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Unformatted text preview: ely large steps, on a scale larger than the drift. M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 4 Solutions 4 0.55
d = −1.0
d = 0.0
d = 1.0 0.5
0.45
0.4 f (n) 0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1 2 3 4 5 6 7 8 9 10 n Figure 1: Plots of the ﬁrst passage probability functions f (n) for the unbiased and biased Cauchy
walks. 1
d = −1.0
d = 0.0
d = 1.0 0.1
0.01
0.001
f (n) 0.0001
1e05
1e06
1e07
1e08
1e09
1e10
1 10 100 1000 10000 100000 n Figure 2: Log plots of the ﬁrst passage probability functions f (n) for the unbiased and biased
Cauchy walks. M. Z. Bazant – 18.366 Random Walks and Diﬀusion – Problem Set 4 Solutions 5 1
d = −1.0
d = 0.0
d = 1.0 0.1 S (n) 0.01 0.001 0.0001 1e05
1 10 100 1000 10000 100000 n Figure 3: Log plots of the survival probability functions S (n) for the unbiased and biased Cauchy
walks. 3 First passage to a sphere To calculate the probability of ﬁrst passage to the sphere, we make use of the electrostatic analogy.
We consider the corresponding problem of a point charge of magnitude q = 1/4πR2 D located at
a distance r0 from the sphere, with the sphere’s surface is kept at zero potential. The probability
of absorption at a point on the sphere’s surface will be given by the magnitude of the electric ﬁeld
there. Let �0 be the the position of the walker, and by symmetry, consider pointing this in the
r
positive z direction. In the absence of the sphere, the electric potential is given by
q
Φ(�) =
r
,
� − �0 
rr
which can be rewritten in terms of spherical coordinates (r, θ, φ) as
q
Φ(�) = �
r
.
2 sin2 θ + (r − r cos θ )2
r
0
To solve for the electric potential in the presence of the sphere, we make use of the image method,
introducing a charge of magnitude v at a location (x, y, z ) = (0, 0, s), to give a solution of the form
q
v
Φ(�) = �
r
+�
.
2 sin2 θ − (r − r cos θ )2
2 sin2 θ − (r cos θ − s)2
r
r
0
In order to set the electric potential to zero on the s...
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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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