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Physical Fluctuomatics Applied Stochastic Process 7th “More is different” and “fluctuation” in physical models. Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University [email protected] http://www.smapip.is.tohoku.ac.jp/~kazu/. - PowerPoint PPT Presentation
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Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 1
Physical FluctuomaticsApplied Stochastic Process
7th “More is different” and “fluctuation” in physical models
Kazuyuki TanakaGraduate School of Information Sciences, Tohoku University
[email protected]://www.smapip.is.tohoku.ac.jp/~kazu/
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 2
Textbooks
Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) , Chapter 5.
ReferencesH. Nishimori: Statistical Physics of Spin Glasses and Information Processing, ---An Introduction, Oxford University Press, 2001. H. Nishimori, G. Ortiz: Elements of Phase Transitions and Critical Phenomena, Oxford University Press, 2011.M. Mezard, A. Montanari: Information, Physics, and Computation, Oxford University Press, 2010.
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 3
More is Different
Atom
Electron
Aomic Nucleus
ProtonNeutron
MoleculeChemical Compound
Substance
Life Material
Community / Society
UniverseParticle Physics
Condensed Matter Physics
More is differentP. W. Anderson
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 4
Probabilistic Model for Ferromagnetic MaterialsProbabilistic Model for
Ferromagnetic Materials
p p
p p
)1,1()1,1()1.1()1.1( PPPP
pPP )1.1()1,1(
11 a
1
12 a
1
11
1 1
p
PP
2
1
)1.1()1,1(
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 5
Probabilistic Model for Ferromagnetic MaterialsProbabilistic Model for
Ferromagnetic Materials
Prior probability prefers to the configuration with the least number of red lines.
> >=
Lines Red of #Lines Blue of # )2
1()( ppaP
p p
11 a 112 a 111 1 1
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 6
More is different in Probabilistic Model for Ferromagnetic Materials
Disordered State
Ordered State
Sampling by Markov Chain Monte Carlo method
p p
Small p Large p
p p
More is different.
p2
1p
2
1
Critical Point(Large fluctuation)
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 7
Model Representation in Statistical Physics
),,,(},,,Pr{ 212211 NNN aaaPaAaAaA
a
aEZ
))(exp(
)(}Pr{ aPaA
))(exp(1
)( aEZ
aP
),,,( 21 NAAAA
Gibbs Distribution Partition Function
)))(exp(ln(ln a
aEZF
Free Energy
Energy Function
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 8
Fundamental Probabilistic Models for Magnetic Materials
a
aEZ
))(exp(
Eji
jiVi
i aaJahaE},{
)(
Translational Symmetry
),( EVJ
J
h h
)(exp1
)( aEZ
aP
),,,( 21 Naaaa
E ： Set of All the neighbouring Pairs of Nodes
1ia 1ia
N
i ai aPa
Nm
1
)(1
Problem: Compute
)'()()'()( aPaPaEaE
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 9
Fundamental Probabilistic Models for Magnetic Materials
Eji
jiVi
i aaJahaE},{
)(
)(exp1
)( aEZ
aP
),,,( ||21 Vaaaa 1ia
Translational Symmetry
),( EV
J
J
h h
1 1 10
1 2 ||
)(lima a a
ih
i
V
aPam
1 1 10
1 2 ||
)())((lim],[Cova a a
jjiih
ji
V
aPmamaaa
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 10
Eji
jiVi
i aaJahaE},{
)(
ai
Vhii aPaam
)(limlim
|0
)(exp1
)( aEZ
aP
),,,( ||21 Vaaaa
1ia
Translational Symmetry
),( EVJ
J
h h
Spontaneous Magnetization
1 1 1||0
1 2 ||
)())((limlim],[Cova a a
jjiiVh
ji
V
aPmamaaa
Fundamental Probabilistic Models for Magnetic Materials
N
Eji },{
Vi
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 11
Finite System and Limit to Infinite System
Eji
jiVi
i aaJahaE},{
)(
)(exp1
)( aEZ
aP
1ia
),( EVJ
J>0
Translational Symmetry
h h
0)(lim)(lim00
a hi
ai
haPaaPa
When |V| is Finite,
a hi
N
ai
Nh
aPa
aPa
)(limlim
)(limlim
0
0
When |V| is taken to the limit to infinity,
),( EVJJ>0
h h
9|| V12|| E
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 12
What happen in the limit to infinite Size System?
Eji
jiVi
i aaJahaE},{
)(
)1)(sinh())(sinh1(
)1)(sinh(0
)(limlim
8/14
0
JJ
J
aPaaa
iNh
i
)(exp1
)( aEZ
aP
1ia ),( EVJ
J>0
h h
Spontaneous Magnetization
2/
0222
0
sin1)1)2(tanh2(2
1)2coth(
)(limlim
dkJJJ
aPaaaaa
jiNh
ji
J
Jk
2cosh
2tanh2
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
J
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
J
Derivative with respect to J diverges
Eji },{
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 13
What happen in the limit to infinite Size System?
Eji
jiVi
i aaJahaE},{
)(
)(exp1
)( aEZ
aP
1ia
),( EVJ
J>0
Translational Symmetry
h h
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
a
ijiiNh
ji
aPaaaa
aa
)())((limlim
],[Cov
0
J
Fluctuations between the neighbouring pairs of nodes have a maximal point at J=0.4406…..
Eji },{
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 14
What happen in the limit to infinite Size System?
),( EVJ
J>0
Translational Symmetry
h h
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
],[Cov ji aa
J
Eji },{
Disordered State Ordered StateIncluding Large Fluctuations
J: small J : large
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 15
What happen in the limit to infinite Size System?
),( EVJ
J>0
Translational Symmetry
h h
4/1|~|],[Cov jiji rraa
Disordered State Ordered StateNear the critical point
J : small J : large
/||
||
1~],[Cov ji rr
jiji e
rraa
|| ji rr
Fluctuations still remain even in large separations between pairs of nodes.
Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 16
Summary
More is different
Probabilistic Model of Ferromagnetic Materials
Fluctuation in Covariance
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