Random walks and diffusion
Master
In Maynard (USA)
Description
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Type
Master
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Location
Maynard (USA)
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Start date
Different dates available
This graduate-level subject explores various mathematical aspects of (discrete) random walks and (continuum) diffusion. Applications include polymers, disordered media, turbulence, diffusion-limited aggregation, granular flow, and derivative securities.
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Start date
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Subjects
- Press
- Probability
- University
Course programme
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.
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.
Overview
History (Pearson, Rayleigh, Einstein, Bachelier)
Normal vs. Anomalous Diffusion
Mechanisms for Anomalous Diffusion
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
Multi-dimensional CLT for Sums of IID Random Vectors
Continuum Derivation Involving the Diffusion Equation
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
Method of Steepest Descent (Saddle-Point Method) for Asymptotic Approximation of Integrals
Application to Random Walks
Example: Asymptotics of the Bernoulli Random Walk
Power-law Tails, Diverging Moments and Singular Characteristic Functions
Additivity of Tail Amplitudes
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
Examples of Random Walks Modeled by Diffusion Equations
Flagellar Bacteria
Run and Tumble Motion, Chemotaxis
Financial Time Series
Additive Versus Multiplicative Processes
Corrections to the Diffusion Equation Approximating Discrete Random Walks with IID Steps
Fat Tails and Riesz Fractional Derivatives
Stochastic Differentials, Wiener Process
Chapman-Kolmogorov Equation, Kramers-Moyall Expansion, Fokker-Planck Equation
Probability Flux
Modified Kramers-Moyall Cumulant Expansion for Identical Steps
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
Probability Generating Functions on the Integers, First Passage and Return on a Lattice, Polya's Theorem
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
General Formulation in One Dimension
Smirnov Density
Minimum First Passage Time of a Set of N Random Walkers
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
Potential Theory using Complex Analysis, Mobius Transformations, First Passage to a Line
First Passage to a Circle, Wedge/Corner, Parabola. Continuous Laplacian Growth, Polubarinova-Galin Equation, Saffman-Taylor Fingers, Finite-time Singularities
Harmonic Measure, Hastings-Levitov Algorithm, Comparison of Discrete and Continuous Dynamics
Overview of Mechanisms for Anomalous Diffusion. Non-identical Steps
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
Laplace Transform
Renewal Theory
Montroll-Weiss Formulation of CTRW
DNA Gel Electrophoresis
CLT for CTRW
Infinite Man Waiting Time, Mittag-Leffler Decay of Fourier Modes, Time-delayed Flux, Fractional Diffusion Equation
"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"
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
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Random walks and diffusion