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**Generalized Jarzynski Equality under Nonequilibrium Feedback**

Takahiro Sagawa University of Tokyo Transmission of Information and Energy in Nonlinear and Complex Systems 2010

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**Collaborators on thermodynamics of information processing**

E. Muneyuki (Chuo Univ.) M. Ueda (Univ. Tokyo) M. Sano (Univ. Tokyo) S. Toyabe (Chuo Univ.) S. D. Liberato (Univ. Tokyo) S. W. Kim (Pusan National Univ. ) 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: Experiment: S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.

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**Brownian Motors and Maxwell’s Demons**

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

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**Szilard Engine: Energetic Maxwell’s Demon**

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

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**Energy Transport Driven by Information Flow**

Demon 1 bit Nanomachine

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**Fundamental Limit of Demon’s Capability**

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

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**Experiment How to realize the Szilard-type Maxwell’s demon?**

Relevant energy is extremely small: Order of To create a clean potential is crucial. 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)**

Conversion rate from information to energy is about 28%. Extracted more work than the conventional bound. Observed the “information-energy conversion” driven by Maxwell’s demon. S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.

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**Generalized Jarzynski Equality**

: work : free-energy difference C. Jarzynski, PRL 78, 2690 (1997) With feedback control TS and M. Ueda, PRL 104, (2010) characterizes the efficacy of feedback control. 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. Without feedback: Szilard engine:

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**Experimental Results (2)**

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

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**Thank you for your attention!**

Summary Fundamental bound of demon’s capacity Generalized Jarzynski equality with feedback Experimental realization of a Szilard-type Maxwell’s demon and verification of the equality “Information- energy conversion” driven by feedback 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. Thank you for your attention!

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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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**Maxwell’s Demon is a Feedback Controller**

Measurement outcome External parameter Thermodynamic system Controller Control protocol can depend on measurement outcomes as .

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

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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**

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

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**Jarzynski Equality (1997) L.H.S. has the information of all cumulants:**

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

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Without feedback How about equality? With feedback

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Backward Processes Without Switching With Switching © Dr. Toyabe

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Backward Protocols Forward: © Dr. Toyabe

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**Example: Szilard Engine**

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

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**Corollaries 1st cumulant: the second law**

2nd 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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