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Discrete diffusion modeling

Explore the theory and simulations of discrete diffusion modeling, including random walks, self-avoiding walks, and characterization of diffusional patterns. Use random walkers to model erosional patterns and analyze them using simple fractal measures. Learn about diffusion-limited processes and erosion as a diffusional process.

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Discrete diffusion modeling

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  1. Discrete diffusion modeling Anders Malthe-Sørenssen

  2. 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

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

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

  5. Singlewalk.avi

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

  7. <r>(t) ? r(t) ? P(r>R;t) ?

  8. <r>(t) ? r(t) ? P(r>R;t) ?

  9. Theory

  10. Geometry of a random walk

  11. Theory

  12. Diffusion fronts Image(latt)

  13. Diffusion fronts Image(latt > 100)

  14. 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

  15. Diffusion limited aggregation

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