Emergence of Nonwhite Noise in Langevin Dynamics with Magnetic Lorentz Force

Abstract

We investigate the low mass limit of Langevin dynamics for a charged Brownian particle driven by the magnetic Lorentz force. In the low mass limit, velocity variables relaxing quickly are coarse-grained out to yield effective dynamics for position variables. Without Lorentz force, the low mass limit is equivalent to the high friction limit. Both cases share the same Langevin equation that is obtained by setting the mass to zero in the original Langevin equation. The equivalence breaks down in the presence of the Lorentz force. The low mass limit turns out to be singular. The system in the low mass limit is different from the system with zero mass. The low mass limit is also different from the large friction limit. We derive the effective equations of motion in the low mass limit. The resulting stochastic differential equation involves a nonwhite noise whose correlation matrix has antisymmetric components. We demonstrate the importance of the nonwhite noise by investigating the heat dissipation by the Brownian particle.