How, Why and When Tsallis Statistical Mechanics Provides Precise Descriptions of Natural Phenomena.

The limit of validity of ordinary statistical mechanics and the pertinence of Tsallis statistics beyond it is explained considering the most probable evolution of complex systems processes. To this purpose we employ a dissipative Landau-Ginzburg kinetic equation that becomes a generic one-dimensional nonlinear iteration map for discrete time. We focus on the Renormalization Group (RG) fixed-point maps for the three routes to chaos. We show that all fixed-point maps and their trajectories have analytic closed-form expressions, not only (as known) for the intermittency route to chaos but also for the period-doubling and the quasiperiodic routes. These expressions have the form of -exponentials, while the kinetic equation's Lyapunov function becomes the Tsallis entropy. That is, all processes described by the evolution of the fixed-point trajectories are accompanied by the monotonic progress of the Tsallis entropy. In all cases the action of the fixed-point map attractor imposes a severe impediment to access the system's built-in configurations, leaving only a subset of vanishing measure available. Only those attractors that remain chaotic have ineffective configuration set reduction and display ordinary statistical mechanics. Finally, we provide a brief description of complex system research subjects that illustrates the applicability of our approach.