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

Rectification of Random Walkers Induced by Energy Flow at Boundaries