Brownian motion across a magnetic field: Langevin approach revisited

In this work we present Langevin's original method to study the Brownian motion of a charged particle across a constant magnetic field. Langevin's strategy allows one to transform the Langevin equation into a deterministic one associated with mean square displacement (MSD), whose solution is explicitly studied 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.