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18.366 Random Walks and Diffusion

Lecture Notes for Spring 2003

Martin Z. Bazant


Note: These notes were written independently by student `scribes' and have received little or no editing by Prof. Bazant.


  1. Introduction (PS notes by K. Titievsky ). Simple random walks; Central Limit Theorem; connection with continuum diffusion via the Bachelier equation for noninteracting walkers; some variations that can produce `anomalous' diffusion: nonidentical steps, correlations, large fluctuations (fat tails), interactions with other walkers or the environment.

    I. Normal Diffusion

  2. Sums of Random Vectors (PS notes by D. Hu). Characteristic functions, convolution theorem, integral for the position PDF after N steps, exact evaluation and long-time (large N) asymptotics for a random walk on a hypercubic lattice in d dimensions.

  3. Cumulants and the Central Limit Theorem. (PS notes by D. Vener) Cumulant generating functions, Central Limit Theorem, Berry-Eseen theorem.

  4. Asymptotics Inside the Central Region and the Continuum Limit. (PS notes by D. Vener) Edgeworth and Gram-Charlier expansions, Hermite polynomials, Kramers-Moyall expansion (PDE) involving moments, modified expansion involving cumulants.

  5. Continuum Approximations . (PS notes by R. Larsen) Corrections to the diffusion (or Fokker-Planck) equation for non-Gaussian transition probabilities, Green function reproducing the Edgeworth expansion, dimensional analysis of continuum limit (time scale >> time step, spatial scale >> step size) when cumulants are finite.

  6. Fat Tails . (PS notes by R. Haghgooie) Corrections to the CLT when some moment (>2) diverges, generalizations of Edgeworth expansion involving Dawson's integral, reduced width of the central region, additivity of power-law tail amplitudes (analogous to cumulants).

  7. Asymptotics Outside the Central Region. (PS notes by N. Savva.) General theory of saddle-point and steepest-descent asymptotics of complex Laplace integrals, applications to random walks, uniformly valid approximations.

  8. Example of Saddle-Point Asymptotics . (Guest lecture by Dion Harmon.) (PS notes by K. Chu.) Detailed analysis of the Bernoulli random walk.

    II. Anomalous Diffusion

  9. Correlations Between Steps . (PS notes by M. Rvachev). Applications (polymers, finance, turbulent diffusion,...), Green-Kubo formula, anomalous diffusion, exponentially decaying correlations, transition from ballistic to diffusive scaling.

  10. Persistent Random Walks and the Telegrapher's Equation. (PS notes by G. Randall) Markov chain for the persistent random walk on the integers; Continuum limits: Diffusion Equation with diffusive scaling, Telegrapher's equation with ballistic scaling.

  11. More on Persistence and Self-Avoidance . ( Draft notes by P. Dechadilok - revision coming) 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.

  12. Really Fat Tails (Levy Flights). (PS notes by G. Randall) 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.

  13. Extreme Events, Levy Stability, and the Continuum Limit. (Notes by M. Slutsky) 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.

  14. Non-identically Distributed Steps . (Notes by R. Larsen) 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.

  15. Non-identically Distributed Steps and Random Waiting Times. (PS Notes by N. Savva) 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.

  16. Continuous-Time Random Walks. ( PS Notes by G. Randall) 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.

  17. Anomalous Sub-Diffusion. (PS Notes by K. Chu) 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.

  18. Non-Markovian Diffusion Equations. (PS Notes by A. Ismail) 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.

  19. Anomalous Diffusion in Disordered Media. (PDF Notes by D. Hu, PS Notes by C. Ma) Physical mechanisms for sub-diffusion; long trapping times: extended objects (polymers), random potential wells and barriers, thermal phase transition from sub- to normal diffusion for exponentially distributed wells, Sinai's problem (random transition rates); fractal geometry: percolation clusters, red bonds, exact renormalization-group analysis of a random walk on the Sierpinski gasket.

  20. Anomalous Diffusion in Fluids. 4/24 Non-separable CTRW, leapers and creepers, Levy walks, turbulent diffusion, Taylor dispersion.

  21. Turbulent Diffusion, Returns and First Passage. 4/29 Levy walk (creeper) models for super-diffusion in turbulent flows, Richardson and Kolmogorov scaling law for homogeneous turbulence; First passage processes: relation between first-passage time distribution and occupation probability, Polya's theorem for eventual return.

  22. First Passage in the Continuum Limit . 5/1 General theory of first-passage-time distribution and moments of the first-passage time for random walks in the continuum limit (normal or anomalous). Exact solutions in one dimension. Smirnov density.

    III. Some Topics in Nonlinear Diffusion

  23. Continuous Laplacian Growth I. 5/6 (PDF and PS notes by A. Ismail) 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.

  24. Continuous Laplacian Growth II. 5/8 (PS notes by T. Savin) 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.

  25. Stochastic Laplacian Growth. 5/13 (PS notes by M. Slutsky) Diffusion-limited aggregation, fractal growth; Hastings-Levitov iterated conformal maps, bump functions; Morphological properties, Laurent coefficients, univalent functions, fractal dimension.

  26. Non-Laplacian Transport-Limited Growth. 5/15 Conformally invariant transport processes, solidification in a background flow, Advection-Diffusion Limited Aggregation, electrodeposition; Continuous and stochastic evolution of conformal maps; Growth on curved surfaces, DLA on a sphere.

    For a review article summarizing some aspects of lectures 25 and 26, see IMA Preprint 1991 by Bazant and Crowdy, published with minor revisions in Chapter 4 of Handbook of Materials Modeling (Kluwer, 2005).

    2005 Lecture Notes


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