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