Introduction

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.

Problems

- 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}r^{2}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.

Java Classes

- TwoDimensionalWalkApp

Updated 28 February 2007.