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# 3 Frozen Disorder: Correlated Effective Medium

[184.1.1] In this section we develop an effective medium approximation to solve eq. (2.1) with initial conditions (2.3) describing the correlated random walk of an A-particle in the frozen background of blockers. [184.1.2] This is possible because the correlated walk on the regular lattice can again be solved exactly. [184.1.3] We will derive a selfconsistent equation similar to that for the generalized diffusion coefficient in the well known effective medium treatment of random walks on a frozen percolating network[24, 25, 26, 27, 28]. [184.1.4] In our case, however, the solution of the selfconsistent equation is only an intermediate step from which the generalized diffusion coefficient has to be calculated[29].

[184.2.1] We start by Laplace transforming eq. (2.1) and inserting the initial conditions. [184.2.2] This gives

 (3.1)

where we have written to shorten the notation. [184.2.3] For a selfconsistent treatment of the disorder we have to compare eq. (3) with the same equation for the regular reference lattice where we allow the kernel to be frequency dependent. [page 185, §0]    [185.0.1] The equation for the regular reference lattice then reads

 (3.2)

[185.0.2] We subtract eq. (3) from eq. (3) and insert a term . [185.0.3] This gives us

 (3.3)

where we have introduced and suppressed the dependence on to further shorten the notation. [185.0.4] We now define the lattice Greens function associated with the reference lattice by

 (3.4)

[185.0.5] Multiplication of eq. (3.3) by , summation over , and use of eq. (3.4) allows us to rewrite eq. (3.3) as

 (3.5)

[185.1.1] As we are dealing with bond percolation it is convenient to switch from site related quantities to bond related ones. [185.1.2] This is done by writing eq. (3.5) for a second site , and then forming the differences . [185.1.3] In terms of the quantities one now has

 (3.6)

where the summations run over all bonds . [page 186, §0]    [186.0.1] As usual[24, 25, 26, 27, 28] one allows only a finite number of bonds (here only one bond) to fluctuate while all other bonds are given their effective medium value. [186.0.2] In this 1-bond-approximation the bond [kl] can be chosen arbitrarily, and we choose it such that it does not touch the starting point of the random walk, i. e.  . [186.0.3] Then eq. (3.6) is easily solved to give

 (3.7)

[186.0.4] Up to this point we have not made use of the fact that the reference lattice is regular. [186.0.5] We assume now a regular lattice for the effective medium such that for all sites. [186.0.6] In this case the solution of eq. (3.4) is recognized as the Greens function for that lattice if one introduces the new spectral variable

 (3.8)

[186.1.1] The self consistent equation is obtained by demanding to choose a frequency dependent medium such that it reproduces on average the behaviour of the original system, i. e.  we demand where denotes the average over all possible conï¬gurations of the bond . Using this condition in eq. (3.7) we find

 (3.9)

where we have also used the symmetry of the Greens function and eq. (3.4) to express in terms of . [186.1.2] The average in eq. (3.9) has to be taken with respect to the probability density which was given in eq. (2.1) as . [186.1.3] Performing the average and introducing the notation for the percolation threshold one finds from eq. (3.9) the selfconsistent equation

with given by eq. (3.8), and . [page 187, §0]    [187.0.1] Partially solving for then leads to the functional equation

 (3.10a)

where

 (3.11a)

[187.0.2] This formulation has the advantage that it displays explicitly the two different branches of the solution. [187.0.3] The decision which branch to use is made by enforcing the correct limiting behaviour of . [187.0.4] This requires that we calculate first the generalized diffusion coefficient . [187.0.5] For that we must solve eq. (2.8). [187.0.6] Fourier-Laplace transforming eq. (2.8) and using the initial conditions of eq. (2.1). [187.0.7] We obtain

 (3.12)

where denotes the wave vector, and is the usual characteristic function of the random walk for the lattice under consideration, e. g.  for the -dimensional simple cubic lattices.

[187.1.1] Now equation (2.11) can be employed to calculate the conductivity . A straightforward calculation using and leads to the generalized diffusion coefficient[29]

 (3.13)

as a function of . [187.1.2] Here denotes , and we have used the index to indicate that eq. (3.13) is valid for the frozen case, i. e. . [187.1.3] With equations (3.13) and (3.10) we have now derived the set of selfconsistent equations for the frozen diffusion coefficient . [187.1.4] It remains to specify a particular lattice, and to solve eq. (3.10) for the case of interest. [page 188, §0]    [188.0.1] That will be done in Section 5 after we have discussed how to make use of these results for the dynamic disorder problem.