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 twodimensional
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
R^{2} is proportional
to N, or equivalently, the root mean square displacement (see Unit 3.7)
is


= N^{1/2}. 

(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:


= N^{s}. 

(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
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:
Use
the properties of logarithms to show that
log2 = log(5^{s})
= slog5 

(8.6) 
or
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.
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HandsOn 36  Random Walk on a Fractal