User:Davidsm22/sandbox

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Geometrical percolation
For describing such a mixture of a dielectric and a metallic component we use the model of bond-percolation. On a regular lattice, the bond between two nearest neighbors can either be occupied with probability $$ p $$ or not occupied with probability $$ 1-p $$. There exists a critical value $$ p_c $$. For occupation probabilities $$ p > p_c $$ an infinite cluster of the occupied bonds is formed. This value $$ p_c $$ is called the percolation threshold. The region near to this percolation threshold can be described by the two critical exponents $$ \nu $$ and $$ \beta $$ (see Percolation critical exponents).

With these critical exponents we have the correlation length, $$ \xi $$

$$ \xi(p) \propto (p_c - p)^{- \nu} $$

and the percolation probability, P:

$$ P(p) \propto (p - p_c)^{\beta} $$

Conductivity near the percolation threshold
In the region around the percolation threshold, the conductivity assumes a scaling form:

$$ \sigma(p) \propto \sigma_m |\Delta p|^t \Phi_{\pm} \left(h|\Delta p|^{-s-t}\right) $$

with $$ \Delta p \equiv p - p_c $$ and $$ h \equiv \frac{\sigma_d}{\sigma_m} $$

At the percolation threshold, the conductivity reaches the value:

$$ \sigma_{DC}(p_c) \propto \sigma_m \left(\frac{\sigma_d}{\sigma_m}\right)^u $$

with $$ u = \frac{t}{t+s} $$