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18.366 Random Walks and Diffusion
2005 Lecture Notes
Martin Z. Bazant
The notes below were written by student "scribes" as homework assignments.
In most cases, Prof. Bazant has reviewed the notes and made revisions or
extersions of the text, but beware many unedited portions still exist.
Note that solutions to the problem sets and
exams also contain detailed discussions (also mostly
edited by Prof. Bazant) of many
topics not appearing in the lectures below.
I. Normal Diffusion
I.A. Fundamental Theory
- Introduction. History; simple analysis of the
isotropic random walk in d dimensions, using the continuum limit;
Bachelier and diffusion
equations; normal versus anomalous 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.
- 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.
- Asymptotics Inside the Central Region
.
Gram-Charlier expansions for random walks, Berry-Esseen theorem,
width of the "central region",
"fat" power-law tails.
- Asymptotics with Fat Tails.
Singular characteristic functions, generalized Gram-Charlier expansions,
Dawson's integral, edge of the central region, additivity of power-law
tails.
- 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.
-
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.
- 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.
- 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.
I.B. Some Finance
- Applications in Finance . Models for financial time series,
additive and multiplicative noise, derivative securities, Bachelier's
fair-game price.
- 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. (Additional
notes)
- Black-Scholes and Beyond .
Riskless hedging and pricing on a binomial tree,
Black-Scholes equation in the continuum limit, risk neutral valuation.
(Additional notes on
"Gram-Charlier" corrections for residual
risk in Bouchaud-Sornette theory; see also Problem Set 3.)
- 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.
I.C. Some Physics
- Applications in Statistical Mechanics.
Random walk in an external force field,
Einstein relation, Boltzmann equilibrium, Ornstein-Uhlenbeck
process, Ehrenfest model.
- 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.)
I.D. First Passage
- 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.
- 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.)
- First Passage in Higher Dimensions.
Return and first passage on a lattice, Polya's theorem,
continuous first passage in
in complicated geometries, moments of the time and the location of
first passage, electrostatic analogy.
I.E. Correlations
- Polymer models: Persistence and
Self-Avoidance.
Random walk models of polymers,
radius of gyration, persistent random walk, self-avoiding walk, Flory's
scaling theory.
- (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.
- (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
(lecture on 4/28/05)
II. Anomalous Diffusion
- Levy Flights. Steps with infinite variance, Levy stability,
Levy distributions, generalized central limit theorems.
(lecture on 4/21/05 by Chris Rycroft)
- 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.
- 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.
- 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.
- 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.
For additional reading, see the...
2003 Lecture Notes
... especially the sections on anomalous diffusion and
diffusion-limited growth.
bazant@mit.edu