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The steps in a random walk can take on discrete or continuous values. Program RandomWalkContinuous allows you to explore how continuous random walks can be described in a similar way as walks with fixed steps as the number of steps $N \gg 1$.

Problem: Simulation of a random walk with variable step length

Use Program RandomWalkContinuous to simulate a random walk in one dimension with a variable step length.

1. The step length is generated at random with a uniform probability between 0 and 1. Calculate the mean displacement and its variance for one step.
2. Compare your analytical results from part (a) to the results of your simulation for $N=1$ ($p=1/2$).
3. How does the variance of the displacement found in the simulation for $N=16$ depend on the variance of the displacement for $N=1$ that you calculated in part (a)?
4. Explore how the histogram changes with the bin width. What is a reasonable choice of the bin width for $N=100$?

Resource

Problem 3.44 in Statistical and Thermal Physics: With Computer Applications, 2nd ed., Harvey Gould and Jan Tobochnik, Princeton University Press (2021).