What is the Fokker-Planck collision operator and how is it derived? On page 7 of Goldston and Towner (1981) they state that "The Fokker-Planck collision operator for pitch-angle scattering is given by
$$\left.\frac{\partial f}{\partial t}\right|_c=\frac{\nu_{ii}}{2}\frac{\partial}{\partial \xi}\left(1-\xi^2\right)\frac{\partial}{\partial \xi}f,$$
where $\xi =v_{||} / v$."
Do you know how this equation is derived? Are there any books or lecture notes you would recommend which explain and give a background to the Fokker-Planck collision operator?
 A: To answer your question I first state some general formulas that you can find e.g. in the classic paper https://doi.org/10.1103/PhysRev.107.1 but also in most books on plasma kinetic theory.
Fokker-Planck Operator in a Plasma
Generally, a Fokker-Planck collision operator has the form
\begin{align}
    C_{\alpha\beta} f_\alpha  = - \frac{\partial}{\partial \mathbf v} \left( \left< \Delta \mathbf v\right>_{\alpha\beta} f_\alpha \right) + \frac{1}{2}\frac{\partial^2}{\partial \mathbf v \partial \mathbf v} : \left( \left< \Delta \mathbf v\Delta \mathbf v \right>_{\alpha\beta} f_\alpha \right),
\end{align}
where we consider collisions of species $\alpha$ with species $\beta$. The Fokker-Planck coefficients are called friction and diffusion coefficients. For example you can write those for a plasma in terms of Rosenbluth potentials,
\begin{align}
    \left< \Delta \mathbf v\right>_{\alpha\beta} &= \Gamma_{\alpha\beta} \frac{\partial H_{\alpha\beta}}{\partial \mathbf v},\\
    \left< \Delta \mathbf v \Delta \mathbf v\right>_{\alpha\beta} &= \Gamma_{\alpha\beta} \frac{\partial^2 G_{\alpha\beta}}{\partial\mathbf v \partial \mathbf v},\\
    H_{\alpha\beta} &= \frac{m_{\alpha}+m_{\beta}}{m_{\beta}} \int \mathrm{d} \mathbf v' f_{\beta}(\mathbf v') |\mathbf v - \mathbf v'|^{-1},\\
    G_{\alpha\beta} &= \int \mathrm{d} \mathbf v' f_{\beta}(\mathbf v') |\mathbf v - \mathbf v'|,\\
    \Gamma_{\alpha\beta} &= \frac{4\pi q_{\alpha}^2 q_{\beta}^2}{m_{\alpha}^2} \ln\Lambda_{\alpha\beta}.
\end{align}
where $q, m$ are charge and mass and $\ln\Lambda_{\alpha\beta}$ is the Coulomb logarithm. The Rosenbluth potentials $H, G$ are evaluated via integrals of over the field particle distribution. Those live in the primed velocity space $\mathbf{v}'$, whereas the "test" particles (which change due to collisions we want to know) have velocity space $\mathbf{v}$.
Pitch Angle Scattering Operator
Assuming the field particles are distributed as Maxwellians,
\begin{align}
f_\beta
&= n_\beta \left(\frac{m_\beta}{2\pi T_\beta}\right)^{3/2} \exp{\left(-\frac{m_\beta \mathbf v^2}{2 T_\beta}\right)},
\end{align}
with density $n$ and temperature $T$ you can analytically calculate the Rosenbluth potentials and Fokker-Planck coefficients. You get the solution in terms of error functions and Gaussians. Then you can Taylor expand those special functions in small velocity $v<<v_{\mathrm{th}}$. You get a term that describes energy diffusion (which we are not interested in here). The second term is the pitch angle operator (or Lorentz collision operator),
\begin{align}
C_{\alpha\beta}^L f_\alpha 
&= \nu(v) \mathcal{L} f_\alpha,\\
\mathcal{L}
&= \frac{1}{2} \frac{\partial}{\partial \mathbf v} \cdot \left( \left( v^2 \mathbb{I}- \mathbf v \mathbf v \right) \cdot\frac{\partial}{\partial \mathbf v} \right),
\end{align}
where $\mathbb{I}$ is the unit tensor. Then you first need to use spherical velocity space coodinates $(v, \theta, \phi)$ and thenpitch angle coordinates $\zeta = v_{||}/v = \cos\theta$, then you get
\begin{align}
\mathcal{L} 
&= \frac{1}{2}\left[\frac{\partial}{\partial\zeta} \left((1-\zeta^2) \frac{\partial f_\alpha}{\partial\zeta}\right) + \frac{1}{(1-\zeta^2)} \frac{\partial^2 f_\alpha}{\partial\phi^2}
\right].
\end{align}
The first term is exactly what you stated in the question (if you choose the ion-ion collision frequency for the prefactor $\nu(v)$).
I hope this helps with your question, I can recommend the excellent script of J. D. Callen which you can find here https://drive.google.com/file/d/1j2Afyq1DO2zeyFf9qTTfFVL9F9Rxzn7s/view. There you can also find a much simpler, more heuristic derivation of the Lorentz collision operator by looking at the scattering of electrons with a stationary ion background.
