Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University

1. Physical FluctuomaticsApplied Stochastic Process 7th “More is different” and “fluctuation” in physical models Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/ Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

2. Kazuyuki Tanaka: Introduction of Image Processing by Probabilistic Models, Morikita Publishing Co., Ltd., 2006 (in Japanese) , Chapter 5. Textbooks References • H. 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 Universe Community / Society Particle Physics Neutron Life Material Proton Substance Aomic Nucleus Condensed Matter Physics Electron Chemical Compound Molecule More is different Atom P. W. Anderson Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

4. Probabilistic Model for Ferromagnetic Materials Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

5. Probabilistic Model for Ferromagnetic Materials > = > Prior probability prefers to the configuration with the least number of red lines. Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

6. More is different in Probabilistic Model for Ferromagnetic Materials Large p Small p Sampling by Markov Chain Monte Carlo method Disordered State Ordered State Critical Point (Large fluctuation) More is different. Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

7. Model Representation in Statistical Physics Gibbs Distribution Partition Function Energy Function Free Energy Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

8. Fundamental Probabilistic Models for Magnetic Materials E：Set of All the neighbouring Pairs of Nodes h h J J Translational Symmetry Problem: Compute Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

9. Fundamental Probabilistic Models for Magnetic Materials h h J J Translational Symmetry Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

10. Fundamental Probabilistic Models for Magnetic Materials h h J J Translational Symmetry Spontaneous Magnetization Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

11. Finite System and Limit to Infinite System h h J J>0 When |V| is Finite, h h J When |V| is taken to the limit to infinity, J>0 Translational Symmetry Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

12. J J What happen in the limit to infinite Size System? h h J J>0 Spontaneous Magnetization Derivative with respect to J diverges Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

13. What happen in the limit to infinite Size System? h h J J>0 Translational Symmetry Fluctuations between the neighbouring pairs of nodes have a maximal point at J=0.4406….. J Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

14. What happen in the limit to infinite Size System? h h J J>0 Translational Symmetry J J : large J: small Disordered State Ordered State Including Large Fluctuations Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

15. What happen in the limit to infinite Size System? h h Fluctuations still remain even in large separations between pairs of nodes. J J>0 Translational Symmetry J : large J : small Ordered State Disordered State Near the critical point Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)

16. Summary • More is different • Probabilistic Model of Ferromagnetic Materials • Fluctuation in Covariance Physics Fluctuomatics / Applied Stochastic Process (Tohoku University)