1. Presented by Rhee, Je-Keun
Ch 4. Simulated Annealing 4.1 ~ 4.4 Adaptive Cooperative Systems, Martin Beckerman, 1997.
Presented by Rhee, Je-Keun

2. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Contents
4.1 Introduction
4.1.1 Energy Landscapes
4.1.2 Multiple Constraints and Selectivity
Searching for Optimal Solutions
4.2 Objectives
4.3 Kinetic Ising Model
4.3.1 The Master Equation
4.3.2 Spin Kinetics
4.4 Order-Disorder Transitions in Binary Alloys
4.4.1 Order-Disorder Transitions
4.4.2 Particle Kinetics and Binary Alloys
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3. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Energy Landscapes
The spontaneous magnetization of two-dimensional Ising ferromagnet can be either positive or negative.
At temperature below the Curie point, there are two minima in the free energy, corresponding to positive or negative ordering.
The energy landscape for a ferromagnet changes dramatically if we allow the sign of the coupling constants Jij to vary in a random way from site to site.
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4. Multiple Constraints and Selectivity
In the traveling salesman, graph partitioning and other NP-hard problems, competing constraints, frustration, and a multiplicity of local energy minima are essential features.
The structure of these problems bears a great similarity to the statistical mechanics description of a dilute disordered magnet known as a spin glass.
The multiple constraints can be appeared in image processing problems.
The digital images can be regarded as highly organized spatial lattice systems, and it is examined by modeling as Markov random fields in Chapter 6.
Those problems involved in the self-organization of the central nervous system.
In the next chapter we will explore an equally dramatic set of findings regarding developmental and functional plasticity in the retinotectal projection of lover vertebrates, and in Chapter 9 we well look at the dynamic modulation of neural firing states.
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5. Searching for Optimal solutions
The search for optimal solutions – that is, for highly ordered, low-energy, stable states – is a particularly difficult one in systems possessing many local minima.
The necessary clues for developing strategies for finding optimal solution
Low-energy, stable states stand out at low temperatures, whereas they are buried among the multitude of disordered states at high temperatures.
The fluctuations are a useful mechanism, in that they allow a system to probe its surrounding solution space for better energy configurations.
The difficulties in obtaining exact, analytic solutions to nearest-neighbor models of lattice cooperativity can be alleviated using the Monte Carlo importance sampling method such as Metropolis algorithm.
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6. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Objectives
The first sampling method is the Glauber heat bath algorithm used in conjunction with the kinetic Ising model. (Section 4.3)
The second set of alternative procedures is related to the treatment of binary alloys. (Section 4.4)
Spin glasses and the general problems encountered in treating systems with frozen-in, or quenched, disorder helped inspire the development of the simulated annealing method. (Section 4.5)
A brief overview of the world of NP-completeness (Section 4.6)
The simulated annealing algorithms (Section 4.7, 4.8)
The microcanonical annealing approach (Section 4.9)
The continuous simulated annealing method (Section 4.10)
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7. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Kinetic Ising Model
Two classes of kinetic Ising models
Due to Glauber, spins may randomly flip from one state to the other. In the spin model neither the total energy nor the magnetization is conserved.
Molecules may exchange positions with one another or migrate to empty lattice sites. Particle (molecule) number, the analog of magnetization, is conserved in these kinetic processes.
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8. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
The Master Equation
Consider a lattice system in which a lattice variable x may assume one of a number of values at each of the i=1,…,N lattice sites.
w({x},t) denote the probability that a system has assumed a particular configuration {x}=(x1,…,.xN) at time t.
p(a,b) is the transition probability from state a to state b.
At equilibrium the time derivative is zero
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9. (C) 2009, SNU Biointelligence Lab, http://bi.snu.ac.kr/
Spin Kinetics
Consider a simple case where the variable for a single lattice site j may change its state.
We regard the lattice variables as spin elements whose interaction are given by an Ising hamilonian.
This results is sometimes called the heat bath algorithm, and for the one-dimensional case, it is known as Glauber dynamics.
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10. An algorithm simulating an Ising spin system
Initialize by arranging spins into the lattice in an ordered arrangement, in a completely disordered arrangement, or randomly.
Select a lattice site
Calculate the transition probability corresponding to flipping the spin
Choose a random number. If it is less than the transition probability, flip the spin; otherwise, do not flip the spin.
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11. Order-Disorder Transitions
Beta brass is an alloy composed of copper and zinc in equal amounts and arranged into a body-centered cubic lattice. In the completely ordered state, all copper atoms are nearest neighbors of zinc atoms, and vice versa.
As the temperature increases, the probability for finding the correct neighbor decreases toward ½, and at temperature above the critical temperature, copper and zinc placements on the lattice are completely random.
There are two types of ordered states in the binary alloys.
The lattice ordering described above resembles an antiferromagnetic domain, and it arise when the interaction between the two kinds of atoms is attractive.
If the interaction is repulsive, or alternatively if one introduces an attraction between like atoms, a different kind of ordering, that of a two-phase system, is produced.
In an phase-separated system, there are regions where atoms of one type only are found. Thus phase separation is analogous to ferromagnetism.
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12. Particle Kinetics and Binary Alloys
Consider a system composed of two different kinds of atoms, called A and B, arranged in a regular lattice.
For the nearest neighbors the interaction energies , and are introduced, and similarly for second nearest neighbors , and are had.
Let us form the quantities
We now generate configurations by interchanging pairs of nearest-neighbor atoms, A on lattice site S and B on site S’.
The energy difference
N are the numbers of first nearest-neighbor B atoms of S and S’, respectively, before the interchange, and similarly for N.
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13. Metropolis algorithm simulating the kinetics of a binary alloy
Initialize by arranging A and B atoms into the lattice in an ordered arrangement, in a completely disordered arrangement, of randomly.
Select a central lattice site at random
Select one of the 12 nearest-neighbor sites at random
If the nearest-neighbor atom selected differs from the central atom, calculate the energy change resulting from an exchange of atoms;
if negative, exchange particles;
if positive, calculate the probability;
select a random number – if less than the probability, switch particles;
otherwise, do not exchange particles, but count as a new configuration.
If the atoms are identical, do not exchange, but count as a new configuration.
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