Computational Physics

Problem Solving with

Computers

Wiley: 2005 2nd Ed

RUBIN H. LANDAU

Oregon State University

MANUEL JOSE PAEZ MEJI A

University of Antioquia

November 2, 2004

1

5. Deterministic Randomness

Random r1, r2, . . . no correlations

0 50 100 150 200 250x

0

50

100

150

200

250

y

0 50 100 150 200 250x

(Left) “bad” generator; (right) drand48

-Random-Number Generators

linear congruent, power residue method

ridef= remainder

(a ri−1 + c

M

).

E.G.: c = 1, a = 4, M = 9

drand48: M = 248, c = B16a =

5DEECE66D16

2

5.2 Problem: A Random Walk: R �√N

N

R

1

23

4

−30 −20 −10 0 10x

−20

−10

0

10

y

0.0 100.0 200.0 300.0N

0.0

100.0

200.0

300.0

Rtheory

simulationstartfinish

(Left) simulated walk; (right) distance

versus N steps

3

6. MC Applications

Radioactive, Spontaneous Decay

0.0 200.0 400.0 600.0 800.0 1000.0 1200.0time

0.0

1.0

2.0

3.0

4.0

5.0lo

g (

N)

N0 =10

N0 =100

N0 =1000

N0 =10000

N0 =100000

exponential stochastic

Input N(0), λ N = Nleft = N(0) InitializeDo till Nleft = 0 all gone

Do for N 1 period0 ≤ r ≤ 1 random #if r ≤ λ, Nleft = Nleft -1 ALGORITHM

Endoprint N, λ*(N-Nleft)N = NleftEndo

4

Rejection Integration

Stone Throwing

von Neumann Weighted Random

Pond

Npond

Npond + Nbox=

Apond

Abox

10-D Integration

I =∫ 1

0dx1 · · ·

∫ 1

0dx10 (x1 + · · · + x10)

2 = 1556

5

24 Fractals

Problem: Fractal Dimensions

Not simple geometric shapes, simple math-

ematical rules: plants, sea shells, poly-

mers, thin films, colloids, and aerosols

15,000 pts Sierpinski gasket, df � 1.585

Hausdorf dimension DH: OK; 1D, 2D,

3D:

M(L)def= ALdf .

6

Self-Affine Connection of Points

(x′, y′) = s(x, y) scaling,

(x′, y′) = (x, y) + (ax, ay), translation,

x′ = x cos θ − y sin θ, rotation

y′ = x sin θ + y cos θ

⇒ self-similar

child : parent :: parent: ancestors

All scales

7

Problem 2: Barnsley’s Fern

Random, beauty, similarity, symmetry

30,000 iterations; each frond similar

(x, y)n+1 =

(0.5,0.27yn),

(−0.139xn + 0.263yn + 0.57

0.246xn + 0.224yn − 0.036), ,

(0.17xn − 0.215yn + 0.408

0.222xn + 0.176yn + 0.0893),

(0.781xn + 0.034yn + 0.1075

−0.032xn + 0.739yn + 0.27),

P =

2%, if r < 0.02,

15%, if 0.02 ≤ r ≤ 0.17,

13%, if 0.17 < r ≤ 0.3,

70%, if 0.3 < r < 1.

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Problem 3: Ballistic Deposition

Thermal evaporate particles (random),deposit on surface film. <Height>∝ T ,surface = fractal

0.0 50.0 100.0 150.0 200.00.0

50.0

100.0

150.0

200.0

250.0

20,000 particles

Random v ⇒ random sites; sticks wherehit, fills holes:

1. Choose a random site r.

2. hr = height of at site r

3.hr =

{hr + 1, if hr ≥ hr−1, hr+1

max[hr−1, hr+1]

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Prob 4: Length Coastline via df

r

L2r

Number boxes size r that covers object

N =L

r∝ 1

r∝ lim

r→0

1

rdf

df = limr→0

logN(r)

log(1/r).

0 20 40 60 80 1000

20

40

60

80

100

df = 1.23 via box counting

10

Problem 5: Correlated Growth, Forests

and Films

Plants: increased growth if others nearby;

deposition + P ∝ distance−2

d

particle i

particle i+1

11

Globular Cluster: Diffusion-Limited

Aggregation

1. seed in the middle

2. release second particle random lo-

cation on circle

3. random walk + discrete x/y jumps

(Brownian motion: diffusion)

4. step length: random Gaussian

5. particle stick to other is within 1

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22 Quantum Monte Carlo:Feynman Path Integrals

Wavefunction ← Classical Trajectory

Hamilton’s principle of least action:

S[x(t)] =∫ tb

tadt [KE(x(t)) − PE(x(t))]

bxa

t a

t b

xx

B

A

t

Huygen’s wavelet principle

ψ(xb, tb) =∫

dxa G(xb, tb;xa, ta)ψ(xa, ta)

Feynman QM postulate: path integral

G(b, a) =∑

pathseiS[b,a]/h, S/h � ∞

13

0.0 4.0 8.0 12.0 t

−2.0

0.0

2.0

x(t

)

−2.0 −1.0 0.0 1.0 2.0 x

0

1000

2000

ψ2 (x

) (c

ount

s)

Long timeShort time

The ground-state wave function of the

harmonic oscillator as determined with

a path-integral calculation.

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