Steady-state distributions of ideal active Brownian particles under confinement and forcing

We develop a formally exact technique for obtaining steady-state distributions of non-interacting active Brownian particles in a variety of systems. Our technique draws on results from the theory of two-way diffusion equations to solve the steady-state Smoluchowski equation for the 1-particle distribution function. The methods are employed to study in detail three scenarios: 1) confinement in a channel, 2) a constant flux steady state, and 3) sedimentation in a uniform external field. In each scenario, known behaviors are reproduced and precisely quantified, and new results are presented. In particular, in the constant flux state we derive an effective diffusivity which interpolates between the ballistic behavior of particle trajectories at short distances and their diffusive behavior at large distances. We also calculate the sedimentation profile of active Brownian particles near a wall, which complements earlier studies on the part far from the wall. Our techniques easily generalize to other active models, including systems whose activity is modeled in terms of Gaussian colored noise.