  An active Brownian particle is a minimal model for a self-propelled colloid
in a dissipative environment. Experiments and simulations show that, in the
presence of boundaries and obstacles, active Brownian particle systems approach
nontrivial nonequilibrium steady states with intriguing phenomenology, such as
accumulation at boundaries, ratchet effects, and long-range depletion
interactions. Nevertheless, theoretical analysis of these phenomena has proven
difficult. Here we address this theoretical challenge in the context of
non-interacting particles in two dimensions, basing our analysis on the
steady-state Smoluchowski equation for the 1-particle distribution function.
Our primary result is an approximation strategy that explicitly connects
asymptotic solutions of the Smoluchowski equation to boundary conditions. We
test this approximation against the exact analytic solution in a 2d planar
geometry as well as numerical solutions in circular and elliptic geometries. We
find good agreement so long as the boundary conditions do not vary too rapidly
with respect to the persistence length of particle trajectories. Our results
are relevant for characterizing long-range flows and depletion interactions in
such systems. In particular, our framework shows how such behaviors are
connected to the breaking of detailed balance at the boundaries.
