1. Generalized Jarzynski Equality under Nonequilibrium Feedback
Takahiro Sagawa
University of Tokyo
Transmission 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 & Experiment
Topic of this talk
Thermodynamic system
Control parameter
Measurement outcome
Controller = Maxwell’s demon

4. 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)

5. Energy Transport Driven by Information Flow
Demon
1 bit
Nanomachine

6. 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.

7. 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

8. Feedback Protocol

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 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:

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!
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!

13. 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

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

15. 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

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

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

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 processes
and

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 law
2nd cumulant: a fluctuation-dissipation theorem
if the work distribution is Gaussian.

20. Without feedback
How about equality?
With feedback

21. Backward Processes
Without Switching
With Switching
© Dr. Toyabe

22. Backward Protocols
Forward:
© Dr. Toyabe

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

24. 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 .