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

▶

$fractals statistical models by example ii$

Jul 12, 2023 356 likes •436 views

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, Pez, & Bordeianu with

SLIDE 1

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

SLIDE 2

Deposition Coastline

Break Over, Back to Work

2 / 5

SLIDE 3

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 ≤ xi = ri ≤ L Sticks, grows if h > neighbors Fills in hole ⇒ max neighbors

hr =    hr + 1, if hr ≥ hr−1, hr > hr+1, max[hr−1, hr+1], if hr < hr−1, hr < hr+1.

3 / 5

SLIDE 4

Deposition Coastline

Length of the British Coastline?

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

4 / 5

SLIDE 5

Deposition Coastline

Box Counting Algorithm

Cover Perimeter L with boxes Length r → 0 (Area, Volume)

40 80 40 100 square (m = 2.00) coastline (m = 1.3) straight line (m = 1.02) log(scale) log(Number boxes)

N(r) =C 1 r df = C′ sdf (1) s ∝ 1 r = scale (2) log N(r) = log C − df log r (3) ⇒ df = − lim

r→0

∆N(r) ∆r (4)

5 / 5

SLIDE 6

Deposition Coastline

Get to Work!

Do this yourself please.

6 / 5