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From Liouville's equation, a phase-space multi-scale transport equation issystematically derived. The proposed phase-space multi-scale transport equationbased on the first principle indicates that the nonlinear stochastic transportis due to the micro-turbulence scattering while the familiar quasilineartransport is due to the long-range correlation of the meso-scale coherentmodes; and more interestingly, it demonstrates a systematic derivation of thefractional transport equation that may provide a new view-angle to understandthe anomalous transport observed in plasma turbulence. The multi-scale kineticssuggests a new approach to confinement improvement through the formation ofnonlinearly self-organized stable large-scale structures.

From Liouville's theorem to quasilinear, nonlinear stochastic, and fractional transport, a multi-scale kinetics of plasma turbulence