  We study the stochastic motion of a particle subject to spatially varying
Lorentz force in the small-mass limit. The limiting procedure yields an
additional drift term in the overdamped equation that cannot be obtained by
simply setting mass to zero in the velocity Langevin equation. We show that
whereas the overdamped equation of motion accurately captures the position
statistics of the particle, it leads to unphysical fluxes in the system that
persist in the long time limit; an anomalous result inconsistent with thermal
equilibrium. These fluxes are calculated analytically from the overdamped
equation of motion and found to be in quantitative agreement with Brownian
dynamics simulations. Our study suggests that the overdamped approximation,
though perfectly suited for position statistics, can yield unphysical values
for velocity-dependent variables such as flux and entropy production.
