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Entropy 2015, 17(9), 6329-6378; doi:10.3390/e17096329

Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids

Weill Cornell Medical College in Qatar, Education City, PO Box 24144, Doha, Qatar
Academic Editor: George Ruppeiner
Received: 16 July 2015 / Revised: 9 September 2015 / Accepted: 10 September 2015 / Published: 15 September 2015
(This article belongs to the Special Issue Geometry in Thermodynamics)
View Full-Text   |   Download PDF [393 KB, uploaded 15 September 2015]

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 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. View Full-Text
Keywords: non-additive entropy; non-extensive statistical mechanics; Tsallis entropy; κ-entropy; convexity; symplectic capacities; Dvoretzky’s theorem non-additive entropy; non-extensive statistical mechanics; Tsallis entropy; κ-entropy; convexity; symplectic capacities; Dvoretzky’s theorem
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Kalogeropoulos, N. Entropies from Coarse-graining: Convex Polytopes vs. Ellipsoids. Entropy 2015, 17, 6329-6378.

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