1. 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

2. (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

3. (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

4. 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

5. 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

6. 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

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

8. (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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. (C) 2009 SNU CSE Biointelligence Lab

16. (C) 2009 SNU CSE Biointelligence Lab