Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 1 Physical Fluctuomatics Applied Stochastic Process 7th “More is different” and.

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