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Angular momentum of an unconfined charged Brownian particle in a static magnetic field within the framework of the generalized Langevin equation
Phys. Rev. E 114, 024149 – Published 24 August, 2026
DOI: https://doi.org/10.1103/297f-hxh9
Abstract
We investigate the angular momentum and kinetic energy of a classical charged Brownian particle in a static magnetic field within the framework of the generalized Langevin equation with Ornstein–Uhlenbeck noise. The mean kinetic energy retains its equilibrium value for all times after the external field is switched on. An analytical expression for the mean time-dependent angular momentum is derived and supported by high-precision numerical calculations. We show that, in the long-time limit, the mean angular momentum approaches a finite nonzero value. While this result might appear to contradict the Bohr–van Leeuwen theorem, we demonstrate that the system considered here does not correspond to a canonical equilibrium ensemble due to the absence of spatial confinement. In particular, although the velocity distribution thermalizes, the position of the particle remains unbounded, and the partition function over position space is not normalizable. The obtained nonzero angular momentum should therefore be interpreted as a property of a nonequilibrium asymptotic dynamical regime within the Langevin description rather than as a violation of equilibrium statistical mechanics. The present results provide a transparent analytical reference for stochastic dynamics of charged particles in magnetic fields and highlight the role of unbounded motion and noncommuting limits in such systems.
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