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1 |
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Introduction
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. |
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I. Normal Diffusion |
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2 |
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Sums of Random Vectors
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. |
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3 |
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Cumulants and the Central Limit Theorem
Cumulant generating functions, Central Limit Theorem, Berry-Eseen theorem. |
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4 |
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Asymptotics Inside the Central Region and the Continuum Limit
Edgeworth and Gram-Charlier expansions, Hermite polynomials, Kramers-Moyall expansion (PDE) involving moments, modified expansion involving cumulants. |
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5 |
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Continuum Approximations
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. |
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6 |
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Fat Tails
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). |
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7 |
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Asymptotics Outside the Central Region
General theory of saddle-point and steepest-descent asymptotics of complex Laplace integrals, applications to random walks, uniformly valid approximations. |
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8 |
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Example of Saddle-Point Asymptotics (Guest lecture)
Detailed analysis of the Bernoulli random walk. |
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II. Anomalous Diffusion |
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9 |
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Correlations Between Steps
Applications (polymers, finance, turbulent diffusion,...), Green-Kubo formula, anomalous diffusion, exponentially decaying correlations, transition from ballistic to diffusive scaling. |
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10 |
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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. |
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11 |
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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. |
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12 |
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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. |
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13 |
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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. |
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14 |
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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. |
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15 |
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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. |
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16 |
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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. |
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17 |
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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. |
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18 |
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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. |
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19 |
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Anomalous Diffusion in Disordered Media
4/17 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. |
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20 |
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Anomalous Diffusion in Fluids
4/24 Non-separable CTRW, leapers and creepers, Levy walks, turbulent diffusion, Taylor dispersion. |
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21 |
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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. |
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22 |
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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. |
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III. Some Topics in Nonlinear Diffusion |
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23 |
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Continuous Laplacian Growth I
5/6 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. |
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24 |
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Continuous Laplacian Growth II
5/8 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. |
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25 |
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Stochastic Laplacian Growth
5/13 Diffusion-limited aggregation, fractal growth; Hastings-Levitov iterated conformal maps, bump functions; Morphological properties, Laurent coefficients, univalent functions, fractal dimension. |
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26 |
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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. |
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IV. Possible Additional Topics for 2005 |
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27 |
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Cooperative Diffusion in Amorphous Materials
Short-range contact forces in dense particle systems, void and spot models for granular drainage, string-like motion in glasses. |
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28 |
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Nonlinear Diffusion Equations
Long-range Coulomb forces in electrochemical transport, Nernst-Planck equations; porous media, Barenblatt's equation; reaction-diffusion equations, fronts, patterns. |
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