Dynamical Foundations of Nonextensive Statistical Mechanics

We construct classes of stochastic differential equations with fluctuating friction forces that generate a dynamics correctly described by Tsallis statistics. These systems generalize the way in which ordinary Langevin equations underlie ordinary statistical mechanics to the more general nonextensive case. As a main example, we construct a dynamical model of velocity fluctuations in a turbulent flow, which generates probability densities that very well fit experimentally measured probability densities in Eulerian and Lagrangian turbulence. Our approach provides a dynamical reason why many physical systems with fluctuations in temperature or energy dissipation rate are correctly described by Tsallis statistics.