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Information Thermodynamics on Causal Networks
Takahiro Sagawa Department of Basic Science, University of Tokyo
In collaboration with
Sosuke Ito Department of Physics, University of Tokyo
The 6th KIAS Conference on Statistical Physics “Nonequilibrium Statistical Physics of Complex Systems” (NSPCS14)
10 July 2014, Seoul, Korea
Collaborators on Information Thermodynamics
• Masahito Ueda (Univ. Tokyo)
• Shoichi Toyabe (LMU Munich)
• Eiro Muneyuki (Chuo Univ.)
• Masaki Sano (Univ. Tokyo)
• Sosuke Ito (Univ. Tokyo)
• Naoto Shiraishi (Univ. Tokyo)
• Sang Wook Kim (Pusan National Univ.)
• Jung Jun Park (National Univ. Singapore)
• Kang-Hwan Kim (KAIST)
• Simone De Liberato (Univ. Paris VII)
• Juan M. R. Parrondo (Univ. Madrid)
• Jordan M. Horowitz (Univ. Massachusetts)
• Jukka Pekola (Aalto Univ.)
• Jonne Koski (Aalto Univ.)
• Ville Maisi (Aalto Univ.)
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
Information Thermodynamics
Information processing at the level of thermal fluctuations
Foundation of the second law of thermodynamics
Application to nanomachines and nanodevices
System Demon
Information
Feedback
Szilard Engine (1929)
Heat bath
T
Initial State Which? Partition
Measurement
Left
Right
Feedback
ln 2
F E TS Free energy: Decrease by feedback Increase
Isothermal, quasi-static expansion B ln 2k T
Work
Can control physical entropy by using information
L. Szilard, Z. Phys. 53, 840 (1929)
Experimental Realizations
• With a colloidal particle Toyabe, TS, Ueda, Muneyuki, & Sano, Nature Physics (2010)
Efficiency: 30% Validation of
• With a single electron Koski, Maisi, TS, & Pekola, to appear in PRL (2014)
Efficiency: 75% Validation of
( )W Fe
( ) 1W F Ie
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
Shannon Information
9
10
1
10
Information content with event : 1
lnkp
k
Shannon information: 1
lnk
k k
H pp
Average
Mutual Information
0 ( )I H M
No information No error
System S Memory M I
( : ) ( ) ( ) ( )I S M H S H M H SM
System S Memory M (measurement device)
Measurement with stochastic errors
0
1
0
1
1
1
Ex. Binary symmetric channel
Correlation between S and M
I
)1ln()1(ln2ln I
Entropy Production
System S
Heat bath B
(inverse temperature β)
Heat Q
Entropy production in the total system: QSS SSB
Change in the Shannon entropy of S
If the initial and the final states are canonical distributions: FWS SB
Free-energy difference
W Work
Averaged heat absorbed by S
Stochastic dynamics of system S (e.g., Langevin system)
Stochastic Entropy Production
Qss SSB
Stochastic entropy production along a trajectory of the system from time 0 to τ
System (phase-space point x)
Heat bath (inverse temperature β)
Q
If the initial and the final states are canonical distributions: FWs SB
W Work
SS Ss
],[ln],[S txPtxs ]0),0([]),([ SSS xsxss
],[ txP : probability distribution at time t
Fluctuation Theorem and Second Law
1SB s
e
Integral fluctuation theorem (Jarzynski equality)
for any initial and final distributions
0SB s
The second law of thermodynamics (Clausius inequality)
QS S FW
Jarzynski, PRL (1997), Seifert, PRL (2005), …
Second law can be expressed by an equality with full cumulants
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
Our Setup
System Y Heat bath B System X Heat bath B
Time evolution of X under the influence of Y with initial and final correlations
X and Y interact with each other only through information exchange
Special Cases: Measurement and Feedback
Measurement Feedback
X: Engine Y: Memory
X: Memory Y: Engine
Stochastic Entropy Production & Mutual Information
XXXB Qss
Entropy production in XB
][]'[
],'[ln],'[''
yPxP
yxPyxdyPdxI f
yx
][][
],[ln],[
yPxP
yxPyxdxdyPI i
xy][][
],[ln
yPxP
yxPI i
xy
Initial correlation
][]'[
],'[ln'
yPxP
yxPI f
yx
Final correlation
General Result
i
xy
f
yx III '
1XB Is
e
Integral fluctuation theorem:
Is XBGeneralized second law:
TS and M. Ueda, PRL 109, 180602 (2012).
Special Case 1: Feedback Control
III rem
1)( remXB
IIse
Feedback: Control protocol depends on the measurement outcome
(Upper bound of) the correlation that is used by feedback
X: Engine Y: Memory
remXB IIs
ITkFW Bext
Special Case 2: Measurement
II
1XB Is
e
X: Memory Y: Engine
Is XB
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
Generalization to Many-body Systems with Complex Information Flow
Sosuke Ito & TS, PRL 111, 180603 (2013).
Characterize the dynamics by Bayesian networks
Bayesian Networks
0x
x2
1x
x1
x2
y1
y0
z0
z1
x0
Node: Event Arrow: Causal relationship
Parents of node x (denoted by “pa(x)”): The set of nodes that have arrows to x
Bayesian Networks
0x
x2
1x
pa(x2 ) = {x1, x0}
x1
x2
y1
y0
z0
z1
x0
pa(x1) = {x0}
pa(x2 ) = {x1, y0}
pa(x1) = {x0, y0}
pa(y1) = {x1, x0, y0, z0}
pa(x0 ) = Æ(Empty set)
1m
2x
1x
),|()|()( 112111 xmxpxmpxp
Simplest Case:
Measurement and Feedback
The path probability
Measurement
Feedback Time
evolution
under
feedback
control
Szilard engine
1x
2x
1m
1x
2x
3x
)( idttxxi )( idttyyi
)( idttzzi
zz
yy
xx
xzfdt
dz
zyfdt
dy
yxfdt
dx
,
,
,
xX (t) = 0,
xX t( )xX '(t ') =DXdXX 'd(t - t ')1y
3y
1z
2z
3z
2y
Multidimensional Langevin Dynamics
X = x, y, z
xi+1 = xi + fx xi, yi( )dt +xxdt
Main Result:
Fluctuation theorem on Causal Networks
,1exp X X
l
lIII trinifin
Iini : Initial correlation between the target system and other systems
Ifin : Final correlation between the system and others
Itrl: Information transfer from the system to others during the dynamics
Iini: Initial correlation
xN
pa(x1)x1
Iini º lnp(x1, pa(x1))
p(x1)p(pa(x1))
Iini : Initial correlation between the system and other systems
Ifin: Final correlation
xN-1
xN
C
x1
cN '
cN '-1cN '-2
cN '-3 cN '-4
Ifin º lnp(xN ,C)
p(xN )p(C)
C is the set of
random variables
that effect on the final state
from outside of X.
Ifin : Final correlation between the system and others
Nj xxaaC ,,\,, 11
a j = xNwith
↓Topological ordering
Iltr : Information transfer
xN-1
xN
C
x1
cN '
cN '-2
cN '-3cN '-4
),,|pa(),,|(
),,|,pa(ln
1111
11tr
cccpcccp
ccccpI
llXll
lllXl
C = cl 1£ l £ N '{ }
),,1( Nl
Itr : Information transfer from X into cl during the dynamics
This term is equivalent to
Transfer entropy [T. Schreiber, PRL 85, 461, 2000.]
NllX xxcc ,,)(pa)(pa 1
↑Topological ordering
cN '-1
Toward Application to Biochemical Signal Transduction
Information thermodynamics gives a useful bound; stronger than the usual second law!
Ito & TS, arXiv:1406.5810
Analogy with communication theory
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
Summary
• Unified framework of information thermodynamics
– Fluctuation theorem for information exchanges
• Generalization to causal networks
– Entropy transfer (information flow) is crucial
• Toward application to signal transduction
Thank you for your attention!
S. Ito & T. Sagawa, arXiv:1406.5810.
S. Ito & T. Sagawa, PRL 111, 180603 (2013).
T. Sagawa & M. Ueda, PRL 109, 180602 (2012).