HandsOn 30 - Random Walk on a Fractal

We now attempt to determine experimentally the average time it takes for a particle to move from one point to another on the Sierpinksi Gasket. The method is as follows: Roll a four-sided die, and move a random walker on the gasket. Keep count of how many steps the walker takes to move from one point to another point. By repeating this procedure, you will find the average time necessary to move specific distances.

Now for the details. "Time'' in this case means the same thing as the "total number of steps'' taken by the random walker. If it makes it easier for you, call the unit of time 1 second, and assume the walker takes 1 step per second. Then 10 steps take 10 seconds, 50 steps take 50 seconds, and so forth. The number of steps-the time-to go from one point to another will most likely be different for different trials because of the randomness of the process, based on the flip of a coin or the throw of a die. Hence we will try to predict the average times to go from one point to another. To emphasize that the times are averages, we use brackets . For example, the average time it takes to go from Point A to Point B is written TAB.

In the following experiment we use a 4-sided die (available in some game stores) to produce random numbers. To use such a die, roll it and look at the number which appears along the bottom edge of the die. This is the result of your roll.

Q8.5: If you have only an ordinary 6-sided die, use four of the faces. That is, if a 5 or 6 comes up, ignore it and roll again. Speculate: Is this really the same as using a 4-sided die?


Figure 8.3: Choosing the direction for the next step from labeled points in Figure 8.2(b). In each case, the direction is given by the number resulting from the roll of a 4-sided die.

To measure the average time it takes to go from one external vertex to another external point on the Sierpinski gasket (for example, from A to B' in Figure 8.2(a)), carry out the following steps. First, place your "walker'' (e.g., penny, pen top, thumb tack) at point A on the gasket. Flip a coin. If the coin comes up heads, move the walker down to the right. If the result is tails, move the walker down to the left. Now there are four possible directions for the next step. Roll the die. Move the walker according to the diagram in Figure 8.3. Select from the three possible orientations the one that fits your current position, and move the walker accordingly.


Q8.6: Compute TAB', the average time it takes to move from point A to either point B'. Use the data from Task 1. Using the data from Task 2, compute TAB'' . How do these two averages compare to what the movement would be like on a square grid?

Q8.7: Compute TC'B'. (Hint: You counted the number of steps necessary to arrive first at C' and then to arrive at either B'. For each trial where you passed through C', subtract the former from the latter to find the number of steps needed to move from C' to either B'.

Q8.8: From Figure 8.2(a), see if you can justify the following equation:
TAB' = 1 + TBB'.
Here we have written 1 for TAB, the time needed to go from A to either point B in Figure 8.2(a). Use this equation to compute TBB' from your data. (With two more such equations, it is possible to solve exactly for the average first arrival time at an external vertex on a Sierpinski gasket.

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