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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

\displaystyle u(u+\gamma _{i}+\omega _{b}-\omega)P_{i}(u)-(u+\gamma _{i}+\omega _{b}-\omega)\delta _{{i0}}-(\omega _{b}-\omega)\frac{1}{z_{i}}\sum _{{j\{ i\}}}A_{{ij}}(\delta _{{j0}}-\delta _{{i0}})
\displaystyle=\omega(u+\gamma _{i})\sum _{{j\{ i\}}}A_{{ij}}\big[P_{j}(u)-P_{i}(u)\big] (3.1)

where we have written P_{i}(u)=P(i,u)=\int^{\infty}_{0}\mathrm{e}^{{-ut}}P(i,t)\mathrm{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 A^{0}(u) to be frequency dependent. [page 185, §0]    [185.0.1] The equation for the regular reference lattice then reads

\displaystyle u(u+\gamma _{i}+\omega _{b}-\omega)P^{0}_{i}(u)-(u+\gamma _{i}+\omega _{b}-\omega)\delta _{{i0}}-(\omega _{b}-\omega)\frac{1}{z_{i}}\sum _{{j\{ i\}}}A^{0}(u)(\delta _{{j0}}-\delta _{{i0}})
\displaystyle=\omega(u+\gamma _{i})\sum _{{j\{ i\}}}A^{0}(u)\big[P^{0}_{j}(u)-P^{0}_{i}(u)\big]. (3.2)

[185.0.2] We subtract eq. (3) from eq. (3) and insert a term A^{0}\big[P_{j}(u)-P_{i}(u)\big]. [185.0.3] This gives us

\left[\frac{u}{\omega A^{0}}\left(1+\frac{\omega _{b}-\omega}{u+\gamma _{i}}\right)+z_{i}\right](P_{i}-P^{0}_{i})-\sum _{{j\{ i\}}}(P_{j}-P^{0}_{j})-\frac{\omega _{b}-\omega}{u+\gamma _{i}}\frac{1}{\omega z_{i}}\sum _{{j\{ i\}}}\Delta _{{ij}}(\delta _{{j0}}-\delta _{{i0}})=\sum _{{j\{ i\}}}\Delta _{{ij}}(P_{j}-P_{i}) (3.3)

where we have introduced \Delta _{{ij}}=(A_{{ij}}-A^{0})/A^{0} 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

\left[\frac{u}{\omega A^{0}}\left(1+\frac{\omega _{b}-\omega}{u+\gamma _{i}}\right)+z_{i}\right]G_{{ik}}-\sum _{{j\{ i\}}}G_{{jk}}=-\delta _{{ik}}. (3.4)

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

(P_{k}-P^{0}_{k})+\sum _{{i,j}}G_{{ik}}\frac{\omega _{b}-\omega}{u+\gamma _{i}}\frac{1}{\omega z_{i}}\Delta _{{ij}}(\delta _{{j0}}-\delta _{{i0}})=-\sum _{{i,j}}G_{{ik}}\Delta _{{ij}}(P_{j}-P_{i}). (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 Q_{{kl}}=P_{k}-P_{l}=-Q_{{lk}}. [185.1.3] In terms of the quantities Q_{{kl}} one now has

Q_{{kl}}=Q^{0}_{{kl}}+\sum _{{[ij]}}(G_{{ik}}-G_{{il}}-G_{{jk}}+G_{{jl}})\Delta _{{ij}}Q_{{ij}}-\frac{1}{2}\sum _{{i,j}}\left[(G_{{ik}}-G_{{il}})\frac{\omega _{b}-\omega}{u+\gamma _{i}}\frac{1}{\omega z_{i}}-(G_{{jk}}-G_{{jl}})\frac{\omega _{b}-\omega}{u+\gamma _{j}}\frac{1}{\omega z_{j}}\right]\Delta _{{ij}}(\delta _{{j0}}-\delta _{{i0}}) (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\neq 0,\, l\neq 0. [186.0.3] Then eq. (3.6) is easily solved to give

Q_{{kl}}=\frac{1}{1-\Delta _{{kl}}(G_{{kk}}+G_{{ll}}-G_{{kl}}-G_{{lk}})}Q^{0}_{{kl}}. (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 z_{i}=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

\tilde{u}=\frac{u}{\omega A^{0}(u)}\left(1+\frac{\omega _{b}-\omega}{u+\gamma}\right) (3.8)

instead of u.

[186.1.1] The self consistent equation is obtained by demanding to choose a frequency dependent medium A^{0}(u) such that it reproduces on average the behaviour of the original system, i. e.  we demand \left\langle Q_{{kl}}\right\rangle=\left\langle Q^{0}_{{kl}}\right\rangle where \left\langle~.~\right\rangle denotes the average over all possible configurations of the bond [kl]. Using this condition in eq. (3.7) we find

1=\left\langle\frac{1}{1+\Delta _{{kl}}\left[\frac{2}{z}+\frac{2\tilde{u}}{z}G_{{ii}}(\tilde{u})\right]}\right\rangle (3.9)

where we have also used the symmetry of the Greens function and eq. (3.4) to express G_{{ij}}(\tilde{u}) in terms of G_{{ii}}(\tilde{u}). [186.1.2] The average in eq. (3.9) has to be taken with respect to the probability density f(A_{{kl}}) which was given in eq. (2.1) as f(A_{{kl}})=(1-p)\delta(A_{{kl}}-1)+p\delta(A_{{kl}}). [186.1.3] Performing the average and introducing the notation p_{c}=2/z for the percolation threshold one finds from eq. (3.9) the selfconsistent equation


with \tilde{u} given by eq. (3.8), and G(u)=G_{{ii}}(u). [page 187, §0]    [187.0.1] Partially solving for A^{0} then leads to the functional equation

A^{0}(u)=\frac{1-p-p_{c}-F(u,A^{0})}{2(1-p_{c})}\left\{ 1\pm\left\{ 1-\frac{4(1-p_{c})F(u,A^{0})}{\big[F(u,A^{0})-1+p+p_{c}\big]^{2}}\right\}^{{1/2}}\right\} (3.10a)


F(u,A^{0})=p_{c}\frac{u}{\omega}\left(1+\frac{\omega _{b}-\omega}{u+\gamma}\right)G\left\{\frac{u}{\omega A^{0}(u)}\left(1+\frac{\omega _{b}-\omega}{u+\gamma}\right)\right\} (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 \sigma. [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({\vec{k}},u)=\frac{\frac{1}{\omega A^{0}}+\frac{\omega _{b}-\omega}{u+\gamma}\frac{1}{\omega}\left(\frac{1}{A^{0}}-1\right)+\frac{\omega _{b}-\omega}{u+\gamma}\frac{1}{\omega}p({\vec{k}})}{\frac{u}{\omega A^{0}}\big(1+\frac{\omega _{b}-\omega}{u+\gamma}\big)+z-zp({\vec{k}})} (3.12)

where {\vec{k}}=(k_{1},\dots,k_{d}) denotes the wave vector, and p({\vec{k}}) is the usual characteristic function of the random walk for the lattice under consideration, e. g. p({\vec{k}})=\frac{1}{d}\sum _{{i=1}}^{d}\cos k_{i} for the d-dimensional simple cubic lattices.

[187.1.1] Now equation (2.11) can be employed to calculate the conductivity \sigma(\omega). A straightforward calculation using p({\vec{k}})\rvert _{{{\vec{k}}=0}}=1 and \nabla _{{\vec{k}}}p({\vec{k}})\rvert _{{{\vec{k}}=0}}=0 leads to the generalized diffusion coefficient[29]

D_{0}(u)=p^{{\prime\prime}}(0)A^{0}(u)\frac{\omega}{z}(b-1+z)\frac{\frac{u}{\omega}+z}{\frac{u}{\omega}+z+2b-2} (3.13)

as a function of u=\mathrm{i}\omega. [187.1.2] Here p^{{\prime\prime}}(0) denotes \left(\nabla _{{\vec{k}}}\right)^{2}p({\vec{k}})\rvert _{{{\vec{k}}=0}}, and we have used the index 0 to indicate that eq. (3.13) is valid for the frozen case, i. e. \tau=\infty. [187.1.3] With equations (3.13) and (3.10) we have now derived the set of selfconsistent equations for the frozen diffusion coefficient D_{0}(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.