Question

An ant is taking a two-dimensional random walk on a flat surface. We will distinguish steps that the ant takes with its feet from the random walk (RW) steps, of length δ, which are made up of lots of footsteps. In a RW step, the ant choose a direction randomly and walks distance δ in that direction. She then choose another direction randomly and walks distance δ in that direction. She repeats this for a total of n RW steps. The surface conveniently has x- and y-coordinates drawn on it, so that we can monitor the ant’s progress along each direction, as well as the distance from the starting point.

The ant walks at a rate of 5 cm/min and normally walks for 30 s in each of its random walk steps.

At the end of each n RW steps, the ant takes a rest for a few minutes and then starts off on another random walk of n steps. Assuming that the ant completes a large number, N, of random walks, all the same, calculate the following:

The average (RMS) distance that the ant moves from the starting point of each walk.

The average (RMS) distance that the ant moves from the starting point of each walk in the x-direction.

The expected total RMS distance from the original starting point after N random walks of n steps each. (This question assumes that the ant would be up to repeating all of the N random walks a large number of times, and the distance would be measured from the start of each set of N walks to the end of each set.)

Answer 1