Anomalous transport in complex systems: strange kinetics and fractal topology

Alexander Milovanov
Department of Space Plasma Physics, Space Science Institute, Russia

This report addresses the fractional (strange) kinetics as a complimentary tool in the description of anomalous transport processes in complex nonlinear dynamical systems. Fractional generalizations of the diffusion and Fokker-Planck-Kolmogorov equations are analyzed. Strange scaling laws for the anomalous transport in stochastic Hamiltonian systems are derived from the fractal structure of the particle chaotic trajectories. A transition to a nonequilibrium stationary state (NESS) is discussed in connection with the universal topological properties of percolating fractal sets near a critical threshold. The fractal topology formalism is presented, and the issue of the percolation constant is introduced. Real-space transport anomalies at the NESS are considered. An extension of the real-space fractional kinetics to the dynamical systems revealing strange stochastic acceleration processes is proposed. A self-consistent nonlinear fractional kinetic equation for the stochastic fractal time accelerations near the marginal NESS is formulated. The ensuing particle pdf is shown to contain a power law nonthermal tail with the exponent ranging from 6 to 7. A comparison with the Earth's magnetotail data is given.