The diffusion equation is ∂u/∂t = η∇2u where u(x, t) is the density and η is the diffusion...

The diffusion equation is ∂u/∂t = η∇2u where u(x, t) is the density and η is the diffusion constant. The model of diffusion by Witten and Sadler can be approximated for numerical integration in two dimensions by considering a two-dimensional square lattice and defining the size of a cluster as the minimum radius that includes all of its particles. Mathematically perform the following:

(a) Place a particle at the center of a 25 × 25 lattice of spacing a.

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(b) Place a particle at a random position away from the center but not adjacent to the center and allow this particle to randomly move one location at a time until it either leaves the lattice or becomes adjacent to the original particle. For the latter eventuality, draw a circle centered on the center of the cluster that just includes these two particles. Call this radius Rmin. After completing this step Rmin = a/2.

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(c) Repeat this process by adding additional particles at random, increasing Rmin if necessary to include all adjacent particles.

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(d) After a reasonable number of particles, N, are aggregated, calculate the fractal dimension, D, by the rule