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Unformatted text preview: m (p, q ) = (1/2, 1/2) and the PDF of the nth step is
pn (r, s) = (δr,−1/3n + δr,1/3n )(δs,−1/3n + δs,1/3n ).
4 M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 6 From this from, it is clear that the variables p and q are independent. We see that the N th step of
the q component can be written as
N QN = = = 1 � 2In − 1
� N 1 � 2In
1 1 1 − 31
2 3 1 − 1
n=1 = 1 +
3N +1 N
3n where the IN are independent random variables that take values of 0 and 1 with equal probability,
as in the previous section. In the limit as N → ∞, we obtain
Q∞ N � In
= . 2n
n=1 Thus, the expansion in base-3 of R∞ is
Q∞ = 0.(2I1 )(2I2 )(2I3 )(2I4 )(2I5 )(2I6 ) . . .
and hence every possible number whose base-3 expansion contains only 0’s and 2’s with no 1’s
occurs with equal probability. The numbers which satisfy this property form the Cantor set, a well
known fractal distribution that can be obtained by taking the unit interval, removing the middle
third, and then recursively removing the third of each new interval. The cumulative distribution
can be written as
for q < 1
⎨ C∞ (3q )/2
for 1 < q < 2
C∞ (q ) = 1/2 3
⎩ C∞ (3q − 2)/2 + 1/2
for 2 < q .
This is referred to as the Cantor function and is shown in ﬁgure 4. It is a type of function referred
to as “Devil’s staircase”. A function f (x) is a Devil’s staircase on the interval [a, b] if it satisﬁes the
• f (x) is continuous on [a, b].
• There exists a set N of measure 0 such that for all x outside of N , f � (x) exists and is zero.
• f (x) in nondecreasing on [a, b].
• f (a) < f (b). M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 2.4 7 Other values of a Figure 5 shows the PDF of the decaying walk for twelve diﬀerent values of a. As a increases, we
see a progression from a sparse PDF (similar to a Cantor set), through a uniform PDF for a = 0.5,
to a PDF approaching a Gaussian as a approaches 1. For the case when a = 2−1/2 , we can write...
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This note was uploaded on 01/23/2014 for the course MATH 18.366 taught by Professor Martinbazant during the Fall '06 term at MIT.
- Fall '06