# Syllabus

 When you click the Amazon logo to the left of any citation and purchase the book (or other media) from Amazon.com, MIT OpenCourseWare will receive up to 10% of this purchase and any other purchases you make during that visit. This will not increase the cost of your purchase. Links provided are to the US Amazon site, but you can also support OCW through Amazon sites in other regions. Learn more.

## 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