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

TL;DR

This paper explores the use of Möbius transforms to analyze q-triplets in statistical mechanics, aiming to classify complex systems and identify new physical properties beyond traditional Boltzmann-Gibbs theory.

Contribution

It introduces a novel framework using Möbius transforms to organize and classify q-indices in complex systems, extending the understanding of non-BG statistical mechanics.

Findings

Möbius transforms can organize q-triplets into classes.

The approach links transformations to known dualities like q --> 2-q and q --> 1/q.

Proposes a classification scheme for complex phenomena based on these transforms.

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.