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Self-affine scalingFamily-Vicsek scaling (1985):L t L f z t L W , /ξ is the roughening exponent, z the dynamic exponent.35Self-affine scaling zLt L f t LW , /u t/Lzt :WL f u L:W t f u 0u z zW Lu L t/L L t const is the growth exponent. Numerically these laws are verified by data collapse.0z36
Ising interfaceAdd next-nearest nearest neighbor interaction:HN E J K i j i ji, j: nni, j:nnnNpunishes curvature introduces stiffness+_+_11 K17 K37Shape of drop• Start with a block of +1 sites attached to wall ofan LL system filled with -1.• Hamiltonian including gravity g:HwithN E J K h Ni j i j j ii, j: nn i, j:nnn j line j( j1)( hLh)h hhjh1 and g L 1L1 L 1• Use Kawasaki dynamics and further do not allowfor disconnected clusters of +1.38
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: Ising interface+++ +NH E J ij+i, j
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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