Generalized Jarzynski Equality under Nonequilibrium Feedback Takahiro Sagawa University of Tokyo Transmission of Information and Energy in Nonlinear and Complex Systems 2010 Collaborators on thermodynamics of information processing E. Muneyuki (Chuo Univ.) M. Ueda (Univ. Tokyo) M. Sano (Univ. Tokyo) S. Toyabe (Chuo Univ.) S. W. Kim (Pusan National Univ. ) Theory: S. D. Liberato (Univ. Tokyo) TS and M. Ueda, Phys. Rev. Lett. 100, 080403 (2008) . TS and M. Ueda, Phys. Rev. Lett. 102, 250602 (2009). TS and M. Ueda, Phys. Rev. Lett. 104, 090602 (2010). S. W. Kim, TS, S. D. Liberato, and M. Ueda, arXiv: 1006.1471. Experiment: S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted. 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). Topic of this talk Second law of thermodynamics with feedback control Theory & Experiment Control parameter Thermodynamic system Controller = Maxwell’s demon Measurement outcome Szilard Engine: Energetic Maxwell’s Demon Initial State kBT ln 2 Partition Heat bath Which? T Measurement Isothermal, quasi-static expansion ln 2 Left Work Information Feedback Right Does the demon contradict the second law? No! Energy cost is needed for the demon itself. L. Szilard, Z. Phys. 53, 840 (1929) Energy Transport Driven by Information Flow Demon 1 bit Nanomachine kBT ln 2 kBT ln 2 TS and M. Ueda, PRL 100, 080403 (2008) Fundamental Limit of Demon’s Capability Information Mutual information I Feedback Work Heat bath Engine F With feedback control Shannon information 0I H Wext No information Error-free Wext F kBTI We have generalized the second law of thermodynamics, in which information contents and thermodynamic variables are treated on an equal footing. Experiment How to realize the Szilard-type Maxwell’s demon? Relevant energy is extremely small: Order of 0.1 kBT To create a clean potential is crucial. A-D: electrodes a 287nm polystyrene bead Realized a spiral-stairs-like potential Feedback Protocol Experimental Results (1) Conversion rate from information to energy is about 28%. F W 0.062kBT I 0.22 F W kBTI 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. Generalized Jarzynski Equality Jarzynski equality e (W F ) 1 W : work F : free-energy difference C. Jarzynski, PRL 78, 2690 (1997) With feedback control e (W F ) TS and M. Ueda, PRL 104, 090602 (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: 1 Szilard engine: ln 2 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. Summary • Fundamental bound of demon’s capacity TS and M. Ueda, PRL 100, 080403 (2008) F W kBTI • Generalized Jarzynski equality with feedback (W F ) TS and M. Ueda, PRL 104, 090602 (2010) e • Experimental realization of a Szilard-type Maxwell’s demon and verification of the equality – “Information- energy conversion” driven by feedback S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted. Thank you for your attention! Thermodynamics of Information Processing • • • • • Maxwell (1871) Szilard (1929) Brillouin (1951) Landauer (1961) Bennett (1982) Topic of this talk Second law of thermodynamics with feedback control Maxwell’s Demon is a Feedback Controller External parameter Thermodynamic system Measurement outcome y Controller Control protocol (t ) can depend on measurement outcomes y as (t ; y ) . 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. Information System Demon Feedback Motivation: Fluctuating Nanomachines Rahav, Horowitz & Jarzynski, PRL (2008) Chernyak & Sinitsyn, PRL (2008) Future Prospects • Quantum Regime • Controlling Bio-/Artificial Nanomachines • Information Thermodynamics in Biology Stochastic Thermodynamics: Setup Classical stochastic dynamics from time 0 to in contact with a heat bath at temperature (kBT )1 (t ) : control protocol of external parameters (volume of the gas etc.) † (t ) ( t ) : time-reversed protocol (r , p) : phase-space point * (r , p) : time-reversal (t ) : trajectory † (t ) * ( t ) : time-reversal P (t ) [(t )] and P † (t ) [† (t )] : probability densities of the forward and backward processes Jarzynski Equality (1997) e W e F C. Jarzynski, PRL 78, 2690 (1997) L.H.S. has the information of all cumulants: 1st cumulant: the second law W F 2nd cumulant: a fluctuation-dissipation theorem W F ( W 2 W 2 ) if the work distribution is Gaussian. Without feedback W F exp( (W F )) 1 How about equality? With feedback W F kBTI ? Backward Processes Without Switching With Switching © Dr. Toyabe Backward Protocols Forward: © Dr. Toyabe Example: Szilard Engine Free-energy difference: F 0 Extracted work: kBT ln 2 P† (t ;L) (L) P† (t ;R ) (R) 1 2 Generalized Jarzynski equality is satisfied: exp( (kBT ln 2 0)) 2 Corollaries exp( (W F )) 1st cumulant: the second law W F ln 2nd cumulant: a fluctuation-dissipation theorem W F ( W 2 W 2 ) ln if the work distribution is Gaussian. Note: the relationship between and I is complicated, because involves the high-order cumulants of I .