Stationary State In Brownian Systems With Lorentz Force

In systems with overdamped dynamics, the Lorentz force reduces the diffusivity of a Brownian particle in the plane perpendicular to the magnetic field. The anisotropy in diffusion implies that the Fokker-Planck equation for the probability distribution of the particle acquires a tensorial coefficient. The tensor, however, is not a typical diffusion tensor due to the antisymmetric elements, which account for the fact that Lorentz force curves the trajectory of a moving charged particle. This gives rise to unusual dynamics with features such as additional Lorentz fluxes and a nontrivial density distribution, unlike a diffusive system. The equilibrium properties are, however, unaffected by the Lorentz force. Here we show that by stochastically resetting the Brownian particle, a nonequilibrium steady state can be created that preserves the hallmark features of dynamics under Lorentz force. We then consider a minimalistic example of a spatially inhomogeneous magnetic field, which shows how Lorentz fluxes fundamentally alter the boundary conditions giving rise to an unusual stationary state.