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Entropy, Volume 10, Issue 3 (September 2008), Pages 131-390

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Research

Open AccessArticle Generalised Exponential Families and Associated Entropy Functions
Entropy 2008, 10(3), 131-149; doi:10.3390/entropy-e10030131
Received: 26 February 2008 / Revised: 1 July 2008 / Accepted: 14 July 2008 / Published: 16 July 2008
Cited by 39 | PDF Full-text (213 KB)
Abstract
A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential families. The well-known result that the inequality of Cram´er and Rao
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A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential families. The well-known result that the inequality of Cram´er and Rao becomes an equality in the case of an exponential family can be generalised. However, this requires the introduction of escort probabilities. Full article
Open AccessArticle Information Equation of State
Entropy 2008, 10(3), 150-159; doi:10.3390/entropy-e10030150
Published: 30 July 2008
Cited by 6 | PDF Full-text (206 KB) | HTML Full-text | XML Full-text
Abstract
Landauer’s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close
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Landauer’s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close to the dark energy value, w = -1, for a wide range of redshifts, 10 > z > 0.8, over one half of cosmic time. A reasonable universe information bit content of only 1087 bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the ‘Why now?’ question we wonder ‘What next?’ as we expect the information equation of state to tend towards w = 0 in the future.c Full article
Open AccessArticle Modeling Non-Equilibrium Dynamics of a Discrete Probability Distribution: General Rate Equation for Maximal Entropy Generation in a Maximum-Entropy Landscape with Time-Dependent Constraints
Entropy 2008, 10(3), 160-182; doi:10.3390/entropy-e10030010
Received: 17 December 2007 / Revised: 30 July 2008 / Accepted: 30 July 2008 / Published: 14 August 2008
Cited by 20 | PDF Full-text (235 KB)
Abstract
A rate equation for a discrete probability distribution is discussed as a route to describe smooth relaxation towards the maximum entropy distribution compatible at all times with one or more linear constraints. The resulting dynamics follows the path of steepest entropy ascent compatible
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A rate equation for a discrete probability distribution is discussed as a route to describe smooth relaxation towards the maximum entropy distribution compatible at all times with one or more linear constraints. The resulting dynamics follows the path of steepest entropy ascent compatible with the constraints. The rate equation is consistent with the Onsager theorem of reciprocity and the fluctuation-dissipation theorem. The mathematical formalism was originally developed to obtain a quantum theoretical unification of mechanics and thermodinamics. It is presented here in a general, non-quantal formulation as a part of an effort to develop tools for the phenomenological treatment of non-equilibrium problems with applications in engineering, biology, sociology, and economics. The rate equation is also extended to include the case of assigned time-dependences of the constraints and the entropy, such as for modeling non-equilibrium energy and entropy exchanges. Full article
Open AccessArticle Entropic Behavior of Binary Carbonaceous Mesophases
Entropy 2008, 10(3), 183-199; doi:10.3390/entropy-e10030183
Received: 28 July 2008 / Revised: 19 August 2008 / Accepted: 19 August 2008 / Published: 23 August 2008
Cited by 9 | PDF Full-text (392 KB) | HTML Full-text | XML Full-text
Abstract
The Maier-Saupe model for binary mixtures of uniaxial discotic nematogens, formulated in a previous study [1], is used to compute and characterize orientational entropy [2] and orientational specific heat. These thermodynamic quantities are used to determine mixture type (ideal or non-ideal) which arise
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The Maier-Saupe model for binary mixtures of uniaxial discotic nematogens, formulated in a previous study [1], is used to compute and characterize orientational entropy [2] and orientational specific heat. These thermodynamic quantities are used to determine mixture type (ideal or non-ideal) which arise due to their different intrinsic properties, determined by the molecular weight asymmetry ΔMw and the molecular interaction parameter β. These molecular properties are also used to characterize the critical concentration where the mixture behaves like a single component system and exhibits the minimum nematic to isotropic (NI) transition temperature (pseudo-pure mixture). A transition within the nematic phase takes place at this specific concentration. According to the Maier-Saupe model, in a single mesogen, entropy at NI transition is a universal value; in this work we quantify the mixing effect on this universal property. The results and analysis provide a new tool to characterize molecular interaction and molecular weight differences in mesogenic mixtures using standard calorimetric measurements. Full article
(This article belongs to the Special Issue Configurational Entropy)
Open AccessArticle Calculation of Differential Entropy for a Mixed Gaussian Distribution
Entropy 2008, 10(3), 200-206; doi:10.3390/entropy-e10030200
Received: 16 June 2008 / Revised: 14 August 2008 / Accepted: 18 August 2008 / Published: 25 August 2008
Cited by 12 | PDF Full-text (1368 KB)
Abstract In this work, an analytical expression is developed for the differential entropy of a mixed Gaussian distribution. One of the terms is given by a tabulated function of the ratio of the distribution parameters. Full article
Open AccessArticle Entropy of Mixing and the Glass Transition of Amorphous Mixtures
Entropy 2008, 10(3), 207-223; doi:10.3390/entropy-e10030207
Received: 24 June 2008 / Revised: 20 August 2008 / Accepted: 20 August 2008 / Published: 26 August 2008
Cited by 41 | PDF Full-text (230 KB) | HTML Full-text | XML Full-text
Abstract
Different equations have been proposed for estimating the glass transition temperature of amorphous mixtures. All such expressions lack a term to account for the effect of the entropy of mixing on the glass transition. An entropy based analysis for the glass transition of
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Different equations have been proposed for estimating the glass transition temperature of amorphous mixtures. All such expressions lack a term to account for the effect of the entropy of mixing on the glass transition. An entropy based analysis for the glass transition of amorphous mixtures is presented. The treatment yields an explicit mixing term in the expression for the glass transition temperature of a mixture. The obtained expression reduces to the Couchman-Karasz equation in the limiting case where the contribution of the entropy of mixing approaches zero. Equivalent expressions are obtained for the glass transition temperature of a mixture of two glass formers as for the effect of a plasticizing liquid diluent on the glass transition temperature of an amorphous material. Full article
(This article belongs to the Special Issue Configurational Entropy)
Open AccessArticle Qualitative Picture of Scaling in the Entropy Formalism
Entropy 2008, 10(3), 224-239; doi:10.3390/entropy-e10030224
Received: 30 June 2008 / Revised: 27 August 2008 / Accepted: 2 September 2008 / Published: 5 September 2008
Cited by 4 | PDF Full-text (203 KB)
Abstract
The properties of an infinite system at a continuous phase transition are characterised by non-trivial critical exponents. These non-trivial exponents are related to scaling relations of the thermodynamic potential. The scaling properties of the singular part of the specific entropy of infinite systems
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The properties of an infinite system at a continuous phase transition are characterised by non-trivial critical exponents. These non-trivial exponents are related to scaling relations of the thermodynamic potential. The scaling properties of the singular part of the specific entropy of infinite systems are deduced starting from the well-established scaling relations of the Gibbs free energy. Moreover, it turns out that the corrections to scaling are suppressed in the microcanonical ensemble compared to the corresponding corrections in the canonical ensemble. Full article
Open AccessArticle Deformed Generalization of the Semiclassical Entropy
Entropy 2008, 10(3), 240-247; doi:10.3390/e10030240
Received: 5 March 2008 / Revised: 25 August 2008 / Accepted: 25 August 2008 / Published: 19 September 2008
Cited by 1 | PDF Full-text (166 KB) | Correction | Supplementary Files
Abstract
We explicitly obtain here a novel expression for the semiclassical Wehrl’s entropy using deformed algebras built up with the q¡coherent states (see Arik and Coon [J.Math.Phys. 17, 524 (1976) and Quesne [J. Phys. A 35, 9213 (2002)]). The generalization is investigated with emphasis
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We explicitly obtain here a novel expression for the semiclassical Wehrl’s entropy using deformed algebras built up with the q¡coherent states (see Arik and Coon [J.Math.Phys. 17, 524 (1976) and Quesne [J. Phys. A 35, 9213 (2002)]). The generalization is investigated with emphasis on i) its behavior as a function of temperature and ii) the results obtained when the deformation-parameter tends to unity. Full article
Open AccessArticle Entropy and Ordering of Hard Rods in One Dimension
Entropy 2008, 10(3), 248-260; doi:10.3390/e10030248
Received: 31 July 2008 / Revised: 30 August 2008 / Accepted: 11 September 2008 / Published: 19 September 2008
Cited by 15 | PDF Full-text (344 KB)
Abstract
We revisit the equilibrium properties of a classical one-dimensional system of hardcore particles in the framework provided by the multiparticle correlation expansion of the configurational entropy. The vanishing of the cumulative contribution of more-than-two-particle correlations to the excess entropy is put in relation
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We revisit the equilibrium properties of a classical one-dimensional system of hardcore particles in the framework provided by the multiparticle correlation expansion of the configurational entropy. The vanishing of the cumulative contribution of more-than-two-particle correlations to the excess entropy is put in relation with the onset of a solidlike behavior at high densities. Full article
Open AccessArticle Axiomatic Characterizations of Information Measures
Entropy 2008, 10(3), 261-273; doi:10.3390/e10030261
Received: 1 September 2008 / Accepted: 12 September 2008 / Published: 19 September 2008
Cited by 68 | PDF Full-text (186 KB)
Abstract
Axiomatic characterizations of Shannon entropy, Kullback I-divergence, and some generalized information measures are surveyed. Three directions are treated: (A) Characterization of functions of probability distributions suitable as information measures. (B) Characterization of set functions on the subsets of {1; : : : ;N}
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Axiomatic characterizations of Shannon entropy, Kullback I-divergence, and some generalized information measures are surveyed. Three directions are treated: (A) Characterization of functions of probability distributions suitable as information measures. (B) Characterization of set functions on the subsets of {1; : : : ;N} representable by joint entropies of components of an N-dimensional random vector. (C) Axiomatic characterization of MaxEnt and related inference rules. The paper concludes with a brief discussion of the relevance of the axiomatic approach for information theory. Full article
Open AccessArticle Residual Entropy, the Third Law and Latent Heat
Entropy 2008, 10(3), 274-284; doi:10.3390/e10030274
Received: 24 July 2008 / Revised: 10 September 2008 / Accepted: 15 September 2008 / Published: 20 September 2008
Cited by 5 | PDF Full-text (270 KB) | HTML Full-text | XML Full-text
Abstract
A novel thermodynamic treatment of residual entropy in crystals, involving the configurational partition function, is suggested, which is consistent with both classical and statistical thermodynamics. It relates residual entropy to the inherent latent heat which would be released upon cooling if the reversible
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A novel thermodynamic treatment of residual entropy in crystals, involving the configurational partition function, is suggested, which is consistent with both classical and statistical thermodynamics. It relates residual entropy to the inherent latent heat which would be released upon cooling if the reversible path were available. The nature of this heat is that if the crystal possessing residual entropy freezes above its Boltzmann’s characteristic temperature of molecular alignment, the difference in energy between different molecular arrangements is overcome by the kT heat bath to form a nearly-ideal solution. However, upon cooling below this characteristic temperature, they would separate with a concomitant release of the corresponding energy, provided the reversible path were available. Full article
(This article belongs to the Special Issue Configurational Entropy)
Open AccessArticle Conformational Entropy of an Ideal Cross-Linking Polymer Chain
Entropy 2008, 10(3), 285-308; doi:10.3390/e10030285
Received: 8 August 2008 / Revised: 10 September 2008 / Accepted: 19 September 2008 / Published: 20 September 2008
Cited by 8 | PDF Full-text (556 KB)
Abstract
We present a novel analytical method to calculate conformational entropy of ideal cross-linking polymers from the configuration integral by employing a Mayer series expansion. Mayer-functions describing chemical bonds within the chain and for cross-links are sharply peaked over the temperature range of interest,
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We present a novel analytical method to calculate conformational entropy of ideal cross-linking polymers from the configuration integral by employing a Mayer series expansion. Mayer-functions describing chemical bonds within the chain and for cross-links are sharply peaked over the temperature range of interest, and, are well approximated as statistically weighted Dirac delta-functions that enforce distance constraints. All geometrical deformations consistent with a set of distance constraints are integrated over. Exact results for a contiguous series of connected loops are employed to substantiate the validity of a previous phenomenological distance constraint model that describes protein thermodynamics successfully based on network rigidity. Full article
(This article belongs to the Special Issue Entropies of Polymers)
Open AccessArticle Prudent Self-Avoiding Walks
Entropy 2008, 10(3), 309-318; doi:10.3390/e10030309
Received: 17 May 2008 / Accepted: 9 September 2008 / Published: 24 September 2008
Cited by 5 | PDF Full-text (164 KB)
Abstract
We have produced extended series for prudent self-avoiding walks on the square lattice. These are subsets of self-avoiding walks. We conjecture the exact growth constant and critical exponent for the walks, and show that the (anisotropic) generating function is almost certainly not differentiably-finite.
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We have produced extended series for prudent self-avoiding walks on the square lattice. These are subsets of self-avoiding walks. We conjecture the exact growth constant and critical exponent for the walks, and show that the (anisotropic) generating function is almost certainly not differentiably-finite. Full article
Open AccessArticle The Entropy Principle from Continuum Mechanics to Hyperbolic Systems of Balance Laws: The Modern Theory of Extended Thermodynamics
Entropy 2008, 10(3), 319-333; doi:10.3390/e10030319
Received: 3 May 2008 / Accepted: 4 September 2008 / Published: 24 September 2008
Cited by 4 | PDF Full-text (224 KB)
Abstract
We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous
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We discuss the different roles of the entropy principle in modern thermodynamics. We start with the approach of rational thermodynamics in which the entropy principle becomes a selection rule for physical constitutive equations. Then we discuss the entropy principle for selecting admissible discontinuous weak solutions and to symmetrize general systems of hyperbolic balance laws. A particular attention is given on the local and global well-posedness of the relative Cauchy problem for smooth solutions. Examples are given in the case of extended thermodynamics for rarefied gases and in the case of a multi-temperature mixture of fluids. Full article
Open AccessArticle Configurons: Thermodynamic Parameters and Symmetry Changes at Glass Transition
Entropy 2008, 10(3), 334-364; doi:10.3390/e10030334
Received: 31 July 2008 / Revised: 9 September 2008 / Accepted: 22 September 2008 / Published: 24 September 2008
Cited by 36 | PDF Full-text (1075 KB) | HTML Full-text | XML Full-text
Abstract
Thermodynamic parameters of configurons – elementary excitations resulting from broken bonds in amorphous materials – are found from viscosity-temperature relationships. Glass-liquid transition phenomena and most popular models are described along with the configuron model of glass transition. The symmetry breaking, which occurs as
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Thermodynamic parameters of configurons – elementary excitations resulting from broken bonds in amorphous materials – are found from viscosity-temperature relationships. Glass-liquid transition phenomena and most popular models are described along with the configuron model of glass transition. The symmetry breaking, which occurs as a change of Hausdorff dimension of bonds, is examined at glass-liquid transition. Thermal history effects in the glass-liquid transition are interpreted in terms of configuron relaxation. Full article
(This article belongs to the Special Issue Configurational Entropy)
Open AccessArticle Entropy Diagnostics for Fourth Order Partial Differential Equations in Conservation Form
Entropy 2008, 10(3), 365-379; doi:10.3390/e10030365
Received: 3 May 2008 / Accepted: 21 September 2008 / Published: 25 September 2008
Cited by 7 | PDF Full-text (201 KB)
Abstract
The entropy evolution behaviour of a partial differential equation (PDE) in conservation form, may be readily discerned from the sign of the local source term of Shannon information density. This can be easily used as a diagnostic tool to predict smoothing and non-smoothing
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The entropy evolution behaviour of a partial differential equation (PDE) in conservation form, may be readily discerned from the sign of the local source term of Shannon information density. This can be easily used as a diagnostic tool to predict smoothing and non-smoothing properties, as well as positivity of solutions with conserved mass. The familiar fourth order diffusion equations arising in applications do not have increasing Shannon entropy. However, we obtain a new class of nonlinear fourth order diffusion equations that do indeed have this property. These equations also exhibit smoothing properties and they maintain positivity. The counter-intuitive behaviour of fourth order diffusion, observed to occur or not occur on an apparently ad hoc basis, can be predicted from an easily calculated entropy production rate. This is uniquely defined only after a technical definition of the irreducible source term of a reaction diffusion equation. Full article
Open AccessArticle The Second Entropy: A Variational Principle for Time-dependent Systems
Entropy 2008, 10(3), 380-390; doi:10.3390/e10030380
Received: 3 May 2008 / Accepted: 4 September 2008 / Published: 25 September 2008
Cited by 5 | PDF Full-text (158 KB)
Abstract
The fundamental optimization principle for non-equilibrium thermodynamics is given. The second entropy is introduced as the quantity that is maximised to determine the optimum state of a non-equilibrium system. In contrast, the principles of maximum or minimum dissipation, which have previously been proposed
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The fundamental optimization principle for non-equilibrium thermodynamics is given. The second entropy is introduced as the quantity that is maximised to determine the optimum state of a non-equilibrium system. In contrast, the principles of maximum or minimum dissipation, which have previously been proposed by Onsager, Prigogine, and others as the variational principle for such systems, are shown to be incapable of fulfilling that rôle. Full article

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