problemset4

# Z bazant 18366 random walks and diusion problem set 4

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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 1e-05 1e-06 1e-07 1e-08 1e-09 1e-10 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 1e-05 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.

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