Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 1 Physical Fluctuomatics Applied Stochastic Process 7th “More is different” and “fluctuation” in physical models Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University

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. References H. Nishimori: Statistical Physics of Spin Glasses and Information Processing, ---An Introduction, Oxford University Press, H. Nishimori, G. Ortiz: Elements of Phase Transitions and Critical Phenomena, Oxford University Press, 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 Proton Neutron Molecule Chemical Compound Substance Life Material Community / Society Universe Particle Physics Condensed Matter Physics More is different 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 Disordered State Ordered State Sampling by Markov Chain Monte Carlo method Small p Large p More is different. Critical Point (Large fluctuation)

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

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

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

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

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

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

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

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

Physics Fluctuomatics / Applied Stochastic Process (Tohoku University) 15 What happen in the limit to infinite Size System? J J>0 Translational Symmetry h h Disordered State Ordered State Near the critical point J : small J : large 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