Unformatted text preview: the random variable as
∞�
� In
�
X∞ =
2n/2
n=1
�
Where the In take values (1, 0), (−1, 0), (0, 1), and (0, −1) with a quarter probability each. This
can be rewritten as
∞�
∞�
√ � I2m−1 � I2m
�∞ =
2
X
+
2m
2m
m=1
m=1
√
�
�
=
2Y1 + Y0
�
�
where Y1 and Y0 are decaying walks with parameter a = 1/2. From part (b), we know that these
are uniformly distributed on the region x + y  < 1, which allows us to easily construct an analytic
solution to the resultant PDF as a convolution two simple PDFs. Similarly, we have
�
X∞ =
�
X∞ = 3
�
i=0
7
� �
2i/4 Yi for a = 2−1/4 �
2i/8 Yi for a = 2−1/8 i=0 �
where the Yi are all decaying walks with parameter a = 1/2; again, this easily allows us to construct
the resultant PDF as a convolution of a ﬁnite number of simple PDFs.
√
Figure 5 also shows the PDF for the case for when a is equal to the golden mean, ( 5 − 1)/2.
For this value, 1 − a = a2 , and this special relationship results in sections of the PDF appearing
selfsimilar. 3 Shock structure We are interested in ﬁnding a traveling wave solution c(x, t) = f (x − v t) of Burgers’ equation
ct + ccx = Dcxx
which satisﬁes the boundary conditions c(−∞, t) = c− and c(∞, t) = c+ . Writing z = x − v t, we
ﬁnd that f (z ) satisﬁes
(−v + f )f � = Df ��
which can be integrated to obtain
f2
= Df � + A
2
for some constant A. Applying the boundary conditions, and assuming f � (z ) → 0 as z → ±∞, we
ﬁnd that
c2
−vc− + − = A
2
c2 −vc+ + + = A 2
−vf + M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 3 Solutions 0.1 a = 0.1 0.4 a = 0.2 8 a = 0.3 0.2
0.3
0.05 0 0.2 0.1 0 0.1
0
0.1
0.1 0.05 0.2
0.3 0.2
0.1 0.4
0.1 0.05 0 0.05 0.2 0.1 a = 1/3 0.6 0.1 0 0.1 0.2 0.4 0.3 0.2 0.1
1 a = 0.4 0 0.1 0.2 0.3 0.4 a = 0.5 0.4
0.4
0.5
0.2
0.2
0 0 0 0.2
0.2
0.5
0.4
0.4
0.6
1
0.4
1.5 0.2 0 0.2 0.6 0.4 0.2 0.4 a = 0.6 1.5 1 0.2 0.4 0.6 √
a = ( 5 + 1)/2 1 2 1 0.5 0 0.5 0 0.5...
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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
 MartinBazant

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