Canonical quantization of classical systems with generalized entropies

We present, in the framework of canonical quantization, a class of nonlinear Schrِdinger equations with a complex nonlinearity describing, in the mean field approximation, systems of collectively interacting particles. The quantum evolution equation is obtained starting from the study of an N-body classical system where the underlined nonlinear kinetics is governed by a kinetic interaction principle (KIP) [20]. The KIP imposes both the form of the generalized entropy associated to the classical system, and determines the Fokker-Planck equation describing the kinetic evolution of the system towards equilibrium.