Random Walks and Diffusion

Study Materials

These lecture notes from previous years were scribed by students who took this class and are used with their permission. The notes were written by the students as homework assignments. In most cases, Prof. Bazant has reviewed the notes and has made revisions or extensions to the text. However, please be advised that many unedited portions still exist. Notes from the 2003 version of this class have been included in the second table below, beginning with lecture nine. They provide supplementary information on the topics of anomalous diffusion and diffusion-limited growth. All of the notes in this section were provided to students taking the 2006 version of this class.

LEC #	TOPICS	NOTES
I. Normal Diffusion: Fundamental Theory
1	Introduction History; Simple Analysis of the Isotropic Random Walk in d Dimensions, Using the Continuum Limit; Bachelier and Diffusion Equations; Normal Versus Anomalous Diffusion	Chris Rycroft (PDF)
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	Ernst van Nierop (PDF)
3	The Central Limit Theorem Multi-dimensional CLT for Sums of IID Random Vectors (Derived by Laplace’s Method of Asymptotic Expansion), Edgeworth Expansion for Convergence to the CLT With Finite Moments	Jacy Bird (PDF)
4	Asymptotics Inside the Central Region Gram-Charlier Expansions for Random Walks, Berry-Esseen Theorem, Width of the “Central Region”, “Fat” Power-law Tails	Erik Allen (PDF)
5	Asymptotics with Fat Tails Singular Characteristic Functions, Generalized Gram-Charlier Expansions, Dawson’s Integral, Edge of the Central Region, Additivity of Power-law Tails	(PDF)
6	Asymptotics Outside the Central Region Additivity of Power-law Tails: Intuitive Explanation, “High-Order” Tauberian Theorem for the Fourier Transform; Laplace’s Method and Saddle-point Method, Uniformly Valid Asymptotics for Random Walks	Mustafa Sabri Kilic (PDF)
7	Approximations of the Bernoulli Random Walk Example of Saddle-point Asymptotics for a Symmetric Random Walk on the Integers, Detailed Comparison with Gram-Charlier Expansion and the Exact Combinatorial Solution	(PDF)
8	The Continuum Limit Application of the Bernoulli Walk to Percentile Order Statistics; Kramers-Moyall Expansion From Bachelier’s Equation for Isotropic Walks, Scaling Analysis, Continuum Derivation of the CLT via the Diffusion Equation	Ernst van Nierop (PDF)
9	Kramers-Moyall Cumulant Expansion Recursive Substitution in Kramers-Moyall Moment Expansion to Obtain Modified Coefficients in Terms of Cumulants, Continuum Derivation of Gram-Charlier Expansion as the Green Function for the Kramers-Moyall Cumulant Expansion	Jacy Bird (PDF)
I. Normal Diffusion: Some Finance
10	Applications in Finance Models for Financial Time Series, Additive and Multiplicative Noise, Derivative Securities, Bachelier’s Fair-game Price	Erik Allen (PDF)
11	Pricing and Hedging Derivative Securities Static Hedge to Minimize Risk, Optimal Trading by Linear Regression of the Random Payoff, Quadratic Risk Minimization, Riskless Hedge for a Binomial Process	J. F. (PDF) Additional Notes (PDF)
12	Black-Scholes and Beyond Riskless Hedging and Pricing on a Binomial Tree, Black-Scholes Equation in the Continuum Limit, Risk Neutral Valuation	Sergiy Sidenko (PDF) Additional notes on “Gram-Charlier” corrections for residual risk in Bouchaud-Sornette theory, by Ken Gosier (PDF) See also Problem Set 3.
13	Discrete versus Continuous Stochastic Processes Discrete Markov Processes in the Continuum Limit, Chapman-Kolomogorov Equation, Kramers-Moyall Moment Expansion, Fokker Planck Equation. Continuous Wiener Processes, Stochastic Differential Equations, Ito Calculus, Applications in Finance	Sergiy Sidenko (PDF)
I. Normal Diffusion: Some Physics
14	Applications in Statistical Mechanics Random Walk in an External Force Field, Einstein Relation, Boltzmann Equilibrium, Ornstein-Uhlenbeck Process, Ehrenfest Model	Kirill Titievsky (PDF)
15	Brownian Motion in Energy Landscapes Kramers Escape Rate From a Trap, Periodic Potentials, Asymmetric Structures, Brownian Ratchets and Molecular Motors (Guest Lecture by Armand Ajdari)	J. F. (PDF)
I. Normal Diffusion: First Passage
16	First Passage in the Continuum Limit General Formula for the First Passage Time PDF, Smirnov Density in One Dimension, First Passage to Boundaries by General Stochastic Processes	Mustafa Sabri Kilic (PDF)
17	Return and First Passage on a Lattice Return Probability in One Dimension, Generating Functions, First Passage and Return on a Lattice, Return of a Biased Bernoulli Walk, Reflection Principle (Guest Lecture by Chris Rycroft)	Ken Kamrin (PDF)
18	First Passage in Higher Dimensions Return and First Passage on a Lattice, Polya’s Theorem, Continuous First Passage in Complicated Geometries, Moments of the Time and the Location of First Passage, Electrostatic Analogy	Kirill Titievsky (PDF)
I. Normal Diffusion: Correlations
19	Polymer Models: Persistence and Self-Avoidance Random Walk Models of Polymers, Radius of Gyration, Persistent Random Walk, Self-avoiding Walk, Flory’s Scaling Theory	Allison Ferguson (PDF)
20	(Physical) Brownian Motion I Ballistic to Diffusive Transition, Correlated Steps, Green-Kubo Relation, Taylor’s Effective Diffusivity, Telegrapher’s Equation as the Continuum Limit of the Persistent Random Walk	Neville Sanjana (PDF)
21	(Physical) Brownian Motion II Langevin Equations, Stratonivich vs. Ito Stochastic Differentials, Multi-dimensional Fokker-Planck Equation, Kramers Equation (Vector Ornstein-Uhlenbeck Process) for the Velocity and Position, Breakdown of Normal Diffusion at Low Knudsen Number, Levy Flight for a Particle Between Rough Parallel Plates	Ken Kamrin (PDF)
II. Anomalous Diffusion
22	Levy Flights Steps with Infinite Variance, Levy Stability, Levy Distributions, Generalized Central Limit Theorems (Guest Lecture by Chris Rycroft)	Neville Sanjana (PDF)
23	Continuous-Time Random Walks Random Waiting Time Between Steps, Montroll-Weiss Theory of Separable CTRW, Formulation in Terms of Random Number of Steps, Tauberian Theorems for the Laplace Transform and Long-time Asymptotics	Chris Rycroft (PDF)
24	Fractional Diffusion Equations Continuum Limits of CTRW; Normal Diffusion Equation for Finite Mean Waiting Time and Finite Step Variance, Exponential Relaxation of Fourier Modes; Fractional Diffusion Equations for Super-diffusion (Riesz Fractional Derivative) and Sub-diffusion (Riemann-Liouville Fractional Derivative); Mittag-Leffler Power-law Relaxation of Fourier Modes	Yuxing Ben (PDF)
25	Large Jumps and Long Waiting Times CTRW Steps with Infinite Variance and Infinite Mean Waiting Time, “Phase Diagram” for Anomalous Diffusion, Polymer Surface Adsorption (Random Walk Near a Wall), Multidimensional Levy Stable Laws	Geraint Jones (PDF)
26	Leapers and Creepers Hughes’ Formulation of Non-separable CTRW, Leapers: Cauchy-Smirnov Non-separable CTRW for Polymer Surface Adsorption, Creepers: Levy Walks for Tracer Dispersion in Homogenous Turbulence	Geraint Jones (PDF)

Notes from the 2003 Version of this Class

LEC #	TOPICS	SCRIBE NOTES
I. Additional Notes on Anomalous Diffusion
9	Correlations Between Steps Applications (Polymers, Finance, Turbulent Diffusion,…), Green-Kubo Formula, Anomalous Diffusion, Exponentially Decaying Correlations, Transition from Ballistic to Diffusive Scaling	Marat Rvachev (PDF)
10	Persistent Random Walks and the Telegrapher’s Equation Markov Chain for the Persistent Random Walk on the Integers; Continuum Limits: Diffusion Equation with Diffusive Scaling, Telegrapher’s Equation with Ballistic Scaling	Greg Randall (PDF)
11	More on Persistence and Self-Avoidance Exact Solution of the Markov Chain Difference Equations by Discrete Fourier Transform, CLT, Green Function for the Telegrapher’s Equation and Transition from Ballistic to Diffusive Scaling (again); Self-Avoiding Walk: Distribution and Scaling of End-to-end Distance, Connectivity Constant and Number of SAWs	Panadda Dechadilok (PDF)
12	Really Fat Tails (Levy Flights) Strong Central Limit Theorems for ‘Slowly’ Diverging Variance, Symmetric Levy Distributions, Asymptotic Expansions, Superdiffusive Scaling; Examples: A Low Density Gas between Two Plates (Knudsen Number » 1), Financial Time Series, Polymer Surface Adsorption	Greg Randall (PDF)
13	Extreme Events, Levy Stability, and the Continuum Limit Extremes of Independent Random Variables, Frechet Distribution for Parent Distributions with Power-law Tails, the Largest Step of a Levy Flight is at the Same Scale as the Final Position; “Renormalization” of Weakly-correlated Steps, Levy Stable Laws, Gnedenko’s Convergence Theorems; Continuum Limit of Levy Flights, Riesz Fractional Derivative	Michael Slutsky (PDF)
14	Non-identically Distributed Steps Formal Continuum Limit with Non-identical Steps and Random Waiting Times, Time-dependent Diffusion Coefficient, Rescaled Time = Total Variance, CLT with Different Scaling; CLT and Berry-Eseen Theorem for Non-identical Variables; Breakdown of the CLT: Power-law Growing/decaying Steps, Exponentially Growing/decaying Steps, Fractal Distributions, Non-recombinant and Recombinant Space-time Trees	Ryan Larsen (PDF)
15	Non-identically Distributed Steps and Random Waiting Times Pseudo-equivalence Between Time-dependent Step Size and Time-dependent Waiting Time Between Steps in the Continuum Limit, Time-dependent Diffusion Coefficient; Geometrically Decaying Step Sizes, Exact Non-Gaussian Solutions; Renewal Theory of Random Waiting Times, Laplace-transform Theory of One-sided Levy Distributions	Nikos Savva (PDF)
16	Continuous-Time Random Walks Separable CTRW, Formulation in Terms of Random Number of Steps in a given Time Interval, Probability Generating Functions and Discrete Convolutions, Variance in Step Size Versus Variance in the Number of Steps Taken, Poisson Process, Exact Solution of the Poisson-Bernoulli CTRW	Greg Randall (PDF)
17	Anomalous Sub-Diffusion Montroll-Weiss Theory of Separable CTRW in Terms of the Random Waiting Time, Moments of the Position, Tauberian Theorems for the Laplace Transform and Long-Time Scaling Laws, Normal Diffusion (CLT + Square-root Scaling); Anomalous Dispersion due to Long Trapping-times with Constant Displacements, Example: Peak Broadening in DNA Gel Electrophoresis; Anomalous Diffusion due to an Infinite Mean Waiting Time, Scalings With and Without Drift	Kevin Chu (PDF)
18	Non-Markovian Diffusion Equations Continuum Limits of CTRW; Normal Diffusion Equation for Finite Mean Waiting Time and Finite Step Variance, Exponential Relaxation of Fourier Modes; Fractional Diffusion Equations for Sub-diffusion, Riemann-Liouville Fractional Derivative, Mittag-Leffler Power-law Relaxation of Fourier Modes, Time-delayed Flux	Ahmed Ismail (PDF)
19	Continuous Laplacian Growth I Diffusion-limited Solidification/Melting, Viscous Fingering in Porous Media or Hele-Shaw Cells; Background from Complex Analysis: Analytic Functions, Conformal Mapping, Potential Theory; Nonlinear Dynamics of Conformal Maps, Polubarinova-Galin Equation for the Time-dependent Map from a Half-plane; Exact Traveling Wave Solutions: Ivantsov Parabola, Saffman-Taylor Fingers	Ahmed Ismail (PDF)
20	Continuous Laplacian Growth II Polubarinova-Galin Equation for the Map From the Unit Circle, ODEs for Laurent Coefficients, Area Theorem; Shraiman-Bensimon Solutions for Circles, Ellipses, and General M-fold Perturbations; Proof of Finite-time Singularity for any Meromorphic Initial Condition	Thierry Savin (PDF)
21	Stochastic Laplacian Growth Diffusion-limited Aggregation, Fractal Growth; Hastings-Levitov Iterated Conformal Maps, Bump Functions; Morphological Properties, Laurent Coefficients, Univalent Functions, Fractal Dimension	Michael Slutsky (PDF)