Diffusion in Disordered Media Nicholas Senno PHYS /12/2013

Random Walk Need to consider relationship between average displacement and time:  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

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

When p = p c the asymptotic behavior changes to ∝ t 0.79

Diffusion cannot occur for clusters generated with p < p c

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.

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 W t+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.

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

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.