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Browse materialsMIT 18.366 · Fall 2006

Random Walks and Diffusion

Subjects: Mathematics · Statistics

Art: Max Brückner (1860 – 1934), Brückner Vielflache Tafel 07 · Public domain

Instructor: Prof. Martin Bazant
Term: Fall 2006
Level: Graduate
What's inside: 1 lecture note · 123 other files · 4 web pages
Materials open: MIT OCW

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Source: MIT OpenCourseWare 18.366 Random Walks and Diffusion, Fall 2006, Prof. Martin Bazant — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

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Exams

Study Materials

41 files

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MIT OpenCourseWare, Prof. Martin Bazant, 18.366 Random Walks and Diffusion, Fall 2006.

License: CC BY-NC-SA 4.0 · Edition: Fall 2006 · Open the original course on OCW ↗

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Random Walks and Diffusion

Before you start

What's here, and what isn't

Not included here

Materials by section

Lecture Notes

Assignments

Exams

Study Materials

Other materials from this course

Lecture 1: Overview History (Pearson, Rayleigh, Einstein, Bachelier) Normal vs. Anomalous Diffusion Mechanisms for Anomalous Diffusion

Lecture 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

Lecture 3: The Central Limit Theorem and the Diffusion Equation Multi-dimensional CLT for Ssums of IID Random Vectors Continuum Derivation Involving the Diffusion Equation

Lecture 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

Lecture 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

Lecture 6: Power-law “Fat Tails” Power-law Tails, Diverging Moments and Singular Characteristic Functions Additivity of Tail Amplitudes

Lecture 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

Lecture 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

Lecture 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

Lecture 10: Weakly Non-identical Steps Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation. Probability Flux Modified Kramers-Moyall Cumulant Expansion for Identical Steps

Lecture 11: Nonlinear Drift Interacting Random Walkers, Concentration-dependent Drift Nonlinear Waves in Traffic Flow, Characteristics, Shocks, Burgers’ Equation Surface Growth, Kardar-Parisi-Zhang Equation

Lecture 12: Nonlinear Diffusion Cole-Hopf Transformation, General Solution of Burgers Equation Concentration-dependent Diffusion, Chemical Potential. Rechargeable Batteries, Steric Effects

Lecture 13: Return Probability on a Lattice Probability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya’s Theorem

Lecture 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

Lecture 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

Lecture 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

Lecture 18: Hitting Probabilities in Two Dimensions Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line

Lecture 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

Lecture 20: Diffusion-limited Aggregation Harmonic Measure, Hastings-Levitov Algorithm, Comparison of Discrete and Continuous Dynamics Overview of Mechanisms for Anomalous Diffusion Non-identical Steps

Lecture 22: Levy Flights Superdiffusion and Limiting Levy Distributions for Steps with Infinite Variance, Examples, Size of the Largest Step, Frechet Distribution

Lecture 23: Continuous-time Random Walks Laplace Transform. Renewal Theory Montroll-Weiss Formulation of CTRW DNA Gel Electrophoresis

Lecture 24: Fractional Diffusion Equations CLT for CTRW Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation

Lecture 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”

Lecture 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