Mobius 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 Mobius 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.