Active Brownian particles in a inhomogeneous activity or magnetic field

We study spherically symmetric active particles in a spatially inhomogeneous activity or magnetic field. In both cases the particles spontaneously orient themselves, which results in an inhomogeneous distribution of particles. Using Green-Kubo approach we obtain analytical expression for the orientation of the particles. We find that the relevant equilibrium correlation function is the self-part of the Van Hove function, which can be approximated accurately. Density does not have a linear response to the activity; however, an expression for the density is derived by using the orientation as input to dynamic density functional theory. All theoretical predictions are validated using Brownian dynamics simulations.