  We derive a bound on generalized currents for Langevin systems in terms of
the total entropy production in the system and its environment. For overdamped
dynamics, any generalized current is bounded by the total rate of entropy
production. We show that this entropic bound on the magnitude of generalized
currents imposes power-efficiency tradeoff relations for ratchets in contact
with a heat bath: Maximum efficiency---Carnot efficiency for a
Smoluchowski-Feynman ratchet and unity for a flashing or rocking ratchet---can
only be reached at vanishing power output. For underdamped dynamics, while
there may be reversible currents that are not bounded by the entropy production
rate, we show that the output power and heat absorption rate are irreversible
currents and thus obey the same bound. As a consequence, a power-efficiency
tradeoff relation holds not only for underdamped ratchets but also for
periodically driven heat engines. For weak driving, the bound results in
additional constraints on the Onsager matrix beyond those imposed by the Second
Law. Finally, we discuss the connection between heat and entropy in a
non-thermal situation where the friction and noise intensity are
state-dependent.
