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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?

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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.

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  • $\begingroup$ Thank you for this really detailed and informative answer. It's exactly what I needed. $\endgroup$
    – Peanutlex
    Commented Feb 3, 2022 at 22:15

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