takahiro sagawa university of tokyo generalized jarzynski equality under nonequilibrium feedback...
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Takahiro SagawaUniversity of Tokyo
Generalized Jarzynski Equality under Nonequilibrium Feedback
Transmission of Information and Energy in Nonlinear and Complex Systems 2010
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)
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).
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.
Energy Transport Driven by Information Flow
B ln 2k TB ln 2k T
Demon
Nanomachine
1 bit
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
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
Feedback Protocol
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.
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.
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.
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.
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
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
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).
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
( , ) 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
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.
How about equality?
?
exp( ( )) 1W F W F
Without feedback
BW F k TI
With feedback
Backward Processes
© Dr. Toyabe
With Switching
Without Switching
Backward Protocols
Forward: © Dr. Toyabe
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
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
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