Diffusion in Disordered Media
Nicholas Senno
PHYS 527
12/12/2013
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
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 = pc the asymptotic behavior changes to <R2> ∝ t0.79
• Diffusion cannot occur for clusters generated with p < pc
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 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.
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.