Efficiency at maximum power of a discrete feedback ratchet

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Efficiency at maximum power is found to be of the same order for a feedback ratchet and for its open-loop counterpart. However, feedback increases the output power up to a factor of five. This increase in output power is due to the increase in energy input and the effective entropy reduction obtained as a consequence of feedback. Optimal efficiency at maximum power is reached for time intervals between feedback actions two orders of magnitude smaller than the characteristic time of diffusion over a ratchet period length. The efficiency is computed consistently taking into account the correlation between the control actions. We consider a feedback control protocol for a discrete feedback flashing ratchet, which works against an external load. We maximize the power output optimizing the parameters of the ratchet, the controller, and the external load. The maximum power output is found to be upper bounded, so the attainable extracted power is limited. After, we compute an upper bound for the efficiency of this isothermal feedback ratchet at maximum power output. We make this computation applying recent developments of the thermodynamics of feedback-controlled systems, which give an equation to compute the entropy reduction due to information. However, this equation requires the computation of the probability of each of the possible sequences of the controller's actions. This computation becomes involved when the sequence of the controller's actions is non-Markovian, as is the case in most feedback ratchets. We here introduce an alternative procedure to set strong bounds to the entropy reduction in order to compute its value. In this procedure the bounds are evaluated in a quasi-Markovian limit, which emerge when there are big differences between the stationary probabilities of the system states. These big differences are an effect of the potential strength, which minimizes the departures from the Markovianicity of the sequence of control actions, allowing also to minimize the departures from the optimal performance of the system. This procedure can be applied to other feedback ratchets and, more in general, to other control systems.
©2016 American Physical Society. We would like to thank Hugo Touchette for comments at the beginning of the redaction of this paper and for stimulating discussions. We also acknowledge Juan Pedro Garcia-Villaluenga for a critical reading of the final version of the manuscript. This work has been supported by the Projects No. GR35/14-920911 (Banco Santander and Universidad Complutense de Madrid, Spain) and No. FIS2010-17440 (Ministerio de Economia y Competitividad, Spain). J.J. also acknowledges the financial support through Grant No. FPU-13/02934 (Ministerio de Educacion, Cultura y Deportes, Spain).
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