Drift kinetic equation

The distribution function is constant along the trajectory of the guiding center in phase space, i.e.,

$$\frac{d f}{d t} = 0.$$ (76)

We consider the case that the distribution function is independent of $\alpha$, i.e., $f = f ( \ensuremath{\boldsymbol{X}}, v_{\parallel}, y, t)$. Then Eq. (76) is written as

$$\frac{\partial f}{\partial t} + \dot{\ensuremath{\boldsymbol{X}}}... ...partial f}{\partial v_{\parallel}} + \dot{y} \frac{\partial f}{\partial y} = 0,$$ (77)

(which is equation (15) in Porcelli's paper) where the guiding center orbits, $\dot{\ensuremath{\boldsymbol{R}}}$, $\dot{v_{\parallel}}$, and $\dot{y}$, are given by Eqs. (66), (27), and (32), i.e.,

$$\dot{\ensuremath{\boldsymbol{X}}} = v_{\parallel} \ensuremath{\bo... ...}^2 \ensuremath{\boldsymbol{\kappa}} - Z e \ensuremath{\boldsymbol{E}} \right),$$ (78)

$$\dot{v_{\parallel}} = \frac{1}{m} \left( - \frac{y}{B} \ensuremat... ...nsuremath{\boldsymbol{\kappa}} \cdot \dot{\ensuremath{\boldsymbol{X}}} \right),$$ (79)

and

$$\dot{y} = \left( \frac{\partial B}{\partial t} + \dot{\ensuremath{\boldsymbol{X}}} \cdot \nabla B \right) \frac{y}{B} .$$ (80)

We note that, besides the independent variables $( \ensuremath{\boldsymbol{X}}, v_{\parallel}, y)$, the right-hand side of the Eqs. (78), (79), and (80) depends on the electromagnetic field. Note that, in the perturbation theory, only the electromagnetic field can be perturbed, the independent variables (variables used as the phase space coordinates) are kept fixed.