Möbius Transforms, Cycles and q-triplets in Statistical Mechanics

Abstract

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.