In this work we present the original Langevin's method to study the Brownian motion of a charged particle across a constant magnetic field. Langevin's strategy allows transforming the Langevin equation into a deterministic one associated with Mean Square Displacement (MSD), whose solution is explicitly study in this work. We calculate the MSD for the $z$-axis, along which the magnetic field is allowed to point, in this direction, the solution is exactly the same proposed by Langevin. While, in the $x$-$y$ plane orthogonal to the magnetic field, the solution for the MSD is obtained with the validity of the Bohr-van Leuuwen theorem in thermal equilibrium.

中文翻译：

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磁场中的布朗运动：重新审视朗之万方法

在这项工作中，我们展示了原始的朗之万方法来研究带电粒子在恒定磁场中的布朗运动。Langevin 的策略允许将 Langevin 方程转换为与均方位移 (MSD) 相关的确定性方程，在这项工作中明确研究了其解决方案。我们计算 $z$ 轴的 MSD，允许磁场指向该轴，在这个方向上，解决方案与 Langevin 提出的完全相同。而在与磁场正交的 $x$-$y$ 平面上，MSD 的解是根据热平衡中 Bohr-van Leuwen 定理的有效性获得的。