Rectification of Fluctuations in an Underdamped Ratchet

We investigate analytically the motion of underdamped particles subject to a deterministic periodic potential and a periodic temperature. Despite the fact that an underdamped particle experiences the temperature modulation many times in its escape out of a well and in its motion along the potential, a net directed current linear in the friction constant is found. If both the potential and the temperature modulation are sinusoidal with a phase lag $\delta$, this current is proportional to $\sin \delta$.