Two-dimensional random walk
A number of random walkers start from the origin of a square lattice and simultaneously move randomly
in one of four directions. You can choose
the probabilities of a step in each of the four directions subject to the normalization condition that the sum of the probabilities is unity.
- Consider the walkers as a swarm of bees that are doing a random walk. Run the simulation and look at the resulting swarm
of bees as they move. Describe the shape of the
swarm if
the probabilities of a step in any of the four directions are equal.
- Roughly estimate the radius of the swarm as a function of time. You
can determine
the coordinates of any point on the swarm by clicking on the point with
the mouse.
- The histogram H(r) counts the number of walkers at a given time that are a radial distance between r and Δr. What is the qualitative dependence of H(r) on r? How does it differ from the histogram H(x) for the one-dimensional random walk? How does the histogram
change with the number of steps?
- You can obtain the histogram data in tabular form under the Views
menu. Use this data to
compute the mean square displacement ∑r r2 P(r), where P(r) is the normalized histogram. Repeat
for a
number of different times and
plot the mean square displacement versus time. Do you obtain the
expected linear behavior for this
plot?
- Change the step probabilities so that the probability to move
left is different than to move
right, but the up and down probabilities are the same. (For example, set
pLeft = 0.15 and pRight = 0.35.)
Estimate the position of
the center of the swarm as a function of time. Does it move linearly
with respect to
time or linearly with respect to the square root of the time? Explain your
result.
Updated 28 February 2007.