Rectification of Random Walkers Induced by Energy Flow at Boundaries

We explore rectification phenomena in a system where two-dimensional random walkers interact with a funnel-shaped ratchet under two distinct classes of reflection rules. The two classes include the angle of reflection exceeding the angle of incidence ($\theta_{reflect} > \theta_{incident}$), or vice versa ($\theta_{reflect} < \theta_{incident}$). These generalized boundary reflection rules are indicative of non-equilibrium conditions due to the introduction of energy flows at the boundary. Our findings reveal that the nature of such particle-wall interactions dictates the system's behavior: the funnel either acts as a pump, directing flow, or as a collector, demonstrating a ratchet reversal. Importantly, we provide a geometric proof elucidating the underlying mechanism of rectification, thereby offering insights into why certain interactions lead to directed motion, while others do not.