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Entropy 2018, 20(6), 453; https://doi.org/10.3390/e20060453

Entropy and Geometric Objects

ACCESS e.V., Intzestr. 5, D-52072 Aachen, Germany
Received: 27 April 2018 / Revised: 30 May 2018 / Accepted: 5 June 2018 / Published: 9 June 2018
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Abstract

Different notions of entropy can be identified in different scientific communities: (i) the thermodynamic sense; (ii) the information sense; (iii) the statistical sense; (iv) the disorder sense; and (v) the homogeneity sense. Especially the “disorder sense” and the “homogeneity sense” relate to and require the notion of space and time. One of the few prominent examples relating entropy to both geometry and space is the Bekenstein-Hawking entropy of a Black Hole. Although this was developed for describing a physical object—a black hole—having a mass, a momentum, a temperature, an electrical charge, etc., absolutely no information about this object’s attributes can ultimately be found in the final formulation. In contrast, the Bekenstein-Hawking entropy in its dimensionless form is a positive quantity only comprising geometric attributes such as an area A—the area of the event horizon of the black hole, a length LP—the Planck length, and a factor 1/4. A purely geometric approach to this formulation will be presented here. The approach is based on a continuous 3D extension of the Heaviside function which draws on the phase-field concept of diffuse interfaces. Entropy enters into the local and statistical description of contrast or gradient distributions in the transition region of the extended Heaviside function definition. The structure of the Bekenstein-Hawking formulation is ultimately derived for a geometric sphere based solely on geometric-statistical considerations. View Full-Text
Keywords: gradient-entropy; contrast; phase-field models; diffuse interfaces; entropy of geometric objects; Bekenstein-Hawking entropy; Heaviside function; Dirac function; 3D delta function gradient-entropy; contrast; phase-field models; diffuse interfaces; entropy of geometric objects; Bekenstein-Hawking entropy; Heaviside function; Dirac function; 3D delta function
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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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Schmitz, G.J. Entropy and Geometric Objects. Entropy 2018, 20, 453.

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