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18.366 Random Walks and Diffusion
2006 Lectures
Martin Z. Bazant
Topics covered in lectures in 2006 are listed below. In some cases, links
are given to new lecture notes by student scribes.
The recommended reading refers to the lectures notes and exam solutions
from previous years available online from MIT's
OpenCourseWare, also available as the course
reader from CopyTech, or to the books listed in the syllabus.
latex template for lecture notes
- Overview. History (Pearson, Rayleigh, Einstein, Bachelier).
Normal vs. anomalous diffusion. Mechanisms for anomalous diffusion.
Reading: 2005 Lecture 1, Hughes.
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. Reading: 2005
Lecture 2, Hughes.
- The Central Limit Theorem and the diffusion equation.
Multi-dimensional CLT for sums of IID random vectors. Continuum derivation
involving the diffusion equation.
Reading: 2005 Lectures 1 and 3.
- 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. Reading: 2005
Lectures 3 and 4, Hughes, Feller.
- 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. Reading: 2005 Lectures 6 and 7, Hughes.
- Power-law "fat tails". Power-law tails, diverging moments and
singular characteristic
functions.
Additivity of tail amplitudes. Reading: 2005 Lectures 5 and 6, Bouchaud and Potters.
- 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.
Reading: 2005 Lecture 5.
- From random walks to diffusion
. Examples of random walks modeled by
diffusion equations. 1. Flagellar bacteria. Run and tumble motion, chemotaxis.
2. Financial time series. Additive
versus multiplicative processes. Reading: 2005 Lecture 10, Bouchaud and Potters.
- 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.
Reading:2005 Lectures 8, 9, 13, Risken.
- Weakly non-identical steps . Chapman-Kolmogorov equation,
Kramers-Moyall expansion, Fokker-Planck equation. Probability flux.
Modified Kramers-Moyall cumulant expansion for identical
steps.
Reading:2005 Lectures 8, 9 13, Risken.
I.B. Nonlinear Diffusion
- Nonlinear drift. Interacting random walkers,
concentration-dependent drift. Nonlinear waves in
traffic flow, characteristics, shocks, Burgers' equation. Surface growth,
Kardar-Parisi-Zhang equation.
- Nonlinear diffusion. Cole-Hopf transformation, general solution
of Burgers equation. Concentration-dependent diffusion, chemical potential. Rechargeable
batteries, steric effects. Reading: Problem set 3 solutions.
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. Reading 2005 Lectures 17 and 18,
Hughes, Redner.
-
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. Reading: Feller.
- First passage in the continuum limit General formulation in one
dimension. Smirnov density. Minimum first passage time of a set of N random
walkers. Reading 2005 Lecture 16 and Exam 2 (problem 2).
- 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.
Reading: 2005 Lecture 18, Redner, Risken.
- 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.
Reading 2003
Lecture 23, an article, Redner.
- Hitting probabilities in two dimensions.
Potential theory using complex analysis, Mobius transformations, first passage to a line.
Reading 2003
Lecture 23,
Redner.
- Applications of conformal mapping. First passage to a circle, wedge/corner,
parabola. Continuous Laplacian growth,
Polubarinova-Galin equation, Saffman-Taylor fingers, finite-time
singularities. Reading:
2003 Lectures 23-24.
- Diffusion-limited aggregation. Harmonic measure, Hastings-Levitov algorithm,
comparison of discrete and continuous dynamics. Overview of mechanisms for anomalous
diffusion. Non-identical steps. Reading
2003 Lectures 25-26, 14-15. a review article.
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.
Reading: 2005 Lectures 19-20, 2003 Lecture 9-11.
- Levy flights . Superdiffusion and
limiting Levy distributions for steps with infinite variance,
examples, size of the
largest step, Frechet distribution. Reading: 2005 Lecture 22, 2003
Lectures 12-13, Hughes.
II.B. Continuous-Time Random Walks
-
Continuous-time random walks . Laplace transform. Renewal
theory. Montroll-Weiss formulation of CTRW. DNA gel electrophoresis.
Reading: 2005 Lecture 23, 2003 Lectures 15-17.
- Fractional diffusion equations. CLT for CTRW. Infinite mean
waiting time, Mittag-Leffler
decay of Fourier modes, time-delayed flux, fractional diffusion equation.
Reading: 2005 Lecture 24, 2003 Lecture 18.
- 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". Reading: 2005 Lectures 25 (revised) and 26,
Hughes.
- 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. Reading: 2005 Lecture 26, Hughes.
bazant@mit.edu