problemset4

Problemset4

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: −vx0 /2D 0 t x0 � 4πDq 3 2 e−x0 /4Dq e−v 2 q/4D dq. Using the substitution u2 = x2 /4Dt and the P´clet number Pe = v x0 /2D, we find e �∞ 2 2 2 2 e−u −Pe /4u du S (t) = 1 − √ e−Pe √ π x/ 4Dt � � �� √ 1 x0 Pe 4Dt = 1− 1 − erf √ + 2 x0 2 4Dt � � �� √ e−2Pe x0 Pe 4Dt + 1 − erf √ − . 2 2 x0 4Dt � � �� √ 1 −Pe−|Pe| |Pe| 4Dt ∼ 1− e 2 − erfc . 2 x0 2 As t → ∞, we get two different behaviors for S (t), depending on the sign of v : � 1 − e−2Pe for Pe > 0 S (t) ∼ x −Pe2 Dt/x2 0 √0 e for Pe ≤ 0 Pe π Dt � 1 − −vx0 /D for v > 0 �e ∼ 4D −v 2 t/4D for v ≤ 0. e πv 2 t From these expressions, we see that if v > 0 then there is a probability of e−vx0 /D of eventual first passage. M. Z. Bazant – 18.366 Random Walks and Diffusion – Problem Set 4 Solutions 1.3 3 Minimum first passage time Let the random variables for the first passage times be T1 , T2 , . . . , TN . Since the walkers are inde­ pendent, we know that P (min{T1 , T2 , . . . , TN } > t) = P (T1 > t, T2 > t, . . . , TN > t) = P (T1 > t)P (T2 > t) . . . P (TN > t) = S (t)N and hence the PDF of the minimum first passage time is given by d S (t)N = f (t)N S (t)N −1 dt where f (t) and S (t) are explicitly given in the previous sections. pn (t) = − 2 2.1 First passage for anomalous walks Unbiased Cauchy walk Appendix A provides a simple C++ code to simulate first passage times for the Cauchy walk. For a large number of trials, it was found that the standard C++ math rand() function was inadequate, and that slight biases in the probabilities around n = 30 could be seen. A second code, listed in appendix B, was therefore written, making use of the more advanced random number generation routines found in the GNU Scientific Library (GSL) [1]. The GSL code was run with 2 × 1010 trials for the case of d = 0.0. Wal...
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