Random Walks and Diffusion

Syllabus

Course Meeting Times

Lectures: 2 sessions / week, 1.5 hours / session

Prerequisite

18.305 (Advanced Analytic Methods in Science and Engineering) or permission of the instructor. A basic understanding of probability, partial differential equations, transforms, complex variables, asymptotic analysis, and computer programming would be helpful, but an ambitious student could take the class to learn some of these topics. Interdisciplinary registration is encouraged.

Problem Sets

There are five problem sets for this course. Solutions should be clearly explained. You are encouraged to work in groups and consult various references (but not solutions to problem sets from a previous term), although you must prepare each solution independently, in your own words.

Midterm Exam

There will be one take-home midterm exam. It will be handed out in class and will be due at the next session.

Final Project

There is no final exam, only a written final-project report, due at the last lecture. The topic must be selected and approved six weeks earlier.

Grading

activities	percentages
Problem Sets	40%
Midterm Exam	30%
Final Project	30%

Topics

1. Normal Diffusion (12+ Lectures)
   • Central Limit Theorem, Asymptotic Approximations, Drift and Dispersion, Fokker-Planck Equation, First Passage, Return, Exploration.
2. Anomalous Diffusion (10+ Lectures)
   • Non-identical Steps, Persistence and Self Avoidance, Levy Flights, Continuous Time Random Walk, Fractional Diffusion Equations, Random Environments.
3. Nonlinear Diffusion (4 Lectures, As Time Permits)
   • Interacting Walkers, Steric Effects, Electrolytes, Porous Media, DLA.

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.

Calendar

LEC #	TOPICS	KEY DATES
1	Overview History (Pearson, Rayleigh, Einstein, Bachelier) Normal vs. Anomalous Diffusion Mechanisms for Anomalous Diffusion
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
3	The Central Limit Theorem and the Diffusion Equation Multi-dimensional CLT for Sums of IID Random Vectors Continuum Derivation Involving the Diffusion Equation
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
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	Problem set 1 due
6	Power-law “Fat Tails” Power-law Tails, Diverging Moments and Singular Characteristic Functions Additivity of Tail Amplitudes
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
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	Problem set 2 due
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
10	Weakly Non-identical Steps Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation Probability Flux Modified Kramers-Moyall Cumulant Expansion for Identical Steps
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
12	Nonlinear Diffusion Cole-Hopf Transformation, General Solution of Burgers Equation Concentration-dependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects
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
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	Problem set 3 due
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
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
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
18	Hitting Probabilities in Two Dimensions Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line	Problem set 4 due
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	Midterm exam out
20	Diffusion-limited Aggregation Harmonic Measure, Hastings-Levitov Algorithm, Comparison of Discrete and Continuous Dynamics Overview of Mechanisms for Anomalous Diffusion. Non-identical Steps	Midterm exam due
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
22	Levy Flights Superdiffusion and Limiting Levy Distributions for Steps with Infinite Variance, Examples, Size of the Largest Step, Frechet Distribution
II.B. Continuous-Time Random Walks
23	Continuous-time Random Walks Laplace Transform Renewal Theory Montroll-Weiss Formulation of CTRW DNA Gel Electrophoresis
24	Fractional Diffusion Equations CLT for CTRW Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation
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”	Problem set 5 due
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