8.2 - The Power Law for a Random Walker on a Sierpinski Gasket

In earlier units you studied the properties of a random walk along a line, and on a two-dimensional square grid. We found that for a random walk along a straight line or on a flat surface, after N steps (or a "time of N'') the mean square distance traveled R2 is proportional to N, or equivalently, the root mean square displacement (see Unit 3.7) is
 = N1/2.
(8.2)

On a fractal, interconnections are not so regular as on a straight line or square lattice. Therefore the relationship between the number of steps and the average distance covered may not be the same. We try to describe this difference by a change in the exponent 1/2 on N in Eq. 8.2 to a different value, as yet undetermined, which we shall call s:
 = Ns.
(8.3)

For the Sierpinski gasket we can derive the exponent s by using a procedure similar to the following: Suppose that in the experiment of HandsOn 8.1, your data showed
 TAB' = 5TAB.
(8.4)
Values of T (the time) in Eq. 8.4 mean "number of steps,'' the same as N in Eq. 8.3. Then Eq. 8.4 and your experiments tell us that to go twice the distance (that is, to go from A to B' which is twice the distance from A to B), it takes on average 5 times longer. For this case, Eq. 8.3 can be written as:
 2 = 5s.
(8.5)

Use the properties of logarithms to show that
 log2 = log(5s) = slog5
(8.6)
or
 s = log2 log5 = 0.431.
(8.7)

This exponent is less than the 1/2 for a random walk on a square grid as shown in Eq. 8.2 above. This result would indicate that movement on a Sierpinski gasket differs from that on a square grid. This movement in fact has been named anomalous diffusion.