Central limit behavior in dissipative and conservative dynamical systems

As it is evident from its indisputable success, Boltzmann-Gibbs statistics is the correct way of thermostatistically approaching ergodic systems. On the other hand, nontrivial ergodicity breakdown and strong correlations typically drag the system into out-of-equilibrium states where Boltzmann-Gibbs statistics fails. For a wide class of such complex systems, it has been shown in recent years that the correct approach seems to use Tsallis statistics instead. Here we discuss how the dynamics of the paradigmatic conservative (area-preserving) standard map exhibits, in an exceptionally clear manner, the crossing from one statistics to the other. Our results unambiguously illustrate the domains of validity of both Boltzmann-Gibbs and Tsallis statistics. We also compare our results to those coming from dissipative dynamical systems.