Diffusion in Disordered Media

1. Diffusion in Disordered Media Nicholas Senno PHYS 527 12/12/2013

2. Random Walk • Need to consider relationship between average displacement and time: <R2>  4Dt • Can define diffusion constant from properties of classical random walk. • However, because each cluster has different structure we need to average over many random walkers per cluster and then again over many clusters

4. Blind Ant • Consider a random walker that can choose to move to any neighboring site with equal probability • If the move is possible it makes the move • If not the ant remains at the current location for the time step ? ? ? ?

5. When p = pc the asymptotic behavior changes to <R2> ∝ t0.79

6. Diffusion cannot occur for clusters generated with p < pc

7. Myopic Ant • What if the ant can see which neighboring sites are available? • Then we can save some computational steps by allowing the ant to move every time.

9. Exact Enumeration • So far we have considered averages over many walkers but what if consider the probability distribution of every random walk? • The probability of being at a cluster site i at a time t+1 (call this number Wt+1(i)) only depends on the probability of the neighboring sites at time t. • This makes exact enumeration a recursive algorithm not a Monte Carlo Simulation.

10. Exact enumeration produces the same results as the myopic ant if enough clusters are averaged over.

11. Diffusion In Random Media • It is possible to define a diffusion constant in analogy to the classical random walk • The blind ant, myopic ant, and exact enumeration methods all give similar results but the implantation of each increases in complexity • The Monte Carlo simulations are good for large systems that scale well while the Recursive approach is better for smaller work stations.