18.366 | Fall 2006 | Graduate

Random Walks and Diffusion

Assignments

The problem sets were due in the lectures noted in the table. 40% of the grade is based on the problem sets.

LEC # ASSIGNMENTS SOLUTIONS
5 Problem Set 1 (PDF) (PDF)
8 Problem Set 2 (PDF) (PDF)
14 Problem Set 3 (PDF) (PDF)
18 Problem Set 4 (PDF) (PDF)
25 Problem Set 5 (PDF) (PDF)

The following assignments are from the Spring 2005 version of the course.

LEC # ASSIGNMENTS TOPICS
4 Problem Set 1 (PDF)

Asymptotics of Rayleigh’s Random Walk, Central Limit Theorem, Gram-Charlier Expansion

Exact Solution for the Position of Cauchy’s Random Walk with Non-identical Steps

Computer Simulation of Pearson’s Random Walk to find the Fraction of Time Spent in the Right Half Plane (“Arcsine Law”) and the First Quadrant

8 Problem Set 2 (PDF)

Percentile Order Statistics, Asymptotics of the Median Versus the Mean

Computer Simulation of the Winding Angle for Pearson’s Random Walk, Logarithmic Scaling and Limiting Distribution

Globally-valid Saddle-point Asymptotics for a Random Walk with Exponentially Distributed Displacements

The Void Model for Granular Drainage, Continuum Limits for the Void Density (Mean Flow Profile) and the Position a Tracer Particle, Exact Similarity Solutions for Parabolic Flow to a Point Orifice

15 Problem Set 3 (PDF)

Modified Kramers-Moyall Expansion for a General Discrete Markov Process

Black-Scholes Formula for a Call Option, Interpretation as Risk Neutral Valuation, Put-call Parity

Continuum Limit of Bouchaud-Sornette Theory for Options with Residual Risk (Corrections to the Black-Scholes Equation)

24 Problem Set 4 (PDF)

Linear Polymer Structure, Random Walk with Exponentially Decaying Correlations, Depending on Temperature

Polymer Surface Adsorption, First Passage to a Plane, Levy Flight for Adsorption Sites, Scalings with the Chain Length

Solution to the Telegrapher’s Equation, Fourier-Laplace Transform, Wave and Diffusion Limits, Exact Green Function

Inelastic Diffusion, Random Walk with Exponentially Decaying Steps, Approach to the Central Limit Theorem

Course Info

Instructor
Departments
As Taught In
Fall 2006
Level
Learning Resource Types
Problem Sets with Solutions
Exams with Solutions
Lecture Notes