problemset4

Z bazant 18366 random walks and diusion problem set 4

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online