  Quantifying energy flows at nanometer scales promises to guide future
research in a variety of disciplines, from microscopic control and
manipulation, to autonomously operating molecular machines. A general
understanding of the thermodynamic costs of nonequilibrium processes would
illuminate the design principles for efficient microscopic machines.
Considerable effort has gone into finding and classifying the deterministic
control protocols that drive a system rapidly between states at minimum
energetic cost. But for autonomous microscopic systems, driving processes are
themselves stochastic. Here we generalize a linear-response framework to
incorporate such protocol variability, deriving a lower bound on the work that
is realized at finite protocol duration, far from the quasistatic limit. Our
findings are confirmed in model systems. This theory provides a thermodynamic
rationale for rapid operation, independent of functional incentives.
