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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 δ, this current is proportional to sinδ.
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Blanter et al. (1998) studied this question.
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