Solutions to Problem Set 3
Chris H. Rycroft
October 30, 2006
1
1.1
Inelastic diusion
The PDF of PN (x)
The PDF of the random variable an Xn is p(xan )an . If Xn has characteristic function p(k )
then an Xn has characteristic function
p(xan )
n
eikx
dx
Lecture 4: Asymptotics in the Central Region
Scribe: Erik Allen
February 10, 2005
Lecture 3 provided a derivation of the Central Limit Theorem - showing that a distributon
PN (x) tends to the multivariate normal distribution in the central region for suci
Lecture 2: Moments, Cumulants, and Scaling
Scribe: Ernst A. van Nierop (and Martin Z. Bazant)
February 4, 2005
Handouts:
Historical excerpt from Hughes, Chapter 2: Random Walks and Random Flights.
Problem Set #1
1
1.1
The Position of a Random Walk
Gener
Lecture 1: Introduction to Random Walks and Diusion
Scribe: Chris H. Rycroft (and Martin Z. Bazant)
Department of Mathematics, MIT
February 1, 2005
History
The term random walk was originally proposed by Karl Pearson in 19051 . In a letter to Na
ture, he
Lecture 3: Central Limit Theorem
Scribe: Jacy Bird
(Division of Engineering and Applied Sciences, Harvard)
February 8, 2003
The goal of todays lecture is to investigate the asymptotic b ehavior of P N ( ) for large N . We
x
use Laplaces Method to show tha
Solutions to Problem Set 5
Edited by Chris H. Rycroft
December 5, 2006
1
Restoring force for highly stretched p olymers
1.1
A globally valid asymptotic approximation
Using the substitution t = ua the expression can be rewritten as
PN (r) =
=
t
sin t N dt
Solutions to Problem Set 4
Chris H. Rycroft
November 10, 2006
1
First passage for biased diusion
1.1
The rst passage time to the origin
The PDF (x, t) of a continuous diusion process with drift velocity v and diusivity D satises a
Fokker-Planck equation
t
Solutions to the Midterm Exam
Martin Z. Bazant
November 16, 2006
1
1.1
Discrete versus continuous steps in a random walk
Finding a generating function and D2 (a)
For pn = C a|n| , the probability generating function is
P (z ) =
pn z n
n=
=C
n
(az ) +
n=0
Solutions to Exam 1
Chris H. Rycroft and Martin Z. Bazant
March 11, 2005
1
Multivariate normal random walk
To calculate the probability density function of XN , we begin by nding the characteristic function
of a single step. We know that
x
x
exp 1 Cn 1
2
18.366 Random Walks and Diusion, Spring 2005, M. Z. Bazant.
Solutions to Exam 2
Martin Z. Bazant
1. Electrochemical Equilibrium.
(a) The FokkerPlanck (or NernstPlanck) equations for diusion and drift of ions in the mean
eld electrostatic potential, , are:
Solutions to Problem Set 1
Edited by Chris H. Rycroft
February 17, 2005
1
Rayleighs Random Walk
We consider an isotropic random walk in 3 dimensions with independent identical displacements of
length a, given by the PDF
(r a)
p( ) =
x
(r = | |)
x
4a2
for
Solutions to Problem Set 2
Edited by Chris H. Rycroft
October 5, 2006
1
Asymptotics of Percentile Order Statistics
Let Xi (i = 1, . . . , N ) be independently identically distributed (IID) continuous random variables
()
with CDF, P (x) = P(Xi x), and PDF,