  We consider the nonequilibrium dynamics of a charged active Brownian particle
in the presence of a space dependent magnetic field. It has recently been shown
that the Lorentz force induces a particle flux perpendicular to density
gradients, thus preventing a diffusive description of the dynamics. Whereas a
passive system will eventually relax to an equilibrium state, unaffected by the
magnetic field, an active system subject to a spatially varying Lorentz force
settles into a nonequilibrium steady state characterized by an inhomogeneous
density and divergence-free bulk fluxes. A macroscopic flux of charged active
particles is induced by the gradient of the magnetic field only and does not
require additional symmetric breaking such as density or potential gradients.
This stands in marked contrast to similar phenomena in condensed matter such as
the classical Hall effect. In a confined geometry we observe circulating
fluxes, which can be reversed by inverting the direction of the magnetic field.
Our theoretical approach, based on coarse-graining of the Fokker-Planck
equation, yields analytical results for the density, fluxes, and polarization
in the steady state, all of which are validated by direct computer simulation.
We demonstrate that passive tracer particles can be used to measure the
essential effects of the Lorentz force on the active particle bath, and we
discuss under which conditions the effects of the flux could be observed
experimentally.
