Introduction to Modeling and Simulation, Spring 2006 – M. Z. Bazant, bazant@mit.edu
1.021J, 2.030J, 3.021J, 10.333J, 18.361J, 22.00J, HST588J
Problem Set 4 due on Wednesday April 12, 2006
Directions: Program in the computer language of your choice (matlab, Mathematica, C/C++,
fortran, etc.). Dion can provide code fragments for the first problem to help you get started. Don’t
worry if you cannot do every part. Turn in the output of your programs which pertain to the
questions, as well as the codes themselves, up to a limit of five pages total. If your codes are longer,
just turn in some key subroutines, and describe briefly how each program works.
1. Percolation. Simulate the growth of percolation clusters starting from the origin on a 2d
square lattice using the Leath algorithm. Do not let clusters grow beyond a circle of radius
R (e.g. R = 25 or 100). Let p be the site occupation probability.
(a) For p = 0.5, calculate SR (p) the probability that a cluster ‘spans’ from the origin to the
circle of radius R. Is p = 0.5 is larger, smaller, or equal to pc ?
(b) Repeat for many values of p in from 0 to 1, and plot the curve, SR (p). Give your best
estimate of pc from this data.
(c) Determine the expected cluster mass, M (r), inside a circle of radius r. Measure the
fractal dimension, Df , as the slope of a best­fit line, log M vs. log r.
2. Diffusion­Limited Aggregation. Simulate off­lattice DLA where “sticky” circular particles
of radius r = 1 are launched one­by­one from a small inner circle of radius R1 > 1. At first,
there is a single seed particle at the origin, and a “sticky cluster” of fixed particles grows by
incorporating new ones as they arrive on its boundary. Each particle undergoes a Pearson
random walk with independent displacements of length a = 0.1 and random angle (uniformly
distributed), which is allowed to move beyond the inner circle. If a particle overlaps with the
cluster, the it is frozen in place (as part of the cluster), and the next one is launched. If a
particle crosses a large outer circle of radius R2 � R1 , then it is replaced at a random position
(uniformly) on the outer circle. Of course, the two circles must grow along with the cluster.
(a) Plot images of a few of your large clusters. What is the largest cluster you were able to
grow? Think about ways to make the code more efficient.
(b) Plot the cluster mass M (r) (number of frozen particles) inside a radius r for the “dead”
interior region (behind the outermost growing tips), averaged over all clusters, on a
log­log plot. Estimate the fractal dimension.