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Diffusion in Disordered Media. Nicholas Senno PHYS 527 12/12/2013. Random Walk. Need to consider relationship between average displacement and time: <R 2 >  4Dt Can define diffusion constant from properties of classical random walk.

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Diffusion in disordered media

Diffusion in Disordered Media

Nicholas Senno

PHYS 527

12/12/2013


Random walk
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
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

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



Myopic ant
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
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
Diffusion In Random Media ant if enough clusters are averaged over.

  • 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.


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