# 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

| uu+γi+ωb-ωPiu-u+γi+ωb-ωδi0-ωb-ω1zi∑jiAijδj0-δi0 | |

| =ωu+γi∑jiAijPju-Piu | | (3.1) |

where we have written Piu=Pi,u=∫0∞e-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

| uu+γi+ωb-ωPi0u-u+γi+ωb-ωδi0-ωb-ω1zi∑jiA0uδj0-δi0 | |

| =ωu+γi∑jiA0uPj0u-Pi0u. | | (3.2) |

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

uωA01+ωb-ωu+γi+ziPi-Pi0-∑jiPj-Pj0-ωb-ωu+γi1ωzi∑jiΔijδj0-δi0=∑jiΔijPj-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

uωA01+ωb-ωu+γi+ziGik-∑jiGjk=-δik. | | (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,jGikωb-ωu+γi1ωziΔijδj0-δi0=-∑i,jGikΔijPj-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

Qkl=Qkl0+∑ijGik-Gil-Gjk+GjlΔijQij-12∑i,jGik-Gilωb-ωu+γi1ωzi-Gjk-Gjlωb-ωu+γj1ωzjΔijδj0-δi0 | | (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. k≠0,l≠0.
[186.0.3] Then eq. (3.6) is easily solved to give

Qkl=11-ΔklGkk+Gll-Gkl-GlkQkl0. | | (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~=uωA0u1+ωb-ωu+γ | | (3.8) |

instead of u.

[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+Δkl2z+2u~zGiiu~ | | (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

A0u=1-p-pc-pcu~Gu~1-pc-pcu~Gu~ | |

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

A0u=1-p-pc-Fu,A021-pc1±1-41-pcFu,A0Fu,A0-1+p+pc21/2 | | (3.10) |

where

Fu,A0=pcuω1+ωb-ωu+γGuωA0u1+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

Pk→,u=1ωA0+ωb-ωu+γ1ω1A0-1+ωb-ωu+γ1ωpk→uωA01+ωb-ωu+γ+z-zpk→ | | (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→=1d∑i=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 ∇k→p(k→)|k→=0=0
leads to the generalized diffusion coefficient[29]

D0u=p′′0A0uωzb-1+zuω+zuω+z+2b-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.