A generalized boundary reflection rule, funnel-shaped geometries, and rectification in systems with active boundaries

Confined active matter exhibits rectification in the presence of funnel-shaped geometries. This phenomenon occurs, for example, when bacteria encountering a boundary, crawl along it. Inspired by such examples, we investigate rectification in a system with funnel-shaped boundaries and with two different classes of reflection rules for randomly-moving particles colliding with the boundary;
(θ_reflect > θ_incident and θ_reflect < θ_incident). This generalized boundary reflection rule can be implemented by an activator at the boundary. For example, one such activator could be an "explosive" reaction when the particle collides with the boundary. In the former case, the funnel acts as a pump and in the latter, it acts like a collector, thereby showing ratchet reversal. We study this system both numerically and analytically. Specifically, we calculate the ratio of number densities across the funnel in steady state. We also discuss implications of our results for potential experiments.