[179.1.1] Consider the random walk of a single particle of type A
in a percolating network on a regular lattice.
[179.1.2] We will always take the lattice constant of the
underlying lattice to be unity.
[179.1.3] For simplicity we consider the case of bond percolation
instead of site percolation.
[179.1.4] That is, the bonds of the regular lattice
are assumed to be blocked be B-particles (blockers) with probability

(2.1) |

where

(2.2) |

[page 180, §0]
[180.0.1] The summation in eq. (2.1a)
runs over the nearest neighbour sites

[180.1.1] Equation (2.1) has to be supplemented by initial conditions
for

(2.3a) | |||

(2.3b) |

where the symbol

[180.2.1] We now derive eq. (2.1) as the equations of motion
for our correlated random walk.
[180.2.2] This will be done by a suitable reformulation of the equations
for the correlated random walk on the regular lattice[19],
and subsequent generalization to the disordered case.
[180.2.3] Consider therefore the random walker on a regular lattice.
[180.2.4] The random walker has a memory of its previous step
and as a consequence its walk is not markovian,
i. e. the transition probabilities are not completely determined
by the currently occupied site.
[180.2.5] However a markovian description can be obtained
by introducing an enlarged state space with internal states
which correspond to the previously occupied sites[21].
[180.2.6] Therefore the central quantity is the probability density

(2.4) |

where the sum runs over all nearest neighbour sites

(2.5) |

where the sum runs over all nearest neighbours

[181.1.1] Equation (2.5) can now be reformulated by first writing it in a more symmetric form. [181.1.2] Using eq. (2.4) we can rewrite eq. (2.5) as

(2.6) |

where

(2.7) |

[181.1.6] Solving eq. (2.6) for

(2.8) |

where the summations, as before,
run over all nearest neighbour sites

[182.1.1] Consider a system where the configuration of accessible sites fluctuates in time.
[182.1.2] We are interested in the case where the B-particles perform a random walk.
[182.1.3] Because we are dealing with bond percolation
this random walk occurs on the dual lattice.
[182.1.4] In an elementary step a blocking bond swings around either one of its end points
through an angle

[182.2.1] Equation (2.1) must be generalized to allow for time dependent transition rates. [182.2.2] Therefore we have to consider an equation of the form

(2.9a) |

where now the coefficients

(2.10a) |

[page 183, §0] [183.0.1] The time dependence of these coefficients could in principle be determined from the many particle master equation for all blockers. [183.0.2] However, because that equation is much too complicated we will approximate the true time dependence by a simple renewal model in Section 4. [183.0.3] Equation (2.9) completes the formulation of the model. [183.0.4] We remark here that other forms of a two step memory are possible and may be useful for applications. [183.0.5] For example one can consider enhanced or reduced transitions continuing in the same direction as the last step. [183.0.6] Such correlations lead to more complicated equations, but they can be treated by the same general approach presented here.

[183.1.1] We conclude this section with the formulas that will be used to calculate
the frequency dependent conductivity

(2.11) |

where

(2.12) |

where