Discrete diffusion modeling
Anders Malthe-Sørenssen
Discrete diffusion modeling
• Random walks
– Theory and simulations
– Central limit theorem
– Self-avoiding walks and polymers
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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