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Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and Complex Systems 2010

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Collaborators on thermodynamics of information processing Theory: TS and M. Ueda, Phys. Rev. Lett. 100, (2008). TS and M. Ueda, Phys. Rev. Lett. 102, (2009). TS and M. Ueda, Phys. Rev. Lett. 104, (2010). S. W. Kim, TS, S. D. Liberato, and M. Ueda, arXiv: S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.Experiment: S. Toyabe (Chuo Univ.) M. Ueda (Univ. Tokyo)M. Sano (Univ. Tokyo) E. Muneyuki (Chuo Univ.) S. W. Kim (Pusan National Univ. ) S. D. Liberato (Univ. Tokyo)

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Brownian Motors and Maxwells Demons Thermodynamic system Control parameter Measurement outcome Controller = Maxwells demon Second law of thermodynamics with feedback control Theory & Experiment Topic of this talk P. Hänggi and F. Marchesoni, Rev. Mod. Phys. 81, 387 (2009). K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009).

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Szilard Engine: Energetic Maxwells Demon Heat bath T Initial State Which? Partition Measurement Left Right Feedback L. Szilard, Z. Phys. 53, 840 (1929) Isothermal, quasi-static expansion Work Information Does the demon contradict the second law? No! Energy cost is needed for the demon itself.

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Energy Transport Driven by Information Flow Demon Nanomachine 1 bit

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Fundamental Limit of Demons Capability Engine Heat bath Work With feedback control TS and M. Ueda, PRL 100, (2008) Information Feedback We have generalized the second law of thermodynamics, in which information contents and thermodynamic variables are treated on an equal footing. No information Error-free Mutual information Shannon information

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Experiment Relevant energy is extremely small: Order of To create a clean potential is crucial. How to realize the Szilard-type Maxwells demon? A-D: electrodes a 287nm polystyrene bead Realized a spiral-stairs-like potential

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Feedback Protocol

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Experimental Results (1) Observed the information-energy conversion driven by Maxwells demon. S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted. Conversion rate from information to energy is about 28%. Extracted more work than the conventional bound.

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Generalized Jarzynski Equality Without feedback: characterizes the efficacy of feedback control. C. Jarzynski, PRL 78, 2690 (1997) : work : free-energy difference Jarzynski equality With feedback control TS and M. Ueda, PRL 104, (2010) Szilard engine: It can be defined independently of L.H.S. The sum of the probabilities of obtaining time-reversed outcomes with the time-reversed control protocol.

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Experimental Results (2) Generalized Jarzynski equality is satisfied. Original Jarzynski equality is violated only in the higher cumulants. S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.

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Summary Fundamental bound of demons capacity Generalized Jarzynski equality with feedback Experimental realization of a Szilard-type Maxwells demon and verification of the equality – Information- energy conversion driven by feedback Thank you for your attention! TS and M. Ueda, PRL 100, (2008) TS and M. Ueda, PRL 104, (2010) S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.

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Thermodynamics of Information Processing Maxwell (1871) Szilard (1929) Brillouin (1951) Landauer (1961) Bennett (1982) Second law of thermodynamics with feedback control Topic of this talk

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Maxwells Demon is a Feedback Controller Control protocol can depend on measurement outcomes as. Thermodynamic system External parameter Measurement outcome Controller

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Maxwells Demon SystemDemon Information Feedback By using the information obtained by the measurement, Maxwells demon can violate the second law on average. J. C. Maxwell, Theory of Heat (1871).

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Motivation: Fluctuating Nanomachines Rahav, Horowitz & Jarzynski, PRL (2008) Chernyak & Sinitsyn, PRL (2008)

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Future Prospects Quantum Regime Controlling Bio-/Artificial Nanomachines Information Thermodynamics in Biology

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Stochastic Thermodynamics: Setup : phase-space point : time-reversal : trajectory : time-reversal : probability densities of the forward and backward processes and : control protocol of external parameters (volume of the gas etc.) : time-reversed protocol Classical stochastic dynamics from time to in contact with a heat bath at temperature

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Jarzynski Equality (1997) 1 st cumulant: the second law L.H.S. has the information of all cumulants: C. Jarzynski, PRL 78, 2690 (1997) 2 nd cumulant: a fluctuation-dissipation theorem if the work distribution is Gaussian.

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How about equality? Without feedback With feedback

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Backward Processes © Dr. Toyabe With Switching Without Switching

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Backward Protocols Forward: © Dr. Toyabe

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Extracted work: Generalized Jarzynski equality is satisfied: Free-energy difference: Example: Szilard Engine

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Corollaries 1 st cumulant: the second law 2 nd cumulant: a fluctuation-dissipation theorem if the work distribution is Gaussian. Note: the relationship between and is complicated, because involves the high-order cumulants of.

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