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Transition probabilityThe proposed configuration Y will beaccepted with probability A(XY) .Total probability of Markov chain is:W(XY) ) = T(XY) A(XY)Master equation:dp( X , t)dt YpYW ( ) ( Y X) p( XW ) ( X Y)Y19Properties of W(XY)• Ergodicity:XY , : W( XY)0• Normality:• Homogeneity:YW( X Y)1 pst( Y) W( Y X) pst( X)Y20
Detailed balancedp( X , t)dt pY ( ) W( Y X) p( X) W( XY)YIn stationary state one should haveequilibrium distribution (Boltzmann)dpst( X , t)0dtY, p ( X) p ( X) p ( Y) W( Y X) p ( X) W( X Y)Yeqstsufficient condition is detailed balance:p ( Y) W( Y X) p ( X) W( X Y)eqYeqeqeq21Metropolis (M(RT) 2 ) peq( Y)AX ( Y) min1,peq( X) Boltzmann:1peq( X) eZTE( X )kTE( Y) E( X)EkTkTAX ( Y) min( 1, e ) min( 1, e )decreasesIf energy always acceptEincreases accept with probability .e - kT22
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: M(RT) 2 algorithmN.C. Metropolis, A
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
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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
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Page 51 and 52: Example RK4Butcher array:its stage
Page 53 and 54: Example q = p = 2y ny t ( 1 )
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Page 59 and 60: 3th order predictor- correctoruse e
Page 61 and 62:
Comparison of methodsΔtfor fixed n
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