Random Walks and Diffusion

Syllabus -- *Revised Feb. 13, 2001*

**Instructor:** Prof. Martin Z. Bazant,
bazant@math.mit.edu,
253-1713.

**Office Hours:** Mondays 2-3pm, Wednesdays 3-4pm,
2-363B

** Three Problem Sets:** Due on Tuesdays... February 27,
March 20, May 1.

** Lecture Summaries and Homework Solutions:** Each student will
write detailed summaries (ideally in latex) of one or two lectures
and/or solutions for some of the homework problems.

** One Final Project:** Topic must be approved by Tuesday
April 24. A brief presentation will be given on May 15 or May 17. A
written report is due by Monday May 21. (No final exam!)

** Grading: ** Grading will be based on the problem sets (30%),
lecture summaries and/or problem solutions (20%), and the final
project presentation (20%) and report (30%). The homeworks will be
self-graded, as soon as the solutions become available. The instructor
will do some quality control.

** Required Books:** None. The course will be based on original
lectures by the instructor and handouts. A review of probability from
Chap. 1 of Bouchaud and Potters (below) is available at MIT CopyTech.

** Recommended Books:**
Barry Hughes, * Random Walks and Random Environments*, Vol. 1
(Oxford, 1996); J. Crank, * Mathematics of Diffusion* (Oxford,
second ed., 1975); D. Stauffer and A. Aharony, * Introduction to
Percolation Theory* (Taylor & Francis, second ed., 1992). H. Risken,
*The Fokker-Planck Equation* (Springer, second ed., 1989).

** Reserved Library Books:** J.-P. Bouchaud and M. Potters, *
Theory of Financial Risks* (Cambridge, 2000); H. C. Berg, *
Random Walks in Biology* (Princeton, 1983); N. G. van Kampen, *
Stochastic Processes in Physics and Chemistry* (Elsevier, 1992);
Barry Hughes, * Random Walks and Random Environments*, Vol. 2
(Oxford, 1996).

**Random Walkers and Diffusion**- Bernoulli, Polya, Pearson, and Rayleigh walks
- Bachelier-Kolmogorov equation, moment expansion, Fokker-Planck equation, Einstein relation
- Green functions, intermediate asymptotics, regular and singular perturbations
- Fourier/Laplace analysis, Central Limit Theorem(s), saddle point method
- Extreme events, order statistics, edge of the central region, fat tails
- Levy flights, anomalous (super)diffusion
- Polya's theorem, return probabilities, first passage times
- Adsorption, capture, growth
- Ito calculus, Langevin/Fokker-Planck "calculus"
- Options pricing and hedging: Black-Scholes and beyond
- Montroll-Weiss continuous-time walks, master equations, anomalous (sub)diffusion
- Queueing theory and Langmuir adsorption

**Random Environments**- Percolation and cluster statistics, spanning probability
- Renormalization groups largest cluster distribution
- Random walks in random environments

**Interacting Random Walkers**- Electrolytes, Debye screening, boundary layers
- Reaction-diffusion fronts