Some Open Points in Nonextensive Statistical mechanics

Statistical mechanics is one of the pillars of contemporary physics. Indeed, it provides in principle the link between the microscopic and the macroscopic description and understanding of the world through a rich variety of mesoscopic instances. More precisely, starting from mechanics (classical, quantum, relativistic) and electromagnetism, it incorporates appropriate concepts of the theory of probabilities, and finally leads, for large enough systems, to thermodynamics. It does so through various intermediate descriptions such as those involving master, Langevin, and Fokker-Planck equations. See Fig. 1 for a schematic set of connections. The statistical mechanical approach of physical phenomena was first introduced by Maxwell, Boltzmann and Gibbs. A crucial step was done by Boltzmann who proposed a specific functional form for the thermodynamic entropy (that had been introduced a few years earlier by Clausius) in terms of the probabilities of the microscopic configurations. Its expression for say a system describable through a discrete set of probabilities {pi} of W possibilities is given by