Computational Physics

12 Introduction

p

x

–2.5

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5

3

–2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2

Figure 1.3. Strange attractor for the Duffing oscillator. Values of the parameters

areF 0 =2.0,ω=2.4,γ=0.1. The initial conditions arex 0 =0.5,x ̇ 0 =0.

The results log[N(b)]and log(b)should be written to a file. For an attractor of

25 000 points, the resulting points lie more or less on a straight line with slope

−Df≈−1.68, for 2≤l≤7.

1.2 [C] In this problem, we consider diffusion limited aggregation.

(a) Write a program for generating DLA clusters on a square lattice of size

150 ×150 (seeSection 1.3). Generate a cluster of about 9000 sites, and write the

sites occupied by this cluster to a file for viewing using a graphics program.

(b) Another definition of the fractal dimension (see Problem 1.1) is obtained by

relating the number of sitesNof the cluster to itsradius of gyration, defined by

Rg=

1

N

∑N

i= 1

(ri−r 0 )^2 ,

where

r 0 =

1

N

∑N

i= 1

ri

is the ‘centre of mass’ of the cluster. Show that the radius of gyration can be

rewritten as

Rg=

1

N

(N

∑

i= 1

r^2 i

)

−r^20.

Use this formula to calculate the radius of gyration after every 200 newly added

sites, and write the values log(Rg), log(N)to a file. Plot this file and fit the results