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260 KiB

Open AccessArticle

Dualistic Hessian Structures Among the Thermodynamic Potentials in the κ-Thermostatistics

by Tatsuaki Wada, Hiroshi Matsuzoe and Antonio M. Scarfone

Entropy 2015, 17(10), 7213-7229; https://doi.org/10.3390/e17107213 - 22 Oct 2015

Cited by 20 | Viewed by 4905

We explore the information geometric structures among the thermodynamic potentials in the κ-thermostatistics, which is a generalized thermostatistics based on the κ-deformed entropy. We show that there exists two different kinds of dualistic Hessian structures: one is associated with the κ-escort expectations and [...] Read more.

We explore the information geometric structures among the thermodynamic potentials in the κ-thermostatistics, which is a generalized thermostatistics based on the κ-deformed entropy. We show that there exists two different kinds of dualistic Hessian structures: one is associated with the κ-escort expectations and the other with the standard expectations. The associated κ-generalized metrics are derived and related to the κ-generalized fluctuation-response relations among the thermodynamic potentials in the κ-thermostatistics. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

314 KiB

Open AccessArticle

Thermodynamic Metrics and Black Hole Physics

by Jan E. Åman, Ingemar Bengtsson and Narit Pidokrajt

Entropy 2015, 17(9), 6503-6518; https://doi.org/10.3390/e17096503 - 22 Sep 2015

Cited by 12 | Viewed by 5923

Abstract

We give a brief survey of thermodynamic metrics, in particular the Hessian of the entropy function, and how they apply to black hole thermodynamics. We then provide a detailed discussion of the Gibbs surface of Kerr black holes. In particular, we analyze its [...] Read more.

We give a brief survey of thermodynamic metrics, in particular the Hessian of the entropy function, and how they apply to black hole thermodynamics. We then provide a detailed discussion of the Gibbs surface of Kerr black holes. In particular, we analyze its global properties and extend it to take the entropy of the inner horizon into account. A brief discussion of Kerr–Newman black holes is included. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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393 KiB

Open AccessArticle

Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids

by Nikos Kalogeropoulos

Entropy 2015, 17(9), 6329-6378; https://doi.org/10.3390/e17096329 - 15 Sep 2015

Cited by 8 | Viewed by 4883

Abstract

We examine the Boltzmann/Gibbs/Shannon SBGS and the non-additive Havrda-Charvát/Daróczy/Cressie-Read/Tsallis Sq and the Kaniadakis κ-entropy Sκ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between [...] Read more.

We examine the Boltzmann/Gibbs/Shannon SBGS and the non-additive Havrda-Charvát/Daróczy/Cressie-Read/Tsallis Sq and the Kaniadakis κ-entropy Sκ from the viewpoint of coarse-graining, symplectic capacities and convexity. We argue that the functional form of such entropies can be ascribed to a discordance in phase-space coarse-graining between two generally different approaches: the Euclidean/Riemannian metric one that reflects independence and picks cubes as the fundamental cells in coarse-graining and the symplectic/canonical one that picks spheres/ellipsoids for this role. Our discussion is motivated by and confined to the behaviour of Hamiltonian systems of many degrees of freedom. We see that Dvoretzky’s theorem provides asymptotic estimates for the minimal dimension beyond which these two approaches are close to each other. We state and speculate about the role that dualities may play in this viewpoint. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

1012 KiB

Open AccessArticle

Metrics and Energy Landscapes in Irreversible Thermodynamics

by Bjarne Andresen

Entropy 2015, 17(9), 6304-6317; https://doi.org/10.3390/e17096304 - 10 Sep 2015

Cited by 11 | Viewed by 5108

Abstract

We describe how several metrics are possible in thermodynamic state space but that only one, Weinhold’s, has achieved widespread use. Lengths calculated based on this metric have been used to bound dissipation in finite-time (irreversible) processes be they continuous or discrete, and described [...] Read more.

We describe how several metrics are possible in thermodynamic state space but that only one, Weinhold’s, has achieved widespread use. Lengths calculated based on this metric have been used to bound dissipation in finite-time (irreversible) processes be they continuous or discrete, and described in the energy picture or the entropy picture. Examples are provided from thermodynamics of heat conversion processes as well as chemical reactions. Even losses in economics can be bounded using a thermodynamic type metric. An essential foundation for the metric is a complete equation of state including all extensive variables of the system; examples are given. Finally, the second law of thermodynamics imposes convexity on any equation of state, be it analytical or empirical. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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237 KiB

Open AccessArticle

Conformal Gauge Transformations in Thermodynamics

by Alessandro Bravetti, Cesar S Lopez-Monsalvo and Francisco Nettel

Entropy 2015, 17(9), 6150-6168; https://doi.org/10.3390/e17096150 - 02 Sep 2015

Cited by 14 | Viewed by 4928

Abstract

In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the [...] Read more.

In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the connection one-form, whilst, as a consequence of the non-integrability of the connection, its curvature is not and, therefore, neither is the associated pseudo-Riemannian geometry. We claim that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all of the elements of the geometric structure of the thermodynamic phase space change under a re-scaling of the connection one-form. We call this transformation of the geometric structure a conformal gauge transformation. As an example, we revisit the change of the thermodynamic representation and consider the resulting change between the two metrics on the thermodynamic phase space, which induce Weinhold’s energy metric and Ruppeiner’s entropy metric. As a by-product, we obtain a proof of the well-known conformal relation between Weinhold’s and Ruppeiner’s metrics along the equilibrium directions. Finally, we find interesting properties of the almost para-contact structure and of its eigenvectors, which may be of physical interest. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

221 KiB

Open AccessArticle

Generalised Complex Geometry in Thermodynamical Fluctuation Theory

by P. Fernández de Córdoba and J. M. Isidro

Entropy 2015, 17(8), 5888-5902; https://doi.org/10.3390/e17085888 - 20 Aug 2015

Cited by 4 | Viewed by 4516

Abstract

We present a brief overview of some key concepts in the theory of generalized complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyze thermodynamical fluctuation theory in the [...] Read more.

We present a brief overview of some key concepts in the theory of generalized complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyze thermodynamical fluctuation theory in the presence of gravitational fields. To illustrate the usefulness of generalized complex geometry, we examine a simplified version of the Unruh effect: the thermalising effect of gravitational fields on the Schroedinger wavefunction. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

1597 KiB

Open AccessArticle

Geometric Interpretation of Surface Tension Equilibrium in Superhydrophobic Systems

by Michael Nosonovsky and Rahul Ramachandran

Entropy 2015, 17(7), 4684-4700; https://doi.org/10.3390/e17074684 - 06 Jul 2015

Cited by 28 | Viewed by 11917

Abstract

Surface tension and surface energy are closely related, although not identical concepts. Surface tension is a generalized force; unlike a conventional mechanical force, it is not applied to any particular body or point. Using this notion, we suggest a simple geometric interpretation of [...] Read more.

Surface tension and surface energy are closely related, although not identical concepts. Surface tension is a generalized force; unlike a conventional mechanical force, it is not applied to any particular body or point. Using this notion, we suggest a simple geometric interpretation of the Young, Wenzel, Cassie, Antonoff and Girifalco–Good equations for the equilibrium during wetting. This approach extends the traditional concept of Neumann’s triangle. Substances are presented as points, while tensions are vectors connecting the points, and the equations and inequalities of wetting equilibrium obtain simple geometric meaning with the surface roughness effect interpreted as stretching of corresponding vectors; surface heterogeneity is their linear combination, and contact angle hysteresis is rotation. We discuss energy dissipation mechanisms during wetting due to contact angle hysteresis, the superhydrophobicity and the possible entropic nature of the surface tension. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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1728 KiB

Open AccessArticle

A Possible Cosmological Application of Some Thermodynamic Properties of the Black Body Radiation in n-Dimensional Euclidean Spaces

by Julian Gonzalez-Ayala, Jennifer Perez-Oregon, Rubén Cordero and Fernando Angulo-Brown

Entropy 2015, 17(7), 4563-4581; https://doi.org/10.3390/e17074563 - 29 Jun 2015

Cited by 2 | Viewed by 4988

Abstract

In this work, we present the generalization of some thermodynamic properties of the black body radiation (BBR) towards an n-dimensional Euclidean space. For this case, the Planck function and the Stefan–Boltzmann law have already been given by Landsberg and de Vos and some [...] Read more.

In this work, we present the generalization of some thermodynamic properties of the black body radiation (BBR) towards an n-dimensional Euclidean space. For this case, the Planck function and the Stefan–Boltzmann law have already been given by Landsberg and de Vos and some adjustments by Menon and Agrawal. However, since then, not much more has been done on this subject, and we believe there are some relevant aspects yet to explore. In addition to the results previously found, we calculate the thermodynamic potentials, the efficiency of the Carnot engine, the law for adiabatic processes and the heat capacity at constant volume. There is a region at which an interesting behavior of the thermodynamic potentials arises: maxima and minima appear for the n—dimensional BBR system at very high temperatures and low dimensionality, suggesting a possible application to cosmology. Finally, we propose that an optimality criterion in a thermodynamic framework could be related to the 3—dimensional nature of the universe. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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Review

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298 KiB

Open AccessReview

Geometry of Multiscale Nonequilibrium Thermodynamics

by Miroslav Grmela

Entropy 2015, 17(9), 5938-5964; https://doi.org/10.3390/e17095938 - 25 Aug 2015

Cited by 17 | Viewed by 4062

Abstract

The time evolution of macroscopic systems can be experimentally observed and mathematically described on many different levels of description. It has been conjectured that the governing equations on all levels are particular realizations of a single abstract equation. We support this conjecture by [...] Read more.

The time evolution of macroscopic systems can be experimentally observed and mathematically described on many different levels of description. It has been conjectured that the governing equations on all levels are particular realizations of a single abstract equation. We support this conjecture by interpreting the abstract equation as a geometrical formulation of general nonequilibrium thermodynamics. Full article

(This article belongs to the Special Issue Geometry in Thermodynamics)

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