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Entropy, Volume 14, Issue 2 (February 2012), Pages 92-389

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Editorial

Jump to: Research, Review

Open AccessEditorial Special Issue: Tsallis Entropy
Entropy 2012, 14(2), 174-176; doi:10.3390/e14020174
Received: 2 February 2012 / Accepted: 2 February 2012 / Published: 3 February 2012
Cited by 4 | PDF Full-text (41 KB) | HTML Full-text | XML Full-text
Abstract
One of the crucial properties of the Boltzmann-Gibbs entropy in the context of classical thermodynamics is extensivity, namely proportionality with the number of elements of the system. The Boltzmann-Gibbs entropy satisfies this prescription if the subsystems are statistically (quasi-) independent, or typically [...] Read more.
One of the crucial properties of the Boltzmann-Gibbs entropy in the context of classical thermodynamics is extensivity, namely proportionality with the number of elements of the system. The Boltzmann-Gibbs entropy satisfies this prescription if the subsystems are statistically (quasi-) independent, or typically if the correlations within the system are essentially local. In such cases the energy of the system is typically extensive and the entropy is additive. In general, however, the situation is not of this type and correlations may be far from negligible at all scales. Tsallis in 1988 introduced an entropic expression characterized by an index q which leads to a non-extensive statistics. Tsallis entropy, Sq, is the basis of the so called non-extensive statistical mechanics, which generalizes the Boltzmann-Gibbs theory. Tsallis statistics have found applications in a wide range of phenomena in diverse disciplines such as physics, chemistry, biology, medicine, economics, geophysics, etc. The focus of this special issue of Entropy was to solicit contributions that apply Tsallis entropy in various scientific fields. [...] Full article
(This article belongs to the Special Issue Tsallis Entropy)

Research

Jump to: Editorial, Review

Open AccessArticle The Rate-Controlled Constrained-Equilibrium Approach to Far-From-Local-Equilibrium Thermodynamics
Entropy 2012, 14(2), 92-130; doi:10.3390/e14020092
Received: 12 October 2011 / Revised: 31 December 2011 / Accepted: 18 January 2012 / Published: 30 January 2012
Cited by 15 | PDF Full-text (596 KB) | HTML Full-text | XML Full-text
Abstract
The Rate-Controlled Constrained-Equilibrium (RCCE) method for the description of the time-dependent behavior of dynamical systems in non-equilibrium states is a general, effective, physically based method for model order reduction that was originally developed in the framework of thermodynamics and chemical kinetics. A [...] Read more.
The Rate-Controlled Constrained-Equilibrium (RCCE) method for the description of the time-dependent behavior of dynamical systems in non-equilibrium states is a general, effective, physically based method for model order reduction that was originally developed in the framework of thermodynamics and chemical kinetics. A generalized mathematical formulation is presented here that allows including nonlinear constraints in non-local equilibrium systems characterized by the existence of a non-increasing Lyapunov functional under the system’s internal dynamics. The generalized formulation of RCCE enables to clarify the essentials of the method and the built-in general feature of thermodynamic consistency in the chemical kinetics context. In this paper, we work out the details of the method in a generalized mathematical-physics framework, but for definiteness we detail its well-known implementation in the traditional chemical kinetics framework. We detail proofs and spell out explicit functional dependences so as to bring out and clarify each underlying assumption of the method. In the standard context of chemical kinetics of ideal gas mixtures, we discuss the relations between the validity of the detailed balance condition off-equilibrium and the thermodynamic consistency of the method. We also discuss two examples of RCCE gas-phase combustion calculations to emphasize the constraint-dependent performance of the RCCE method. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics)
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Open AccessArticle Control Parameters for Boundary-Layer Instabilities in Unsteady Shock Interactions
Entropy 2012, 14(2), 131-160; doi:10.3390/e14020131
Received: 7 November 2011 / Revised: 18 January 2012 / Accepted: 18 January 2012 / Published: 31 January 2012
Cited by 1 | PDF Full-text (469 KB) | HTML Full-text | XML Full-text
Abstract
This article presents the computation of a set of control parameters for the deterministic prediction of laminar boundary-layer instabilities induced by an imposed unsteady shock interaction. The objective of the study is exploratory in nature by computing a supersonic flight environment for [...] Read more.
This article presents the computation of a set of control parameters for the deterministic prediction of laminar boundary-layer instabilities induced by an imposed unsteady shock interaction. The objective of the study is exploratory in nature by computing a supersonic flight environment for flow over a blunt body and the deterministic prediction of the spectral entropy rates for the boundary layer subjected to an unsteady pressure disturbance. The deterministic values for the spectral entropy rate within the instabilities are determined for each control parameter. Computational results imply that the instabilities are of a span-wise vortex form, that the maximum vertical velocity wave vector components are produced in the region nearest the wall and that extended transient coherent structures are produced in the boundary layer at a vertical location slightly below the mid-point of the boundary layer. Full article
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Open AccessArticle An Estimation of the Entropy for a Double Exponential Distribution Based on Multiply Type-II Censored Samples
Entropy 2012, 14(2), 161-173; doi:10.3390/e14020161
Received: 20 November 2011 / Revised: 14 January 2012 / Accepted: 18 January 2012 / Published: 31 January 2012
Cited by 2 | PDF Full-text (248 KB) | HTML Full-text | XML Full-text
Abstract
In many life-testing and reliability studies, the experimenter might not always obtain complete information on failure times for all experimental units. Multiply Type-II censored sampling arises in a life-testing experiment whenever the experimenter does not observe the failure times of some units [...] Read more.
In many life-testing and reliability studies, the experimenter might not always obtain complete information on failure times for all experimental units. Multiply Type-II censored sampling arises in a life-testing experiment whenever the experimenter does not observe the failure times of some units placed on a life-test. In this paper, we obtain estimators for the entropy function of a double exponential distribution under multiply Type-II censored samples using the maximum likelihood estimation and the approximate maximum likelihood estimation procedures. We compare the proposed estimators in the sense of the mean squared errors by using Monte Carlo simulation. Full article
Open AccessArticle Disentangling Complexity from Randomness and Chaos
Entropy 2012, 14(2), 177-212; doi:10.3390/e14020177
Received: 27 December 2011 / Revised: 26 January 2012 / Accepted: 31 January 2012 / Published: 7 February 2012
Cited by 4 | PDF Full-text (1682 KB) | HTML Full-text | XML Full-text
Abstract
This study aims to disentangle complexity from randomness and chaos, and to present a definition of complexity that emphasizes its epistemically distinct qualities. I will review existing attempts at defining complexity and argue that these suffer from two major faults: a tendency [...] Read more.
This study aims to disentangle complexity from randomness and chaos, and to present a definition of complexity that emphasizes its epistemically distinct qualities. I will review existing attempts at defining complexity and argue that these suffer from two major faults: a tendency to neglect the underlying dynamics and to focus exclusively on the phenomenology of complex systems; and linguistic imprecisions in describing these phenomenologies. I will argue that the tendency to discuss phenomenology removed from the underlying dynamics is the main root of the difficulties in distinguishing complex from chaotic or random systems. In my own definition, I will explicitly try to avoid these pitfalls. The theoretical contemplations in this paper will be tested on a sample of five models: the random Kac ring, the chaotic CA30, the regular CA90, the complex CA110 and the complex Bak-Sneppen model. Although these modelling studies are restricted in scope and can only be seen as preliminary, they still constitute on of the first attempts to investigate complex systems comparatively. Full article
(This article belongs to the Special Issue Arrow of Time)
Open AccessArticle Quantitative Comparison of Conformational Ensembles
Entropy 2012, 14(2), 213-232; doi:10.3390/e14020213
Received: 1 January 2012 / Revised: 27 January 2012 / Accepted: 3 February 2012 / Published: 10 February 2012
Cited by 5 | PDF Full-text (878 KB) | HTML Full-text | XML Full-text
Abstract
A number of measures have been used in the structural biology literature to compare the shapes or conformations of biological macromolecules. However, the issue of how to compare two ensembles of conformations has received far less attention. Herein, the problem of how [...] Read more.
A number of measures have been used in the structural biology literature to compare the shapes or conformations of biological macromolecules. However, the issue of how to compare two ensembles of conformations has received far less attention. Herein, the problem of how to quantitatively compare two such ensembles is addressed in several different ways using concepts from probability and information theory. Ultimately, such metrics could be used in the evaluation of structure-prediction algorithms and the analysis of how conformational mobility is inhibited by bound ligands. Full article
(This article belongs to the Special Issue Loop Entropy)
Open AccessArticle Scientific Élan Vital: Entropy Deficit or Inhomogeneity as a Unified Concept of Driving Forces of Life in Hierarchical Biosphere Driven by Photosynthesis
Entropy 2012, 14(2), 233-251; doi:10.3390/e14020233
Received: 12 December 2011 / Revised: 22 January 2012 / Accepted: 7 February 2012 / Published: 10 February 2012
Cited by 5 | PDF Full-text (513 KB) | HTML Full-text | XML Full-text
Abstract
Life is considered something different from non-living things, but no single driving force can account for all the different aspects of life, which consists of different levels of hierarchy, such as metabolism, cell physiology, multi-cellular development and organization, population dynamics, ecosystem, and [...] Read more.
Life is considered something different from non-living things, but no single driving force can account for all the different aspects of life, which consists of different levels of hierarchy, such as metabolism, cell physiology, multi-cellular development and organization, population dynamics, ecosystem, and evolution. Although free energy is evidently the driving force in biochemical reactions, there is no established relationship between metabolic energy and spatiotemporal organization of living organisms, or between metabolic energy and genetic information. Since Schrödinger pointed out the importance of exporting entropy in maintaining life, misunderstandings of entropy notion have been obstacles in constructing a unified view on the driving forces of life. Here I present a simplified conceptual framework for unifying driving forces of life at various different levels of hierarchy. The key concept is “entropy deficit”, or simply, ‘inhomogeneity’, which is defined as the difference of maximal possible entropy and actual entropy. This is equivalent to information content in genetic information and protein structure, and is also defined similarly for non-homogeneous structures in ecosystems and evolution. Entropy deficit or inhomogeneoity is a unified measure of all driving forces of life, which could be considered a scientific equivalent to ‘élan vital’ of Bergson. Full article
(This article belongs to the Special Issue Concepts of Entropy and Their Applications)
Open AccessArticle Classical and Quantum Models in Non-Equilibrium Statistical Mechanics: Moment Methods and Long-Time Approximations
Entropy 2012, 14(2), 291-322; doi:10.3390/e14020291
Received: 21 December 2011 / Revised: 25 January 2012 / Accepted: 9 February 2012 / Published: 15 February 2012
Cited by 1 | PDF Full-text (497 KB) | HTML Full-text | XML Full-text
Abstract
We consider non-equilibrium open statistical systems, subject to potentials and to external “heat baths” (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann’s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials [...] Read more.
We consider non-equilibrium open statistical systems, subject to potentials and to external “heat baths” (hb) at thermal equilibrium at temperature T (either with ab initio dissipation or without it). Boltzmann’s classical equilibrium distributions generate, as Gaussian weight functions in momenta, orthogonal polynomials in momenta (the position-independent Hermite polynomialsHn’s). The moments of non-equilibrium classical distributions, implied by the Hn’s, fulfill a hierarchy: for long times, the lowest moment dominates the evolution towards thermal equilibrium, either with dissipation or without it (but under certain approximation). We revisit that hierarchy, whose solution depends on operator continued fractions. We review our generalization of that moment method to classical closed many-particle interacting systems with neither a hb nor ab initio dissipation: with initial states describing thermal equilibrium at T at large distances but non-equilibrium at finite distances, the moment method yields, approximately, irreversible thermalization of the whole system at T, for long times. Generalizations to non-equilibrium quantum interacting systems meet additional difficulties. Three of them are: (i) equilibrium distributions (represented through Wigner functions) are neither Gaussian in momenta nor known in closed form; (ii) they may depend on dissipation; and (iii) the orthogonal polynomials in momenta generated by them depend also on positions. We generalize the moment method, dealing with (i), (ii) and (iii), to some non-equilibrium one-particle quantum interacting systems. Open problems are discussed briefly. Full article
(This article belongs to the Special Issue Arrow of Time)
Open AccessArticle Filter-Type Variable Selection Based on Information Measures for Regression Tasks
Entropy 2012, 14(2), 323-343; doi:10.3390/e14020323
Received: 8 December 2011 / Revised: 9 February 2012 / Accepted: 12 February 2012 / Published: 17 February 2012
Cited by 4 | PDF Full-text (296 KB) | HTML Full-text | XML Full-text
Abstract
This paper presents a supervised variable selection method applied to regression problems. This method selects the variables applying a hierarchical clustering strategy based on information measures. The proposed technique can be applied to single-output regression datasets, and it is extendable to multi-output [...] Read more.
This paper presents a supervised variable selection method applied to regression problems. This method selects the variables applying a hierarchical clustering strategy based on information measures. The proposed technique can be applied to single-output regression datasets, and it is extendable to multi-output datasets. For single-output datasets, the method is compared against three other variable selection methods for regression on four datasets. In the multi-output case, it is compared against other state-of-the-art method and tested using two regression datasets. Two different figures of merit are used (for the single and multi-output cases) in order to analyze and compare the performance of the proposed method. Full article
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Open AccessArticle Interventionism in Statistical Mechanics
Entropy 2012, 14(2), 344-369; doi:10.3390/e14020344
Received: 13 January 2012 / Revised: 9 February 2012 / Accepted: 9 February 2012 / Published: 17 February 2012
PDF Full-text (135 KB) | HTML Full-text | XML Full-text
Abstract
I defend the idea that the fact that no system is entirely isolated (“Interventionism”) can be used to explain the successful use of the microcanonical distribution in statistical mechanics. The argument turns on claims about what is needed for an adequate explanation [...] Read more.
I defend the idea that the fact that no system is entirely isolated (“Interventionism”) can be used to explain the successful use of the microcanonical distribution in statistical mechanics. The argument turns on claims about what is needed for an adequate explanation of this fact: I argue in particular that various competing explanations do not meet reasonable conditions of adequacy, and that the most striking lacuna in Interventionism—its failure to explain the “arrow of time”—is no real defect. Full article
(This article belongs to the Special Issue Arrow of Time)
Open AccessArticle Optimal Design of ORC Systems with a Low-Temperature Heat Source
Entropy 2012, 14(2), 370-389; doi:10.3390/e14020370
Received: 5 December 2011 / Revised: 30 January 2012 / Accepted: 8 February 2012 / Published: 21 February 2012
Cited by 16 | PDF Full-text (399 KB) | HTML Full-text | XML Full-text
Abstract
A numerical model of subcritical and trans-critical power cycles using a fixed-flowrate low-temperature heat source has been validated and used to calculate the combinations of the maximum cycle pressure (Pev) and the difference between the source temperature and the maximum [...] Read more.
A numerical model of subcritical and trans-critical power cycles using a fixed-flowrate low-temperature heat source has been validated and used to calculate the combinations of the maximum cycle pressure (Pev) and the difference between the source temperature and the maximum working fluid temperature (DT) which maximize the thermal efficiency (ηth) or minimize the non-dimensional exergy losses (β), the total thermal conductance of the heat exchangers (UAt) and the turbine size (SP). Optimum combinations of Pev and DT were calculated for each one of these four objective functions for two working fluids (R134a, R141b), three source temperatures and three values of the non-dimensional power output. The ratio of UAt over the net power output (which is a first approximation of the initial cost per kW) shows that R141b is the better working fluid for the conditions under study. Full article
(This article belongs to the Special Issue Advances in Applied Thermodynamics)

Review

Jump to: Editorial, Research

Open AccessReview Modeling Structures and Motions of Loops in Protein Molecules
Entropy 2012, 14(2), 252-290; doi:10.3390/e14020252
Received: 26 December 2011 / Revised: 10 January 2012 / Accepted: 3 February 2012 / Published: 13 February 2012
Cited by 11 | PDF Full-text (3695 KB) | HTML Full-text | XML Full-text
Abstract
Unlike the secondary structure elements that connect in protein structures, loop fragments in protein chains are often highly mobile even in generally stable proteins. The structural variability of loops is often at the center of a protein’s stability, folding, and even biological [...] Read more.
Unlike the secondary structure elements that connect in protein structures, loop fragments in protein chains are often highly mobile even in generally stable proteins. The structural variability of loops is often at the center of a protein’s stability, folding, and even biological function. Loops are found to mediate important biological processes, such as signaling, protein-ligand binding, and protein-protein interactions. Modeling conformations of a loop under physiological conditions remains an open problem in computational biology. This article reviews computational research in loop modeling, highlighting progress and challenges. Important insight is obtained on potential directions for future research. Full article
(This article belongs to the Special Issue Loop Entropy)
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