Derivation of a reduced kinetic equation using Lie-transform techniques

The asymptotic elimination of fast time scales from a general kinetic equation, of the form: ∂tf+z·∂xf = C[f], facilitates the study of the long time behavior of its solution f(z,t). Here z describe the single-particle Hamiltonian dynamics and the operator C, which may possess nonlinear functional dependence on f, describes processes (such as discrete-particle effects, resonant wave-particle effects, or effects due to external sources) which cause changes in f as it is convectively transported along a Hamiltonian phase-space trajectory. When a fast time scale is associated with z through the dependence on a fast angle θ (whose frequency θ = Ω satisfies ε ≡ 1/Ωτ much-lt 1, where τ is a slow time scale of interest), a near-identity phase-space transformation Tε:z→Z (carried out with Lie-transform techniques) yields reduced Hamiltonian dynamical equations Zε which are θ-independent. The corresponding transformed kinetic equation is derived. Averaging this equation over the fast angle θ yields a kinetic equation for left-angle F right-angle, the θ-averaged part of F. In general, the θ-dependence of Cε couples the kinetic equations for left-angle F right-angle and F, the θ-dependent part of F. One solves for the Fourier coefficient Fl (associated with eilθ) as a functional of left-angle F right-angle. One obtains a reduced kinetic equation for left-angle F right-angle: dR left-angle F right-angle/dt = CR[left-angle F right-angle]. General expressions for CR are given, as well as expressions for the guiding-center and oscillation-center phase-space transformations of a linear Fokker-Planck operator. A discussion of the relationship with Mynick's work is presented