Presentation on theme: "Generalized Jarzynski Equality under Nonequilibrium Feedback"— Presentation transcript:
1 Generalized Jarzynski Equality under Nonequilibrium Feedback Takahiro SagawaUniversity of TokyoTransmission of Information and Energy in Nonlinear and Complex Systems 2010
2 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.
3 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 & ExperimentTopic of this talkThermodynamic systemControl parameterMeasurement outcomeController = Maxwell’s demon
4 Szilard Engine: Energetic Maxwell’s Demon Heat bath TWhich?PartitionInitial StateInformationIsothermal,quasi-staticexpansionWorkMeasurementLeftRightFeedbackDoes the demon contradict the second law?No! Energy cost is needed for the demon itself.L. Szilard, Z. Phys. 53, 840 (1929)
5 Energy Transport Driven by Information Flow Demon1 bitNanomachine
6 Fundamental Limit of Demon’s Capability TS and M. Ueda, PRL 100, (2008)Fundamental Limit of Demon’s CapabilityInformationFeedbackNo informationError-freeMutual informationShannon informationWorkHeat bathEngineWith feedback controlWe have generalized the second law of thermodynamics, in which information contents and thermodynamic variables are treated on an equal footing.
7 Experiment How to realize the Szilard-type Maxwell’s demon? Relevant energy is extremely small: Order ofTo create a clean potential is crucial.A-D: electrodesa 287nm polystyrene beadRealized a spiral-stairs-like potential
9 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.
10 Generalized Jarzynski Equality : work: free-energy differenceC. Jarzynski, PRL 78, 2690 (1997)With feedback controlTS 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:
11 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.
12 Thank you for your attention! SummaryFundamental bound of demon’s capacityGeneralized Jarzynski equality with feedbackExperimental realization of a Szilard-type Maxwell’s demon and verification of the equality“Information- energy conversion” driven by feedbackTS 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!
13 Thermodynamics of Information Processing Maxwell (1871)Szilard (1929)Brillouin (1951)Landauer (1961)Bennett (1982)Second law of thermodynamics with feedback controlTopic of this talk
14 Maxwell’s Demon is a Feedback Controller Measurement outcomeExternal parameterThermodynamic systemControllerControl protocol can depend on measurement outcomes as .
15 Maxwell’s DemonJ. C. Maxwell, “Theory of Heat” (1871).By using the “information” obtained by the measurement, “Maxwell’s demon” can violate the second law on average.SystemDemonInformationFeedback
16 Motivation: Fluctuating Nanomachines Rahav, Horowitz & Jarzynski, PRL (2008)Chernyak & Sinitsyn, PRL (2008)
18 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 processesand
19 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 law2nd cumulant: a fluctuation-dissipation theoremif the work distribution is Gaussian.
20 Without feedbackHow about equality?With feedback
24 Corollaries 1st cumulant: the second law 2nd cumulant: a fluctuation-dissipation theoremif the work distribution is Gaussian.Note: the relationship between and is complicated, because involves the high-order cumulants of .