Random Walks and Diffusion

Lecture Notes

Topics covered in lectures in 2006 are listed below. In some cases, links are given to new lecture notes by student scribes. All scribed lecture notes are used with the permission of the student named in the file. The recommended reading refers to the lectures notes and exam solutions from previous years or to the books listed below. Lecture notes from previous years are also found in the study materials section.

Recommended Texts

Hughes, B. Random Walks and Random Environments. Vol. 1. Oxford, UK: Clarendon Press, 1996. ISBN: 0198537883.

Redner, S. A Guide to First Passage Processes. Cambridge, UK: Cambridge University Press, 2001. ISBN: 0521652480.

Risken, H. The Fokker-Planck Equation. 2nd ed. New York, NY: Springer-Verlag, 1989. ISBN: 0387504982.

Further Readings

Bouchaud, J. P., and M. Potters. Theory of Financial Risks. Cambridge, UK: Cambridge University Press, 2000. ISBN: 0521782325.

Crank, J. Mathematics of Diffusion. 2nd ed. Oxford, UK: Clarendon Press, 1975. ISBN: 0198533446.

Rudnick, J., and G. Gaspari. Elements of the Random Walk. Cambridge, UK: Cambridge University Press, 2004. ISBN: 0521828910.

Spitzer, F. Principles of the Random Walk. 2nd ed. New York, NY: Springer-Verlag, 2001. ISBN: 0387951547.

LEC #	TOPICS	2006 Lecture NOTES	2006 READINGS
1	Overview History (Pearson, Rayleigh, Einstein, Bachelier) Normal vs. Anomalous Diffusion Mechanisms for Anomalous Diffusion		2005 Lecture 1 (PDF) Hughes
I. Normal Diffusion
I.A. Linear Diffusion
2	Moments, Cumulants, and Scaling Markov Chain for the Position (in d Dimensions), Exact Solution by Fourier Transform, Moment and Cumulant Tensors, Additivity of Cumulants, “Square-root Scaling” of Normal Diffusion		2005 Lecture 2 (PDF) Hughes
3	The Central Limit Theorem and the Diffusion Equation Multi-dimensional CLT for Ssums of IID Random Vectors Continuum Derivation Involving the Diffusion Equation		2005 Lecture 1 (PDF) 2005 Lecture 3 (PDF)
4	Asymptotic Shape of the Distribution Berry-Esseen Theorem Asymptotic Analysis Leading to Edgeworth Expansions, Governing Convergence to the CLT (in one Dimension), and more Generally Gram-Charlier Expansions for Random Walks Width of the Central Region when Third and Fourth Moments Exist		2005 Lecture 3 (PDF) 2005 Lecture 4 (PDF) Hughes Feller
5	Globally Valid Asymptotics Method of Steepest Descent (Saddle-Point Method) for Asymptotic Approximation of Integrals Application to Random Walks Example: Asymptotics of the Bernoulli Random Walk		2005 Lecture 6 (PDF) 2005 Lecture 7 (PDF) Hughes
6	Power-law “Fat Tails” Power-law Tails, Diverging Moments and Singular Characteristic Functions Additivity of Tail Amplitudes		2005 Lecture 5 (PDF) 2005 Lecture 6 (PDF) Bouchaud and Potters
7	Asymptotics with Fat Tails Corrections to the CLT for Power-law Tails (but Finite Variance) Parabolic Cylinder Functions and Dawson’s Integral A Numerical Example Showing Global Accuracy and Fast Convergence of the Asymptotic Approximation	(PDF) (Courtesy of Damian Burch. Used with permission.) Numerical Example (PDF) (Courtesy of Chris H. Rycroft. Used with permission.)	2005 Lecture 5 (PDF)
8	From Random Walks to Diffusion Examples of Random Walks Modeled by Diffusion Equations Flagellar Bacteria Run and Tumble Motion, Chemotaxis Financial Time Series Additive Versus Multiplicative Processes	(PDF) (Courtesy of Daniel Rudoy. Used with permission.)	2005 Lecture 10 (PDF) Bouchaud and Potters
9	Discrete Versus Continuous Stochastic Processes Corrections to the Diffusion Equation Approximating Discrete Random Walks with IID Steps Fat Tails and Riesz Fractional Derivatives Stochastic Differentials, Wiener Process	(PDF) (Courtesy of Kwai Hung Henry Lam. Used with permission.)	2005 Lecture 8 (PDF) 2005 Lecture 9 (PDF) 2005 Lecture 13 (PDF) Risken
10	Weakly Non-identical Steps Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation. Probability Flux Modified Kramers-Moyall Cumulant Expansion for Identical Steps		2005 Lecture 8 (PDF) 2005 Lecture 9 (PDF) 2005 Lecture 13 (PDF) Risken
I.B. Nonlinear Diffusion
11	Nonlinear Drift Interacting Random Walkers, Concentration-dependent Drift Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers’ Equation Surface Growth, Kardar-Parisi-Zhang Equation	(PDF) (Courtesy of Lou Odette. Used with permission.)
12	Nonlinear Diffusion Cole-Hopf Transformation, General Solution of Burgers Equation Concentration-dependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects		Problem set 3 solutions
I.C. First Passage and Exploration
13	Return Probability on a Lattice Probability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya’s Theorem	(PDF) (Courtesy of Chris H. Rycroft. Used with permission.)	2005 Lecture 17 (PDF) 2005 Lecture 18 (PDF) Hughes Redne
14	The Arcsine Distribution Reflection Principle and Path Counting for Lattice Random Walks, Derivation of the Discrete Arcsine Distribution for the Fraction of Time Spent on One Side of the Origin, Continuum Limit	(PDF) (Courtesy of Chris H. Rycroft. Used with permission.)	Feller
15	First Passage in the Continuum Limit General Formulation in One Dimension Smirnov Density Minimum First Passage Time of a Set of N Random Walkers		2005 Lecture 16 (PDF) Exam 2 (problem 2)
16	First Passage in Arbitrary Geometries General Formulation in Higher Dimensions, Moments of First Passage Time, Eventual Hitting Probability, Electrostatic Analogy for Diffusion, First Passage to a Sphere		2005 Lecture 18 (PDF) Redner Risken
17	Conformal Invariance Conformal Transformations (Analytic Functions of the Plane, Stereographic Projection from the Plane to a Sphere,…), Conformally Invariant Transport Processes (Simple Diffusion, Advection-diffusion in a Potential Flow,…), Conformal Invariance of the Hitting Probability	(PDF) (Courtesy of Yee Lok Wong. Used with permission.)	2003 Lecture 23 (PDF) An Article Redner
18	Hitting Probabilities in Two Dimensions Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line		2003 Lecture 23 (PDF) Redner
19	Applications of Conformal Mapping First Passage to a Circle, Wedge/Corner, Parabola. Continuous Laplacian Growth, Polubarinova-Galin Equation, Saffman-Taylor Fingers, Finite-time Singularities		2003 Lecture 23 (PDF) 2003 Lecture 24 (PDF)
20	Diffusion-limited Aggregation Harmonic Measure, Hastings-Levitov Algorithm, Comparison of Discrete and Continuous Dynamics Overview of Mechanisms for Anomalous Diffusion Non-identical Steps		2003 Lectures 25 (PDF) 2003 Lecture 14 (PDF) 2003 Lecture 15 (PDF) A Review Article
II. Anomalous Diffusion
II.A. Breakdown of the CLT
21	Polymer Models: Persistence and Self-avoidance Random Walk to Model Entropic Effects in Polymers, Restoring Force for Stretching; Persistent Random Walk to Model Bond-bending Energetic Effects, Green-Kubo Relation, Persistence Length, Telegrapher’s Equation; Self-avoiding Walk to Model Steric Effects, Fisher-Flory Estimate of the Scaling Exponent		2005 Lectures 19 (PDF) 2005 Lecture 20 (PDF) 2003 Lecture 9 (PDF) 2003 Lecture 10 (PDF) 2003 Lecture 11 (PDF)
22	Levy Flights Superdiffusion and Limiting Levy Distributions for Steps with Infinite Variance, Examples, Size of the Largest Step, Frechet Distribution		2005 Lecture 22 (PDF) 2003 Lecture 12 (PDF) 2003 Lecture 13 (PDF) Hughes
II.B. Continuous-Time Random Walks
23	Continuous-time Random Walks Laplace Transform. Renewal Theory Montroll-Weiss Formulation of CTRW DNA Gel Electrophoresis	(PDF) (Courtesy of Michael Vahey. Used with permission.)	2005 Lecture 23 (PDF) 2003 Lecture 15 (PDF) 2003 Lecture 16 (PDF) 2003 Lecture 17 (PDF)
24	Fractional Diffusion Equations CLT for CTRW Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation		2005 Lecture 24 (PDF) 2003 Lecture 18 (PDF)
25	Non-separable Continuous-time Random Walks “Phase Diagram” for Anomalous Diffusion: Large Steps Versus Long Waiting Times Application to Flagellar Bacteria. Hughes’ General Formulation of CTRW with Motion between “turning points”		2005 Lecture 25 (PDF) 2005 Lecture 26 (PDF) Hughes
26	Leapers and Creepers Hughes’ Leaper and Creeper Models Leaper Example: Polymer Surface Adsorption Sites and Cross-sections of a Random Walk Creeper Examples: Levy Walks, Bacterial Motion, Turbulent Dispersion		2005 Lecture 26 (PDF) Hughes