From nolinear statistical mechanics to nonlinear quantum mechanics - Concepts and applications

We briefly review a perspective along which the Boltzmann-Gibbs statistical mechanics, the strongly chaotic dynamical systems, and the Schroedinger, Klein-Gordon and Dirac partial differential equations are seen as linear physics, and are characterized by an index q = 1. We exhibit in what sense q {\neq} 1 yields nonlinear physics, which turn out to be quite rich and directly related to what is nowadays referred to as complexity, or complex systems. We first discuss a few central points like the distinction between additivity and extensivity, and the Central Limit Theorem as well as the large-deviation theory. Then we comment the case of gravitation (which within the present context corresponds to q {\neq} 1, and to similar nonlinear approaches), with special focus onto the entropy of black holes. Finally we briefly focus on recent nonlinear generalizations of the Schroedinger, Klein-Gordon and Dirac equations, and mention various illustrative predictions, verifications and applications within physics (in both low- and high-energy regimes) as well as out of it.