Discrete diffusion modeling Anders Malthe-Sørenssen

Discrete diffusion modeling Random walks Theory and simulations Central limit theorem Self-avoiding walks and polymers Characterization of diffusional patterns Ant in labyrinth Diffusion-limited processes Erosion as a diffusional process

Practical exercise Create algorithm for Using random walkers to model erosional patterns Characterize the erosional patterns using simple fractal measures

Diffusion – random walks Brownian motion Random jumps at constants rate Algorithmic: lattice based

Singlewalk.avi

Statistics of a random walk <r>(t) ? r(t) ? P(r>R;t) ?

<r>(t) ? r(t) ? P(r>R;t) ?

<r>(t) ? r(t) ? P(r>R;t) ?

Theory

Geometry of a random walk

Theory

Diffusion fronts Image(latt)

Diffusion fronts Image(latt > 100)

Physical processes with random walks Diffusion limited processes: Diffusion of material is limiting growth or reaction processes: example DLA Diffusion as basis for physical processes Diffusion Limited erosion Drainage networks

Diffusion limited aggregation