsallis Entropy
Entropy 2011, 13(10), 1865-1881; doi:10.3390/e13101865
Received: 1 September 2011 / Revised: 28 September 2011 / Accepted: 30 September 2011 / Published: 14 October 2011
Cited by 11 | PDF Full-text (443 KB)
Abstract
Over the last couple of decades nonextensive Tsallis entropy has shown remarkable applicability to describe nonequilibrium physical systems with large variability and multifractal structure. Herein, we review recent results from the application of Tsallis statistical mechanics to the detection of dynamical changes [...] Read more.
Over the last couple of decades nonextensive Tsallis entropy has shown remarkable applicability to describe nonequilibrium physical systems with large variability and multifractal structure. Herein, we review recent results from the application of Tsallis statistical mechanics to the detection of dynamical changes related with the occurrence of magnetic storms. We extend our review to describe attempts to approach the dynamics of magnetic storms and solar flares by means of universality through Tsallis statistics. We also include a discussion of possible implications on space weather forecasting efforts arising from the verification of Tsallis entropy in the complex system of the magnetosphere. Full article
(This article belongs to the Special Issue Tsallis Entropy)
Open AccessArticle On the Growth Rate of Non-Enzymatic Molecular Replicators
Entropy 2011, 13(10), 1882-1903; doi:10.3390/e13101882
Received: 27 September 2011 / Accepted: 13 October 2011 / Published: 21 October 2011
Cited by 3 | PDF Full-text (581 KB) | HTML Full-text | XML Full-text
Abstract
It is well known that non-enzymatic template directed molecular replicators X + nO -> 2X exhibit parabolic growth d[X]/dt -> k[X]1/2. Here, we analyze the dependence of the effective replication rate constant k [...] Read more.
It is well known that non-enzymatic template directed molecular replicators X + nO -> 2X exhibit parabolic growth d[X]/dt -> k[X]1/2. Here, we analyze the dependence of the effective replication rate constant k on hybridization energies, temperature, strand length, and sequence composition. First we derive analytical criteria for the replication rate k based on simple thermodynamic arguments. Second we present a Brownian dynamics model for oligonucleotides that allows us to simulate their diffusion and hybridization behavior. The simulation is used to generate and analyze the effect of strand length, temperature, and to some extent sequence composition, on the hybridization rates and the resulting optimal overall rate constant k. Combining the two approaches allows us to semi-analytically depict a replication rate landscape for template directed replicators. The results indicate a clear replication advantage for longer strands at lower temperatures in the regime where the ligation rate is rate limiting. Further the results indicate the existence of an optimal replication rate at the boundary between the two regimes where the ligation rate and the dehybridization rates are rate limiting. Full article
(This article belongs to the Special Issue Emergence in Chemical Systems)

Review

Jump to: Research

Open AccessReview The Nonadditive Entropy Sq and Its Applications in Physics and Elsewhere: Some Remarks
Entropy 2011, 13(10), 1765-1804; doi:10.3390/e13101765
Received: 15 August 2011 / Revised: 11 September 2011 / Accepted: 19 September 2011 / Published: 28 September 2011
Cited by 42 | PDF Full-text (1415 KB)
Abstract
The nonadditive entropy Sq has been introduced in 1988 focusing on a generalization of Boltzmann–Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which [...] Read more.
The nonadditive entropy Sq has been introduced in 1988 focusing on a generalization of Boltzmann–Gibbs (BG) statistical mechanics. The aim was to cover a (possibly wide) class of systems among those very many which violate hypothesis such as ergodicity, under which the BG theory is expected to be valid. It is now known that Sq has a large applicability; more specifically speaking, even outside Hamiltonian systems and their thermodynamical approach. In the present paper we review and comment some relevant aspects of this entropy, namely (i) Additivity versus extensivity; (ii) Probability distributions that constitute attractors in the sense of Central Limit Theorems; (iii) The analysis of paradigmatic low-dimensional nonlinear dynamical systems near the edge of chaos; and (iv) The analysis of paradigmatic long-range-interacting many-body classical Hamiltonian systems. Finally, we exhibit recent as well as typical predictions, verifications and applications of these concepts in natural, artificial, and social systems, as shown through theoretical, experimental, observational and computational results. Full article
(This article belongs to the Special Issue Tsallis Entropy)

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