### One-dimensional random walk with variable step length

Introduction

An example of a continuous random variable is the displacement from the origin of a one-dimensional random walker that steps at random to the right with probability p, but with a step length that is chosen at random between zero and the maximum step length a. The continuous nature of the step length means that the displacement x of the walker is a continuous variable.

We can record the number of times H(x) that the displacement of the walker from the origin after N steps is in a bin of width Δx between x and x + Δx. If the number of walkers that is sampled is sufficiently large, we would find that H(x) is proportional to the estimated probability that a walker is in a bin of width Δx a distance x from the origin after N steps. To obtain the probability, we divide H(x) by the total number of walkers.

In practice, the choice of the bin width is a compromise. If Δx is too big, the features of the histogram would be lost. If Δx is too small, many of the bins would be empty for a given number of walkers, and our estimate of the number of walkers in each bin would be less accurate.

Problems

1. Describe the qualitative features of the histogram for the displacement x after N steps. Does the qualitative features of the histogram change as the number of walkers is increased?
2. How does the variance depend on N for fixed p?
3. The simulation uses a step length with a uniform probability between 0 and 1. Calculate analytically the mean displacement and the variance for one step. (Remember that a step can be either to the left or the right and is of variable length.) Compare your analytical result to the result of the simulation for N = 1.
4. How does the variance found in the simulation after N steps compare to the variance for one step?
5. Explore the dependence of the histogram on the bin width. What is a reasonable choice of the bin width for N = 100?

References

- Harvey Gould and Jan Tobochnik, Thermal and Statistical Physics, Chapter 3.

Java Classes

- VariableStepLengthWalkApp

Updated 18 March 2007.