Course Meeting Times
Lectures: 2 sessions / week, 1.5 hours / session
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.
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.
There will be one take-home midterm exam. It will be handed out in class and will be due at the next session.
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.
- Normal Diffusion (12+ Lectures)
- Central Limit Theorem, Asymptotic Approximations, Drift and Dispersion, Fokker-Planck Equation, First Passage, Return, Exploration.
- Anomalous Diffusion (10+ Lectures)
- Non-identical 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.
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.
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||KEY DATES|
History (Pearson, Rayleigh, Einstein, Bachelier)
Normal vs. Anomalous Diffusion
Mechanisms for Anomalous Diffusion
|I. Normal Diffusion|
|I.A. Linear Diffusion|
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
The Central Limit Theorem and the Diffusion Equation
Multi-dimensional CLT for Sums of IID Random Vectors
Continuum Derivation Involving the Diffusion Equation
Asymptotic Shape of the Distribution
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
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|
Power-law "Fat Tails"
Power-law Tails, Diverging Moments and Singular Characteristic Functions
Additivity of Tail Amplitudes
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
From Random Walks to Diffusion
Examples of Random Walks Modeled by Diffusion Equations
Run and Tumble Motion, Chemotaxis
Financial Time Series
Additive Versus Multiplicative Processes
|Problem set 2 due|
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
Weakly Non-identical Steps
Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation
Modified Kramers-Moyall Cumulant Expansion for Identical Steps
|I.B. Nonlinear Diffusion|
Interacting Random Walkers, Concentration-dependent Drift
Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers' Equation
Surface Growth, Kardar-Parisi-Zhang Equation
Cole-Hopf Transformation, General Solution of Burgers Equation
Concentration-dependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects
|I.C. First Passage and Exploration|
Return Probability on a Lattice
Probability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya's Theorem
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|
First Passage in the Continuum Limit
General Formulation in One Dimension
Minimum First Passage Time of a Set of N Random Walkers
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
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
Hitting Probabilities in Two Dimensions
Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line
|Problem set 4 due|
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|
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|
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
Superdiffusion and Limiting Levy Distributions for Steps with Infinite Variance, Examples, Size of the Largest Step, Frechet Distribution
|II.B. Continuous-Time Random Walks|
Continuous-time Random Walks
Montroll-Weiss Formulation of CTRW
DNA Gel Electrophoresis
Fractional Diffusion Equations
CLT for CTRW
Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation
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|
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