Adaptive Cooperative Systems Chapter 3 Coperative Lattice Systems

Slides:



Advertisements
Similar presentations
Introduction Landau Theory Many phase transitions exhibit similar behaviors: critical temperature, order parameter… Can one find a rather simple unifying.
Advertisements

Ising model in the zeroth approximation Done by Ghassan M. Masa’deh.
Metals: Bonding, Conductivity, and Magnetism (Ch. 6)
Ch 11. Sampling Models Pattern Recognition and Machine Learning, C. M. Bishop, Summarized by I.-H. Lee Biointelligence Laboratory, Seoul National.
The Markov property Discrete time: A time symmetric version: A more general version: Let A be a set of indices >k, B a set of indices
Entanglement in Quantum Critical Phenomena, Holography and Gravity Dmitri V. Fursaev Joint Institute for Nuclear Research Dubna, RUSSIA Banff, July 31,
Image Segmentation Using Physical Models Yuliya Kopylova CS 867 Computer Vision.
Content Origins of Magnetism Kinds of Magnetism Susceptibility and magnetization of substances.
Monte Carlo Simulation of Ising Model and Phase Transition Studies
Phase Transformations: – many different kinds of phase transitions: dimension, microscopic origin… – cooperativity (dominos’ effect) play a key role Example:
Chap.3 A Tour through Critical Phenomena Youjin Deng
Magnetism III: Magnetic Ordering
The Ising Model of Ferromagnetism by Lukasz Koscielski Chem 444 Fall 2006.
Carbon Nanorings, Lattice Gross-Neveu models of Polyacetylene and the Stability of Quantum Information Michael McGuigan Brookhaven National Laboratory.
Presentation in course Advanced Solid State Physics By Michael Heß
Monte Carlo Simulation of Ising Model and Phase Transition Studies By Gelman Evgenii.
Physics Fluctuomatics (Tohoku University) 1 Physical Fluctuomatics 7th~10th Belief propagation Appendix Kazuyuki Tanaka Graduate School of Information.
Lecture 11: Ising model Outline: equilibrium theory d = 1
Solutions and Mixtures Chapter 15 # Components > 1 Lattice Model  Thermody. Properties of Mixing (S,U,F,  )
The Ising Model Mathematical Biology Lecture 5 James A. Glazier (Partially Based on Koonin and Meredith, Computational Physics, Chapter 8)
Daniel Ariosa Ecole Polytechnique Fédérale de Lausanne (EPFL) Institut de Physique de la Matière Complexe CH-1015 Lausanne, Switzerland and Hugo Fort Instituto.
8. Selected Applications. Applications of Monte Carlo Method Structural and thermodynamic properties of matter [gas, liquid, solid, polymers, (bio)-macro-
Artificial Intelligence Chapter 3 Neural Networks Artificial Intelligence Chapter 3 Neural Networks Biointelligence Lab School of Computer Sci. & Eng.
Adaptive Cooperative Systems Chapter 8 Synaptic Plasticity 8.11 ~ 8.13 Summary by Byoung-Hee Kim Biointelligence Lab School of Computer Sci. & Eng. Seoul.
Graduate School of Information Sciences, Tohoku University
From J.R. Waldram “The Theory of Thermodynamics”.
6.4 Random Fields on Graphs 6.5 Random Fields Models In “Adaptive Cooperative Systems” Summarized by Ho-Sik Seok.
Some physical properties of disorder Blume-Emery-Griffiths model S.L. Yan, H.P Dong Department of Physics Suzhou University CCAST
Ginzburg-Landau theory of second-order phase transitions Vitaly L. Ginzburg Lev Landau Second order= no latent heat (ferromagnetism, superfluidity, superconductivity).
Eutectic Phase Diagram NOTE: at a given overall composition (say: X), both the relative amounts.
Ch 6. Markov Random Fields 6.1 ~ 6.3 Adaptive Cooperative Systems, Martin Beckerman, Summarized by H.-W. Lim Biointelligence Laboratory, Seoul National.
Computational Physics (Lecture 10) PHY4370. Simulation Details To simulate Ising models First step is to choose a lattice. For example, we can us SC,
Chapter 7 in the textbook Introduction and Survey Current density:
1 (c) SNU CSE Biointelligence Lab, Chap 3.8 – 3.10 Joon Shik Kim BI study group.
Ch 2. THERMODYNAMICS, STATISTICAL MECHANICS, AND METROPOLIS ALGORITHMS 2.6 ~ 2.8 Adaptive Cooperative Systems, Martin Beckerman, Summarized by J.-W.
Bayesian Brain - Chapter 11 Neural Models of Bayesian Belief Propagation Rajesh P.N. Rao Summary by B.-H. Kim Biointelligence Lab School of.
Biointelligence Laboratory, Seoul National University
Biointelligence Laboratory, Seoul National University
Ginzburg Landau phenomenological Theory
Computational Physics (Lecture 10)
Statistical-Mechanical Approach to Probabilistic Image Processing -- Loopy Belief Propagation and Advanced Mean-Field Method -- Kazuyuki Tanaka and Noriko.
Kinetic-Molecular Theory
Biointelligence Laboratory, Seoul National University
Biointelligence Laboratory, Seoul National University
MAGNETIC MATERIALS. MAGNETIC MATERIALS – Introduction MAGNETIC MATERIALS.
Phase diagram of a mixed spin-1 and spin-3/2 Ising ferrimagnet
Ising Model of a Ferromagnet
Model systems with interaction
In “Adaptive Cooperative Systems” Summarized by Ho-Sik Seok
Presented by Rhee, Je-Keun
Artificial Intelligence Chapter 3 Neural Networks
Presented by Rhee, Je-Keun
Ferromagnetism.
Biointelligence Laboratory, Seoul National University
Artificial Intelligence Chapter 3 Neural Networks
Quantum Mechanical Considerations
Mean Field Approximation
Section 1 The Kinetic-Molecular Theory of Matter
Adaptive Cooperative Systems Chapter 6 Markov Random Fields
Lattice gas with interactions
Artificial Intelligence Chapter 3 Neural Networks
Chapter 5 Language Change: A Preliminary Model (2/2)
Artificial Intelligence Chapter 3 Neural Networks
Biointelligence Laboratory, Seoul National University
Wiess field model of Paramagnetism. Wiess field model of paramagnetism In the ferromagnetic materials the magnetic moments (spins) are magnetized spontaneously.
Institute for Theoretical Physics,
Thermodynamics and Statistical Physics
Kensuke Homma / Hiroshima Univ. from PHENIX collaboration
Ginzburg-Landau theory
Presentation transcript:

Adaptive Cooperative Systems Chapter 3 Coperative Lattice Systems 3.6 ~ 3.7 Summary by Byoung-Hee Kim Biointelligence Lab School of Computer Sci. & Eng. Seoul National University

(C) 2009 SNU CSE Biointelligence Lab Outline 3.6 Strong cooperativity and Peierls’s argument 3.7 Mean-field theory The Weiss molecular field equation The Bragg-Williams (random mixing) method The Bethe approximation and Synopsis of mean-field results (C) 2009 SNU CSE Biointelligence Lab

(C) 2009 SNU CSE Biointelligence Lab Overview ‘Strong cooperativity’  differences in behavior between one- and two-dimensional Ising models (C) 2009 SNU CSE Biointelligence Lab

Cooperativity and Free Energy To minimize A (free energy) energy minimization: ordered states (all spins are aligned) entropy maximization: disordered states Cooperative ordering vs. entropy disordering c.o stronger : spontaneous magnetism Entropy in an Ising system? (C) 2009 SNU CSE Biointelligence Lab

Ising System as a Collection of Distinct Regions Entropy in an Ising system The number of ways this partitioning can be accomplished Removing borders reduces the entropy 1-d Ising chain There is no value for the ratio for which the energy gain 2-d Ising lattice There is always a temperature for which the energy gain is greater than the entropic loss The extent of the nearest-neighbor connectivity is an important aspect of the cooperativity 1-d case: insufficient number of nearest neighbors (C) 2009 SNU CSE Biointelligence Lab

Ising System as a Collection of Distinct Regions In 2-D Ising lattice Ω: multiplicity factor L: the length of a boundary Peierls proof of spontaneous magnetization in a 2-D Ising ferromagnet (C) 2009 SNU CSE Biointelligence Lab

(C) 2009 SNU CSE Biointelligence Lab 3.7 Mean-Field Theory (C) 2009 SNU CSE Biointelligence Lab

(C) 2009 SNU CSE Biointelligence Lab Ovierview Mean-field approach is an important tool in the study of cooperative phenomena Detailed effects of a set of neighboring elements  single, effective field Related contents Spatial lattice models in image construction (Ch. 6) Construction of biologically motivated NNs (Ch. 8) Collective dynamics of oscillator communities (Ch. 9) (C) 2009 SNU CSE Biointelligence Lab

List of Mean-Field Approaches Weiss molecular field approximation Random mixing argument (Bragg and Williams, 1934, 1935) First-order approximation (Bethe, Peierls, and Weiss) Higher-order methods (Kramers-Wannier, Kikuchi) Lattice gas model (C) 2009 SNU CSE Biointelligence Lab

The Weiss Molecular Field Equation A local field for an individual spin element si The total field (interaction energy) Average local field (neglecting fluctuations in orientations in space) Self-contained field equation At low temperatures, <s> = 1 (or -1) In the vicinity of Tc (C) 2009 SNU CSE Biointelligence Lab

Tc as the Threshold for the Onset of Spontaneous Magnetization Curie temperature Critical exponent Significance of the critical temperature When T>Tc, the only self-consistent solution is <s>=0 When T<Tc, we have a positive-negative pair of nontrivial solutions as well  Tc is the threshold for the onset of spontaneous magnetization (C) 2009 SNU CSE Biointelligence Lab

The Bragg-Williams (Random Mixing) Method The basic idea of the mean-field approaches is to avoid the difficult task of evaluating the density of states by eliminating N12 (see Section 3.1, Ising ferromagnet) Assumption: the spin orientations are completely random Internal energy: - above the Tc: the spontaneous magnetization and the internal vanishes - below: (C) 2009 SNU CSE Biointelligence Lab

Mean-Field Approach: Zeroth-order Approximation Main result in this section: the existence of a critical point below which there are two nontrivial solutions to the mean-field equation Mean-field theory shows The existence of a spontaneous magnetization The singular behavior of the specific heat The existence of a phase transition, marking the onset of long-range order in the lattice system Modeling: the cooperative influences in the lattice elements Effect: captures the essential properties of the assembly The mean-field approach is a zeroth-order approximation (C) 2009 SNU CSE Biointelligence Lab

Bethe-Peierls-Weiss approximation: first-order method We treat more accurately the interactions of a given spin with its nearest neighbors (C) 2009 SNU CSE Biointelligence Lab

(C) 2009 SNU CSE Biointelligence Lab

(C) 2009 SNU CSE Biointelligence Lab