Fractals & Statistical Models by Example II Nonlinear - PowerPoint PPT Presentation

Deposition Coastline Fractals & Statistical Models by Example II Nonlinear Computational Science in Action Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 5

Deposition Coastline Break Over, Back to Work 2 / 5

Deposition Coastline E.G. 3: Ballistic Deposition of Film Grow Film Particle-by-Particle 20K points, 200 sites Hot filament ⇒ random Evaporated particles stick Line L , 200 sites 0 ≤ x i = r i ≤ L Sticks, grows if h > neighbors Fills in hole ⇒ max neighbors  h r + 1 , if h r ≥ h r − 1 , h r > h r + 1 ,  h r = max [ h r − 1 , h r + 1 ] , if h r < h r − 1 , h r < h r + 1 .  3 / 5

Deposition Coastline Length of the British Coastline? 1967 Mandelbrot: “How long is the coast of Britain?” ⇒ Length = perimeter Problem: d f = ? Length self similar = ? Map maker: ruler r ⇒ L r ≃ Nr Colorado, Wyoming = easy Geometric: L r → L 2D = finite perimeter Nature: L ( r ) = Mr 1 − d f Coasts = geographic d f > 1, L → r → 0 ∞ Coasts � = geometric Finite size, ∞ perimeter Math � = physics: quantum, Appear self similar Compton sizes 4 / 5 Model coast = fractal

Deposition Coastline Box Counting Algorithm Cover Perimeter L with boxes Length r → 0 (Area, Volume) 100 log(Number boxes) square (m = 2.00) 40 coastline (m = 1.3) straight line (m = 1.02) log(scale) 0 0 40 80 � 1 � d f = C ′ s d f N ( r ) = C (1) log N ( r ) = log C − d f log r (3) r ∆ N ( r ) s ∝ 1 d f = − lim (4) ⇒ = scale (2) ∆ r r → 0 r 5 / 5

Deposition Coastline Get to Work! Do this yourself please. 6 / 5