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Canonical Monte CarloE(X) ) = energy of configuration XProbability for system to be in X is given byBoltzmann factor:Z T is the partition function:1peq( X) eZTE( X )kTZTXE( X )kTe p ( ) eqX 1X15Problem of samplingQT ( ) Q( X) peq( X)XThe distribution ofenergy E aroundthe average < E > Tgets sharper withincreasing size.Choosing configurations equally distributedover energy would be very ineffective.16
M(RT) 2 algorithmN.C. Metropolis, A.W. Rosenbluth,M.N. Rosenbluth, , A.H. Teller and E. Teller (1953)importance sampling through a Markov chain:X 1 X 2 …..where the probability for a configuration is p eq (X)Markov chain: X t only depends on X t -1Properties of Markov chainStart in configuration X and propose a newconfiguration Y with probability T (XY).1. Ergodicity: One must be able to reach anyconfiguration Y after a finite number of steps.2. Normalization:3. Reversibility:YT( X Y)1T( X Y) T( Y X)1718
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: Not smooth integralsg(x)ax ineed mo
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 and 32: Dielectric breakdown modelη = 0.16
Page 33 and 34: Simulated Annealing (SA)S. Kirckpat
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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