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

where we have written Pi⁢u=P⁢i,u=∫0∞e-u⁢t⁢P⁢i,t⁢d⁢t 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 A0⁢u to be frequency dependent. [page 185, §0]    [185.0.1] The equation for the regular reference lattice then reads

[185.0.2] We subtract eq. (3) from eq. (3) and insert a term A0⁢Pj⁢u-Pi⁢u. [185.0.3] This gives us

where we have introduced Δi⁢j=Ai⁢j-A0/A0 and suppressed the dependence on u to further shorten the notation. [185.0.4] We now define the lattice Greens function associated with the reference lattice by

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

[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 l, and then forming the differences Qk⁢l=Pk-Pl=-Ql⁢k. [185.1.3] In terms of the quantities Qk⁢l one now has

where the summations run over all bonds i⁢j. [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.  k≠0,l≠0. [186.0.3] Then eq. (3.6) is easily solved to give

[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 zi=z 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

instead of u.

[186.1.1] The self consistent equation is obtained by demanding to choose a frequency dependent medium A0⁢u such that it reproduces on average the behaviour of the original system, i. e.  we demand Qk⁢l=Qk⁢l0 where ⟨.⟩ denotes the average over all possible configurations of the bond k⁢l. Using this condition in eq. (3.7) we find

where we have also used the symmetry of the Greens function and eq. (3.4) to express Gi⁢j⁢u~ in terms of Gi⁢i⁢u~. [186.1.2] The average in eq. (3.9) has to be taken with respect to the probability density f⁢Ak⁢l which was given in eq. (2.1) as f⁢Ak⁢l=1-p⁢δ⁢Ak⁢l-1+p⁢δ⁢Ak⁢l. [186.1.3] Performing the average and introducing the notation pc=2/z for the percolation threshold one finds from eq. (3.9) the selfconsistent equation

with u~ given by eq. (3.8), and G⁢u=Gi⁢i⁢u. [page 187, §0]    [187.0.1] Partially solving for A0 then leads to the functional equation

where

[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 σ. [187.0.4] This requires that we calculate first the generalized diffusion coefficient D. [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

where k→=k1,…,kd denotes the wave vector, and p⁢k→ is the usual characteristic function of the random walk for the lattice under consideration, e. g. p⁢k→=1d⁢∑i=1dcos⁡ki for the d-dimensional simple cubic lattices.

[187.1.1] Now equation (2.11) can be employed to calculate the conductivity σ⁢ω. A straightforward calculation using p(k→)|k→=0=1 and ∇k→p(k→)|k→=0=0 leads to the generalized diffusion coefficient[29]

as a function of u=i⁢ω. [187.1.2] Here p′′⁢0 denotes (∇k→)2p(k→)|k→=0, and we have used the index 0 to indicate that eq. (3.13) is valid for the frozen case, i. e. τ=∞. [187.1.3] With equations (3.13) and (3.10) we have now derived the set of selfconsistent equations for the frozen diffusion coefficient D0⁢u. [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.