Frimain-Chen equation was obtained by first transforming to the guiding-center coordinates and then using the classical perturbation expansion for the distribution function (to separate gyro-phase independent part form the gyro-phase dependent one). Note that the guiding-center transform does not involve the perturbed field.
The modern form of the nonlinear gyrokinetic equation is in the total-f form. This way of deriving the gyrokinetic equation is to use coordinate transforms to eliminate gyro-phase dependence of the total distribution function (rather than splitting the distribution function itself) and thus obtain an equation for the resulting gyro-phase independent distribution function (called gyro-center distribution function). The coordinate transform involves the perturbed fields besides the equilibrium field.
The resulting equation for the gyro-center distribution function is given by (see Baojian’s paper)
| (428) |
where
| (429) |
| (430) |
Here the independent variables are gyro-center position X, magnetic moment μ and parallel velocity v∥.
The gyro-phase dependence of the particle distribution can be recovered by the inverse transformation of the transformation used before. The pull-back transformation (inverse gyro-center transform) gives rise to the polarization density term. (phase-space-Lagrangian Lie perturbation method (Littlejohn, 1982a, 1983), I need to read these two papers.).
The learning curve of Lie tansform gyrokinetic theory seems steep. I tried to learn it but failed. To derive the Frieman-Chen equation, you just need some patience. Meanwhile, patience is all that you need to learn any mathematic stuff, but some requires more patience than others, and Lie transform gyrokinetic is one that requires more for me.
