Exercise 2-4: A group of random walkers start at radius r in the annular domain. Those...

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Exercise 2-4: A group of random walkers start at radius r in the annular domain. Those that reach the inner boundary at r = a are captured, while those that first reach the outer boundary at r = b escape. r = a r = b Figure 1: Annulus with random walkers (black dots) (a) You will solve this problem using Laplace's equation. Why is Laplace's equation an appro- priate choice for this problem? (b) Derive the 2D Laplacian operator in polar coordinates, using the following relationship between x, y and r, 0: x = r cos(0) y = r sin(0) (c) Set up and solve Laplace's equation in polar coordinates. (Hint: Propose a solution of the form P = ra.) (d) Solve for the probability of escape: P(r = b)/P(r = a). Exercise 2-4: A group of random walkers start at radius r in the annular domain. Those that reach the inner boundary at r = a are captured, while those that first reach the outer boundary at r = b escape. r = a r = b Figure 1: Annulus with random walkers (black dots) (a) You will solve this problem using Laplace's equation. Why is Laplace's equation an appro- priate choice for this problem? (b) Derive the 2D Laplacian operator in polar coordinates, using the following relationship between x, y and r, 0: x = r cos(0) y = r sin(0) (c) Set up and solve Laplace's equation in polar coordinates. (Hint: Propose a solution of the form P = ra.) (d) Solve for the probability of escape: P(r = b)/P(r = a).