Revisiting the ratchet principle: Lessons from active and passive stochastic dynamics

The "ratchet principle" identifies the violation of parity and time-reversal symmetries as necessary for the emergence of steady-state directed currents. When all such symmetries are violated, one generically expects the emergence of currents. We study stochastic systems in the presence of asymmetric fluctuation sources, which violate these symmetries and yet fail to display steady currents. We show that this stems from a hidden conservation law for momentum. For underdamped and overdamped Brownian dynamics, we show that thermal fluctuations cannot power the momentum sources required to sustain directed currents, even when time-reversal symmetry is broken due to an inhomogeneous temperature field. While active Brownian and run-and-tumble particles display interaction-induced directed currents in asymmetric activity landscapes, we show that effective momentum conservation prevents this in Active Ornstein-Uhlenbeck particles: not all inhomogeneous active fluctuations can power transport. For each of the systems considered in this article, we numerically test for the emergence of interaction-induced directed currents. We then characterize time-reversal (a)symmetry in position space using a combination of path-integral and operator methods. When the existence of effective momentum conservation is ruled out, we develop perturbation theories to characterize the onset of interaction-induced directed currents.