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arXiv · 1911.00594

M\"{o}bius transforms, cycles and q-triplets in statistical mechanics

Jean-Pierre Gazeau·Constantino Tsallis
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Abstract

In the realm of Boltzmann-Gibbs (BG) statistical mechanics and its q-generalisation for complex systems, we analyse observed sequences of q-triplets, or q-doublets if one of them is the unity, in terms of cycles of successive M\"obius transforms of the line preserving unity ( q=1 corresponds to the BG theory). Such transforms have the form q --> (aq + 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.

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Jean-Pierre Gazeau, Constantino Tsallis. 2019-11-01. M\"{o}bius transforms, cycles and q-triplets in statistical mechanics. https://doi.org/10.3390/e21121155

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