kazuyuki tanaka graduate school of information sciences, tohoku university
DESCRIPTION
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 PresentationTRANSCRIPT
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