### Two-dimensional random walk

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

1. 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.
2. 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.
3. 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?
4. 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?
5. 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.