The present paper considers the time evolution of a charged test particle of mass $m$ in a constant temperature heat bath of a second charged particle of mass $M$. The time dependence of the distribution function of the test particles is given by a Fokker-Planck equation with a diffusion coefficient for Coulomb collisions as well as a diffusion coefficient for wave-particle interactions. For the mass ratio $m/M\to 0$, the steady distribution is a Kappa distribution which has been employed in space physics to fit observed particle energy spectra. The time dependence of the distribution functions with some initial value is expressed in terms of the eigenvalues and eigenfunctions of the linear Fokker-Planck operator and also interpreted with the transformation to a Schrödinger equation. We also consider the explicit time dependence of the distribution function with a discretization of the Fokker-Planck equation. We study the stability of the Kappa distribution to Coulomb collisions.

中文翻译：

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库克弛豫和波粒扩散的Fokker-Planck方程：谱解和Kappa分布对库仑碰撞的稳定性

本文考虑了质量带电测试粒子的时间演化 $米$ 在质量第二带电粒子的恒温热浴中 $\mathrm{中号}$。由Fokker-Planck方程给出测试粒子分布函数的时间依赖性，该方程具有库仑碰撞的扩散系数以及波粒相互作用的扩散系数。对于质量比$米/\mathrm{中号}\to 0$，稳态分布是一种Kappa分布，已在空间物理学中用于拟合观测到的粒子能谱。分布函数与某些初始值的时间相关性用线性Fokker-Planck算子的特征值和特征函数表示，并通过变换为Schrödinger方程进行解释。我们还考虑了Fokker-Planck方程离散化后分布函数的显式时间依赖性。我们研究了Kappa分布对库仑碰撞的稳定性。