takahiro sagawa university of tokyo generalized jarzynski equality under nonequilibrium feedback...
TRANSCRIPT
![Page 1: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/1.jpg)
Takahiro SagawaUniversity of Tokyo
Generalized Jarzynski Equality under Nonequilibrium Feedback
Transmission of Information and Energy in Nonlinear and Complex Systems 2010
![Page 2: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/2.jpg)
Collaboratorson thermodynamics of information processing
Theory: 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.
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)
![Page 3: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/3.jpg)
Brownian Motors and Maxwell’s Demons
Thermodynamic systemControl parameter
Measurement outcome
Controller = Maxwell’s 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).
![Page 4: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/4.jpg)
Szilard Engine:Energetic Maxwell’s Demon
Heat bath
TInitial State Which?Partition
Measurement
Left
Right
Feedback
L. Szilard, Z. Phys. 53, 840 (1929)
Isothermal,quasi-static expansionB ln 2k T
Work
ln 2
Information
Does the demon contradict the second law?
No! Energy cost is needed for the demon itself.
![Page 5: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/5.jpg)
Energy Transport Driven by Information Flow
B ln 2k TB ln 2k T
Demon
Nanomachine
1 bit
![Page 6: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/6.jpg)
Fundamental Limit of Demon’s Capability
EngineHeat bath
FextW
Work
ext BW F k TI With feedback control
TS and M. Ueda, PRL 100, 080403 (2008)
IInformation
Feedback
We have generalized the second law of thermodynamics, in which information contents and thermodynamic variables are treated on an equal footing.
0 I H No information Error-free
Mutual information
Shannon information
![Page 7: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/7.jpg)
Experiment
Relevant energy is extremely small: Order of B0.1 k T
To create a clean potential is crucial.
How to realize the Szilard-type Maxwell’s demon?
A-D: electrodesa 287nm polystyrene bead
Realized a spiral-stairs-like potential
![Page 8: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/8.jpg)
Feedback Protocol
![Page 9: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/9.jpg)
Experimental Results (1)
Observed the “information-energy conversion” driven by Maxwell’s demon.
S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.
Conversion rate from information to energy is about 28%.
0.22I B0.062F W k T
BF W k TI
Extracted more work than the conventional bound.
![Page 10: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/10.jpg)
Generalized Jarzynski Equality
1 Without feedback:
characterizes the efficacy of feedback control.
( ) 1W Fe C. Jarzynski, PRL 78, 2690 (1997)
W : work
F : free-energy difference
Jarzynski equality
( )W Fe With feedback control
TS and M. Ueda, PRL 104, 090602 (2010)
ln 2 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.
![Page 11: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/11.jpg)
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.
![Page 12: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/12.jpg)
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
( )W Fe
BF W k TI
Thank you for your attention!
TS and M. Ueda, PRL 100, 080403 (2008)
TS and M. Ueda, PRL 104, 090602 (2010)
S. Toyabe, TS, M. Ueda, E. Muneyuki, and M. Sano, submitted.
![Page 13: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/13.jpg)
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
![Page 14: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/14.jpg)
Maxwell’s Demon is aFeedback Controller
Control protocol can depend on measurement outcomes as .
( )t y ( ; )t y
Thermodynamic systemExternal parameter y
Measurement outcome
Controller
![Page 15: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/15.jpg)
Maxwell’s Demon
System Demon
Information
Feedback
By using the “information” obtained by the measurement, “Maxwell’s demon” can violate the second law on average.
J. C. Maxwell, “Theory of Heat” (1871).
![Page 16: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/16.jpg)
Motivation: Fluctuating Nanomachines
Rahav, Horowitz & Jarzynski, PRL (2008)Chernyak & Sinitsyn, PRL (2008)
![Page 17: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/17.jpg)
Future Prospects
• Quantum Regime
• Controlling Bio-/Artificial Nanomachines
• Information Thermodynamics in Biology
![Page 18: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/18.jpg)
Stochastic Thermodynamics: Setup
( , ) r p : phase-space point* ( , ) r p : time-reversal( )t : trajectory† *( ) ( )t t : time-reversal
( )[ ( )]tP t : probability densities of the forward and backward processes
†
†
( )[ ( )]
tP t
and
( )t : control protocol of external parameters (volume of the gas etc.)† ( ) ( )t t : time-reversed protocol
Classical stochastic dynamics from time to in contact with a heat bath at temperature 1
B( )k T 0
![Page 19: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/19.jpg)
Jarzynski Equality (1997)
W F1st cumulant: the second law
W Fe e
L.H.S. has the information of all cumulants:
C. Jarzynski, PRL 78, 2690 (1997)
22( )W F W W 2nd cumulant: a fluctuation-dissipation theorem
if the work distribution is Gaussian.
![Page 20: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/20.jpg)
How about equality?
?
exp( ( )) 1W F W F
Without feedback
BW F k TI
With feedback
![Page 21: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/21.jpg)
Backward Processes
© Dr. Toyabe
With Switching
Without Switching
![Page 22: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/22.jpg)
Backward Protocols
Forward: © Dr. Toyabe
![Page 23: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/23.jpg)
B ln 2k T
Extracted work:
Bexp( ( ln 2 0)) 2k T Generalized Jarzynski equality is satisfied:
† †( ;L) ( ;R )(L) (R) 1
t tP P
2
0F
Free-energy difference:
Example: Szilard Engine
![Page 24: Takahiro Sagawa University of Tokyo Generalized Jarzynski Equality under Nonequilibrium Feedback Transmission of Information and Energy in Nonlinear and](https://reader034.vdocuments.us/reader034/viewer/2022051614/551739a155034603568b60bd/html5/thumbnails/24.jpg)
Corollaries
exp( ( ))W F
lnW F 1st cumulant: the second law
22( ) lnW F W W 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 .
I I