Competing active and passive rectification mechanisms in a discrete ratchet model of run-and-tumble particles

We investigate the thermodynamic behavior of a hybrid Brownian motor operating in a discrete three-state ratchet potential, where a particle, modeled as a run-and-tumble active agent, is subject to both thermal asymmetry and internal active propulsion. The model combines features of passive heat engines and active matter systems by allowing the particle to self-propel with velocity v0 and to switch between internal orientations σ=±1 at generally asymmetric rates α+ and α−. We derive exact time-dependent and steady-state solutions for the probability distributions, particle current, entropy, and entropy production. Our analysis reveals that, while asymmetric switching breaks time-reversal symmetry and modulates current magnitude, it does not by itself induce current reversal. The direction of transport emerges from the interplay between propulsion, thermal gradients, ratchet asymmetry, and load. We show that the system interpolates continuously between passive and active motor regimes: in the absence of propulsion, the efficiency is bounded by the Carnot limit; with propulsion, it can approach unity in the quasistatic limit. This work provides the first exact time-resolved solution of a discrete-state active ratchet model, offering quantitative insights into nonequilibrium energy conversion in systems that integrate active matter dynamics with passive rectification.