D_{eff} = P_{NC} D_{fast} + (1-P_{NC}) D_{slow}.

Consider a small stripe of material around depth z. Let the local 3D particle density within this z-stripe be rho. We neglect any possible correlations between the positions of different particles within the stripe, i.e. we assume spatial Poisson statistics with average density rho. Then the fraction P_NC is the probability that, for an arbitrary particle, the nearest neighbor is found in a distance larger than r_c. The average number of particles in a sphere of radius r_c is given by

\overline{n} = \rho \;\frac{4}{3}\pi r^3_c.

According to Poisson statistics, the probability that n=0 is

P_{NC} = e^{-\overline{n}} = e^{ - \rho \;\frac{4}{3}\pi r^3_c} = e^{- \rho / \rho_c},

with

\rho_c = \frac{3}{4 \pi}\; r^{-3}_c.

Thus we can define a density dependent diffusion constant:

D_{eff}(\rho) \;= \; (e^{-\rho / \rho_c}) D_{fast} \;+\; (1-e^{-\rho / \rho_c}) D_{slow}.

The figure below shows an example:

Using this, we can write our model as a non-linear Fokker-Planck equation for the position and time-dependent particle density:

\frac{d}{dt} \rho(z,t)\; = \;D_{eff}(\rho(z))\; \frac{\partial^2}{\partial x^2} \;\rho(z,t).

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