This study advocates the application of fractional dynamics to the description of anomalous acceleration processes in self-organized turbulent systems. Such processes (termed “strange” accelerations) involve both the non-Markovian fractal time acceleration events associated with a generalized stochastic Fermi mechanism, and the velocity-space Levy flights identified with nonlocal violent accelerations in turbulent media far from the (quasi)equilibrium. The “strange” acceleration processes are quantified by a fractional extension of the velocity-space transport equation with fractional time and phase space derivatives. A self-consistent nonlinear fractional kinetic equation is proposed for the stochastic fractal time accelerations near the turbulent nonequilibrium saturation state. The ensuing self-consistent energy distribution reveals a power-law superthermal tail $\psi \left(\mathcal{E}\right)\propto {\mathcal{E}}^{-\eta }$ with slope $6<~\eta <~7$ depending on the type of acceleration process (persistent or antipersistent). The results obtained are in close agreement with observational data on the Earth’s magnetotail.

• Received 25 April 2001

DOI:https://doi.org/10.1103/PhysRevE.64.052101