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Clustering of clustersFractal dimension is d f ≈ 1.42 in 2d, d f ≈ 1.7 in 3d.dynamical scaling:2zns s f s/twith dynamical exponent z .Simulated Annealing63• P.J.M. Van Laarhoven and E.H. Aarts,„Simulated Annealing: Theory andApplications“ (Kluwer, 1987)• A. Das and B.K. Chakrabarti (eds.),„Quantum Annealing and RelatedOptimization Methods“ Lecture Notes No.679 (Springer, 1990)64
Simulated Annealing (SA)S. Kirckpatrick, C.D. Gelatt and M.P. Vecci, 1983SA is a stochastic optimization technique.Given a finite set S of solutionsand a cost function F: S Search a global minimum:s* S* { s S : F(s) F(t) t S }.Difficult when S is very big (like |S| = n!).65Travelling SalesmanGiven n cities σ i and the travelling costs c(σ i ,σ j ).We search for the cheapest trip through allcities.• S = { permutations of {1,..,n} }• F() = i=1..n c(σ i ,σ j ) für SFinding the best trajectory is a NP-complex problem,i.e. time to solve grows faster than any polynomial of n.66
Page 1 and 2: Monte Carlo MethodsNicholas Constan
Page 3 and 4: Calculation of π1x 2 + y 2 = 1 124
Page 5 and 6: Error of integrationconventional me
Page 7 and 8: Not smooth integralsg(x)ax ineed mo
Page 9 and 10: M(RT) 2 algorithmN.C. Metropolis, A
Page 11 and 12: Detailed balancedp( X , t)dt pY ( )
Page 13 and 14: The Ising ModelBinary variables:i1,
Page 15 and 16: Binary mixtures(lattice gas)Conside
Page 17 and 18: Ising interface+++ +NH E J ij+i, j
Page 19 and 20: Ising interfaceAdd next-nearest nea
Page 21 and 22: Irreversible growth• T. Vicsek,
Page 23 and 24: Random deposition with surface diff
Page 25 and 26: Ballistic depositionParticles fall
Page 27 and 28: DLA clustersfractaldimensiond f = 1
Page 29 and 30: Dielectric breakdown57Dielectric br
Page 31: Dielectric breakdown modelη = 0.16
Page 35 and 36: Slow coolingT 0T 1T 20Different coo
Page 37 and 38: Solving equationsFinding the soluti
Page 39 and 40: Bisection methodTake two starting v
Page 41 and 42: N-dimensionalequationsBex a N-dimen
Page 43 and 44: N-dimensionalsecant methodIf the de
Page 45 and 46: Euler methodStart with y 0 and iter
Page 47 and 48: The order of a methodIf the error a
Page 49 and 50: Fourth order Runge - Kuttak1tf( y ,
Page 51 and 52: Example RK4Butcher array:its stage
Page 53 and 54: Example q = p = 2y ny t ( 1 )
Page 55 and 56: Adaptive time-stepΔtδ expectedDef
Page 57 and 58: Richardson extrapolationCalculate f
Page 59 and 60: 3th order predictor- correctoruse e
Page 61 and 62: Comparison of methodsΔtfor fixed n

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