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Open AccessArticle

Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

1
Astroparticles and Cosmology (UMR 7164), Sorbonne Paris Cité, Univ Paris Diderot, 75205 Paris, France
2
Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology of Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, Brazil
3
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
4
Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080 Vienna, Austria
*
Author to whom correspondence should be addressed.
Entropy 2019, 21(12), 1155; https://doi.org/10.3390/e21121155
Received: 24 October 2019 / Revised: 21 November 2019 / Accepted: 22 November 2019 / Published: 26 November 2019
(This article belongs to the Special Issue The Ubiquity of Entropy)
In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analysed sequences of q-triplets, or q-doublets if one of them was the unity, in terms of cycles of successive Möbius transforms of the line preserving unity ( q = 1 corresponds to the BG theory). Such transforms have the form q ( a q + 1 a ) / [ ( 1 + a ) q a ] , where a is a real number; the particular cases a = 1 and a = 0 yield, respectively, q ( 2 q ) and q 1 / q , currently known as additive and multiplicative dualities. This approach seemingly enables the organisation of various complex phenomena into different classes, named N-complete or incomplete. The classification that we propose here hopefully constitutes a useful guideline in the search, for non-BG systems whenever well described through q-indices, of new possibly observable physical properties. View Full-Text
Keywords: non-additive entropy; q-statistics; Möbius transform; complex systems non-additive entropy; q-statistics; Möbius transform; complex systems
MDPI and ACS Style

Gazeau, J.P.; Tsallis, C. Möbius Transforms, Cycles and q-triplets in Statistical Mechanics. Entropy 2019, 21, 1155.

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