Angular Momentum of a Classical Charged Brownian Particle in a Static Magnetic Field

Within the framework of the original Langevin theory of Brownian motion, we derive an analytical expression that simultaneously determines both the time-dependent diffusion coefficient and the mean angular momentum of a classical charged Brownian particle in a static magnetic field. While the diffusion coefficient and the associated mean square displacement are consistent with wellestablished results, the derived angular momentum, which is, based on the approach used, as theoretically sound as the diffusion coefficient, determines the magnetic moment induced by the motion of the particle in the unbounded plane perpendicular to the magnetic field and corrects previous expressions in the literature. In the long-time limit after the magnetic field is switched on, the mean angular momentum approaches a finite, nonzero value, while the kinetic energy of the particle remains the same at all times, corresponding to the equipartition theorem in equilibrium. The physical interpretation of this result is discussed in relation to the Bohr–van Leeuwen theorem. The present results provide a transparent analytical reference for stochastic dynamics of charged particles in magnetic fields and may serve as a basis for extensions, including memory effects or confinement.