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
0I 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 .