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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

 u⁢u+γi+ωb-ω⁢Pi⁢u-u+γi+ωb-ω⁢δi⁢0-ωb-ω⁢1zi⁢∑j⁢iAi⁢j⁢δj⁢0-δi⁢0 =ω⁢u+γi⁢∑j⁢iAi⁢j⁢Pj⁢u-Pi⁢u (3.1)

where we have written Piu=Pi,u=0e-utPi,tdt 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 A0u to be frequency dependent. [page 185, §0]    [185.0.1] The equation for the regular reference lattice then reads

 u⁢u+γi+ωb-ω⁢Pi0⁢u-u+γi+ωb-ω⁢δi⁢0-ωb-ω⁢1zi⁢∑j⁢iA0⁢u⁢δj⁢0-δi⁢0 =ω⁢u+γi⁢∑j⁢iA0⁢u⁢Pj0⁢u-Pi0⁢u. (3.2)

[185.0.2] We subtract eq. (3) from eq. (3) and insert a term A0Pju-Piu. [185.0.3] This gives us

 uω⁢A0⁢1+ωb-ωu+γi+zi⁢Pi-Pi0-∑j⁢iPj-Pj0-ωb-ωu+γi⁢1ω⁢zi⁢∑j⁢iΔi⁢j⁢δj⁢0-δi⁢0=∑j⁢iΔi⁢j⁢Pj-Pi (3.3)

where we have introduced Δij=Aij-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

 uω⁢A0⁢1+ωb-ωu+γi+zi⁢Gi⁢k-∑j⁢iGj⁢k=-δi⁢k. (3.4)

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

 Pk-Pk0+∑i,jGi⁢k⁢ωb-ωu+γi⁢1ω⁢zi⁢Δi⁢j⁢δj⁢0-δi⁢0=-∑i,jGi⁢k⁢Δi⁢j⁢Pj-Pi. (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 l, and then forming the differences Qkl=Pk-Pl=-Qlk. [185.1.3] In terms of the quantities Qkl one now has

 Qk⁢l=Qk⁢l0+∑i⁢jGi⁢k-Gi⁢l-Gj⁢k+Gj⁢l⁢Δi⁢j⁢Qi⁢j-12⁢∑i,jGi⁢k-Gi⁢l⁢ωb-ωu+γi⁢1ω⁢zi-Gj⁢k-Gj⁢l⁢ωb-ωu+γj⁢1ω⁢zj⁢Δi⁢j⁢δj⁢0-δi⁢0 (3.6)

where the summations run over all bonds ij. [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.  k0,l0. [186.0.3] Then eq. (3.6) is easily solved to give

 Qk⁢l=11-Δk⁢l⁢Gk⁢k+Gl⁢l-Gk⁢l-Gl⁢k⁢Qk⁢l0. (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 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

 u~=uω⁢A0⁢u⁢1+ωb-ωu+γ (3.8)

[186.1.1] The self consistent equation is obtained by demanding to choose a frequency dependent medium A0u such that it reproduces on average the behaviour of the original system, i. e.  we demand Qkl=Qkl0 where . denotes the average over all possible conï¬gurations of the bond kl. Using this condition in eq. (3.7) we find

 1=11+Δk⁢l⁢2z+2⁢u~z⁢Gi⁢i⁢u~ (3.9)

where we have also used the symmetry of the Greens function and eq. (3.4) to express Giju~ in terms of Giiu~. [186.1.2] The average in eq. (3.9) has to be taken with respect to the probability density fAkl which was given in eq. (2.1) as fAkl=1-pδAkl-1+pδAkl. [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

 A0⁢u=1-p-pc-pc⁢u~⁢G⁢u~1-pc-pc⁢u~⁢G⁢u~

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

 A0⁢u=1-p-pc-F⁢u,A02⁢1-pc⁢1±1-4⁢1-pc⁢F⁢u,A0F⁢u,A0-1+p+pc21/2 (3.10)

where

 F⁢u,A0=pc⁢uω⁢1+ωb-ωu+γ⁢G⁢uω⁢A0⁢u⁢1+ωb-ωu+γ (3.11)

[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

 P⁢k→,u=1ω⁢A0+ωb-ωu+γ⁢1ω⁢1A0-1+ωb-ωu+γ⁢1ω⁢p⁢k→uω⁢A0⁢1+ωb-ωu+γ+z-z⁢p⁢k→ (3.12)

where k=k1,,kd denotes the wave vector, and pk is the usual characteristic function of the random walk for the lattice under consideration, e. g. pk=1di=1dcoski for the d-dimensional simple cubic lattices.

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

 D0⁢u=p′′⁢0⁢A0⁢u⁢ωz⁢b-1+z⁢uω+zuω+z+2⁢b-2 (3.13)

as a function of u=iω. [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 D0u. [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.