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 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
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
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 ScalingMarkov 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 EquationMultidimensional CLT for Sums of IID Random Vectors Continuum Derivation Involving the Diffusion Equation 

4 
Asymptotic Shape of the DistributionBerryEsseen 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 AsymptoticsMethod 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 TailsCorrections 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 DiffusionExamples 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 ProcessesCorrections to the Diffusion Equation Approximating Discrete Random Walks with IID Steps Fat Tails and Riesz Fractional Derivatives Stochastic Differentials, Wiener Process 

10 
Weakly Nonidentical StepsChapmanKolmogorov Equation, KramersMoyall Expansion, FokkerPlanck Equation Probability Flux Modified KramersMoyall Cumulant Expansion for Identical Steps 

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

12 
Nonlinear DiffusionColeHopf 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 LatticeProbability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya’s Theorem 

14 
The Arcsine DistributionReflection 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 LimitGeneral Formulation in One Dimension Smirnov Density Minimum First Passage Time of a Set of N Random Walkers 

16 
First Passage in Arbitrary GeometriesGeneral Formulation in Higher Dimensions, Moments of First Passage Time, Eventual Hitting Probability, Electrostatic Analogy for Diffusion, First Passage to a Sphere 

17 
Conformal InvarianceConformal 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 DimensionsPotential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line 
Problem set 4 due 
19 
Applications of Conformal MappingFirst Passage to a Circle, Wedge/Corner, Parabola. Continuous Laplacian Growth, PolubarinovaGalin Equation, SaffmanTaylor Fingers, Finitetime Singularities 
Midterm exam out 
20 
Diffusionlimited AggregationHarmonic 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 SelfavoidanceRandom 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 FlightsSuperdiffusion 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 WalksLaplace Transform Renewal Theory MontrollWeiss Formulation of CTRW DNA Gel Electrophoresis 

24 
Fractional Diffusion EquationsCLT 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 CreepersHughes’ Leaper and Creeper Models Leaper Example: Polymer Surface Adsorption Sites and Crosssections of a Random Walk Creeper Examples: Levy Walks, Bacterial Motion, Turbulent Dispersion 