Brownian ratchet: In a periodic but asymmetric potential, thermal noise drives directed net motion — not violating thermodynamics because the system is maintained out of equilibrium (e.g., by ATP or flashing).
The asymmetric sawtooth potential has a steep left wall (α) and gentle right slope. Particles undergo overdamped Langevin dynamics: ẋ = −V'(x)/γ + √(2kT/γ) ξ(t).
Increase asymmetry and watch net flux emerge. At α=0.5 (symmetric), flux vanishes — no rectification.