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