# Anomalous fluxes in overdamped Brownian dynamics with Lorentz force

## TL;DR

This paper investigates the effects of Lorentz force on overdamped Brownian particles, revealing that the standard overdamped approximation produces unphysical fluxes despite accurately capturing position statistics.

## Contribution

It demonstrates that the overdamped limit introduces an additional drift term and unphysical fluxes, highlighting limitations of the common approximation in systems with Lorentz forces.

## Key findings

- Overdamped equation captures position statistics accurately.
- Unphysical fluxes persist in the long-time limit.
- Analytical flux calculations match simulations.

## Abstract

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.