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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 takehome 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 finalproject report, due at the last lecture. The topic must be selected and approved six weeks earlier.
Grading
Grading criteria.
activities 
percentages 
Problem Sets 
40% 
Midterm Exam 
30% 
Final Project 
30% 
Topics

Normal Diffusion (12+ Lectures)

Central Limit Theorem, Asymptotic Approximations, Drift and Dispersion, FokkerPlanck Equation, First Passage, Return, Exploration.

Anomalous Diffusion (10+ Lectures)

Nonidentical Steps, Persistence and Self Avoidance, Levy Flights, Continuous Time Random Walk, Fractional Diffusion Equations, Random Environments.

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 FokkerPlanck Equation. 2nd ed. New York, NY: SpringerVerlag, 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: SpringerVerlag, 2001. ISBN: 0387951547.
Calendar
Course 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, "Squareroot Scaling" of Normal Diffusion


3 
The Central Limit Theorem and the Diffusion Equation
Multidimensional CLT for Sums of IID Random Vectors
Continuum Derivation Involving the Diffusion Equation


4 
Asymptotic Shape of the Distribution
BerryEsseen Theorem
Asymptotic Analysis Leading to Edgeworth Expansions, Governing Convergence to the CLT (in one Dimension), and more Generally GramCharlier Expansions for Random Walks
Width of the Central Region when Third and Fourth Moments Exist


5 
Globally Valid Asymptotics
Method of Steepest Descent (SaddlePoint Method) for Asymptotic Approximation of Integrals
Application to Random Walks
Example: Asymptotics of the Bernoulli Random Walk

Problem set 1 due 
6 
Powerlaw "Fat Tails"
Powerlaw Tails, Diverging Moments and Singular Characteristic Functions
Additivity of Tail Amplitudes


7 
Asymptotics with Fat Tails
Corrections to the CLT for Powerlaw 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 Nonidentical Steps
ChapmanKolmogorov Equation, KramersMoyall Expansion, FokkerPlanck Equation
Probability Flux
Modified KramersMoyall Cumulant Expansion for Identical Steps


I.B. Nonlinear Diffusion 
11 
Nonlinear Drift
Interacting Random Walkers, Concentrationdependent Drift
Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers' Equation
Surface Growth, KardarParisiZhang Equation


12 
Nonlinear Diffusion
ColeHopf Transformation, General Solution of Burgers Equation
Concentrationdependent 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, Advectiondiffusion 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, PolubarinovaGalin Equation, SaffmanTaylor Fingers, Finitetime Singularities

Midterm exam out 
20 
Diffusionlimited Aggregation
Harmonic Measure, HastingsLevitov Algorithm, Comparison of Discrete and Continuous Dynamics
Overview of Mechanisms for Anomalous Diffusion. Nonidentical Steps

Midterm exam due 
II. Anomalous Diffusion 
II.A. Breakdown of the CLT 
21 
Polymer Models: Persistence and Selfavoidance
Random Walk to Model Entropic Effects in Polymers, Restoring Force for Stretching; Persistent Random Walk to Model Bondbending Energetic Effects, GreenKubo Relation, Persistence Length, Telegrapher's Equation; Selfavoiding Walk to Model Steric Effects, FisherFlory 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. ContinuousTime Random Walks 
23 
Continuoustime Random Walks
Laplace Transform
Renewal Theory
MontrollWeiss Formulation of CTRW
DNA Gel Electrophoresis


24 
Fractional Diffusion Equations
CLT for CTRW
Infinite Man Waiting Time, MittagLeffler Decay of Fourier Modes, Timedelayed Flux, Fractional Diffusion Equation


25 
Nonseparable Continuoustime 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 Crosssections of a Random Walk
Creeper Examples: Levy Walks, Bacterial Motion, Turbulent Dispersion

