Lectures in Milano University Hiroyuki Sagawa, Univeristy of Aizu March, 2008
Information Thermodynamics on Causal...
Transcript of Information Thermodynamics on Causal...
![Page 1: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/1.jpg)
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
![Page 2: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/2.jpg)
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.)
![Page 3: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/3.jpg)
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
![Page 4: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/4.jpg)
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
![Page 5: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/5.jpg)
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)
![Page 6: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/6.jpg)
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
![Page 7: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/7.jpg)
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
![Page 8: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/8.jpg)
Shannon Information
9
10
1
10
Information content with event : 1
lnkp
k
Shannon information: 1
lnk
k k
H pp
Average
![Page 9: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/9.jpg)
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
![Page 10: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/10.jpg)
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)
![Page 11: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/11.jpg)
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
![Page 12: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/12.jpg)
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
![Page 13: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/13.jpg)
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
![Page 14: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/14.jpg)
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
![Page 15: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/15.jpg)
Special Cases: Measurement and Feedback
Measurement Feedback
X: Engine Y: Memory
X: Memory Y: Engine
![Page 16: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/16.jpg)
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
![Page 17: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/17.jpg)
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).
![Page 18: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/18.jpg)
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
![Page 19: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/19.jpg)
Special Case 2: Measurement
II
1XB Is
e
X: Memory Y: Engine
Is XB
![Page 20: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/20.jpg)
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
![Page 21: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/21.jpg)
Generalization to Many-body Systems with Complex Information Flow
Sosuke Ito & TS, PRL 111, 180603 (2013).
Characterize the dynamics by Bayesian networks
![Page 22: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/22.jpg)
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
![Page 23: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/23.jpg)
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)
![Page 24: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/24.jpg)
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
![Page 25: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/25.jpg)
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
![Page 26: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/26.jpg)
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
![Page 27: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/27.jpg)
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
![Page 28: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/28.jpg)
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
![Page 29: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/29.jpg)
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
![Page 30: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/30.jpg)
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
![Page 31: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/31.jpg)
Outline • Introduction
• Information and Entropy
• Second Law with Measurement and Feedback
• Information Thermodynamics on Causal Networks
• Summary
![Page 32: Information Thermodynamics on Causal Networkshome.kias.re.kr/psec/nspcs14/강연자료/17_Takahiro_Sagawa.pdf · Information Thermodynamics on Causal Networks Takahiro Sagawa Department](https://reader035.fdocuments.in/reader035/viewer/2022081613/5f92007a290070032b559e93/html5/thumbnails/32.jpg)
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).