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2 Formulation of the Model

2.1 Correlated Hopping in a Frozen Percolation Network

[179.1.1] Consider the random walk of a single particle of type A in a percolating network on a regular lattice. [179.1.2] We will always take the lattice constant of the underlying lattice to be unity. [179.1.3] For simplicity we consider the case of bond percolation instead of site percolation. [179.1.4] That is, the bonds of the regular lattice are assumed to be blocked be B-particles (blockers) with probability p. [179.1.5] If a bond is blocked by a B-particle it cannot be crossed by the A-particle (walker). [179.1.6] We assume that the walker has a memory of its previous step. [179.1.7] It returns with a transition rate \omega _{b}, to the previously visited site, and jumps with a rate \omega to any other of the nearest neighbour sites. [179.1.8] The ratio b=\omega _{b}/\omega is a measure of the strength of the memory correlations. [179.1.9] For b>1 the walker returns preferentially to its previously visited site, and we will refer to this case as “enhanced reversals”. [179.1.10] In the case b<1 the walker tends to avoid the previously visited site and this will be termed “reduced reversals”. [179.1.11] As usual, we are interested in the autocorrelation function P(i,t)=P\left({\vec{r}}_{i},t\middle|r_{0},0\right), i. e.  the probability density to find the walker at site i at time t if it started from site 0 at time 0. [179.1.12] We will show below that the problem can be formulated as a system of second order equations for the P(i,t) which reads

\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(i,t)+(\gamma _{i}+\omega _{b}-\omega)\frac{\mathrm{d}}{\mathrm{d}t}P(i,t)=\omega\sum _{{j\{ i\}}}A_{{ij}}\frac{\mathrm{d}}{\mathrm{d}t}\big[P(j,t)-P(i,t)\big]+\omega\gamma _{i}\sum _{{j\{ i\}}}A_{{ij}}\big[P(j,t)-P(i,t)\big] (2.1a)

where \gamma _{i}=\omega _{b}+\omega(z_{i}-1) and z_{i} is the coordination number of site i. [179.1.13] The symmetric quantities A_{{ij}}=A_{{ji}} represent the bond disorder and are defined as

A_{{ij}}=\begin{cases}1&\text{if the bond }[ij]\text{ is vacant},\\ 0&\text{if the bond }[ij]\text{ is blocked}.\end{cases} (2.2a)

[page 180, §0]    [180.0.1] The summation in eq. (2.1a) runs over the nearest neighbour sites j of site i. [180.0.2] Note that in the uncorrelated case, b=1, eq. (2.1) reduces to the usual master equation for a random walk on a bond percolation network if one replaces P(i,t) by the sum of P and its derivative.

[180.1.1] Equation (2.1) has to be supplemented by initial conditions for P(i,t) and its derivative. [180.1.2] Special attention has to be paid to the condition on \frac{\mathrm{d}}{\mathrm{d}t}P(i,t) and its derivative. [180.1.3] The correct choice is

\displaystyle P(i,0+) \displaystyle=\delta _{{i0}} (2.3a)
\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}P(i,0+) \displaystyle=\big[\omega _{b}+\omega(z_{i}-1)\big]\frac{1}{z_{i}}\sum _{{j\{ i\}}}A_{{ij}}\big[P(j,0+)-P(i,0+)\big]=\gamma _{i}/z_{i}\sum _{{j\{ i\}}}A_{{ij}}(\delta _{{j0}}-\delta _{{i0}}) (2.3b)

where the symbol 0+ stands for the limit t\to 0 from above. [180.1.4] Note that \gamma _{i}/z_{i} is the average transition rate out of the starting point.

[180.2.1] We now derive eq. (2.1) as the equations of motion for our correlated random walk. [180.2.2] This will be done by a suitable reformulation of the equations for the correlated random walk on the regular lattice[19], and subsequent generalization to the disordered case. [180.2.3] Consider therefore the random walker on a regular lattice. [180.2.4] The random walker has a memory of its previous step and as a consequence its walk is not markovian, i. e.  the transition probabilities are not completely determined by the currently occupied site. [180.2.5] However a markovian description can be obtained by introducing an enlarged state space with internal states which correspond to the previously occupied sites[21]. [180.2.6] Therefore the central quantity is the probability density P(i,j,t) to find the walker at site i at time t given that it arrived at i via a direct transition from site j. [180.2.7] Thus j labels the previously occupied site or history. [page 181, §0]    [181.0.1] Then the symmetric probablity density P(i,t) is obtained from P(i,j,t) by a summation over all possible histories

P(i,t)=\sum _{{j\{ i\}}}P(i,j,t) (2.4)

where the sum runs over all nearest neighbour sites j of site i. [181.0.2] The conditional probability densities P(i,j,t) obey the master equation

\frac{\mathrm{d}}{\mathrm{d}t}P(i,j,t)=\omega _{b}\big[P(j,i,t)-P(i,j,t)\big]+\omega\sum _{{k\neq i}}\big[P(j,k,t)-P(i,j,t)\big] (2.5)

where the sum runs over all nearest neighbours k of site j except for site i on the regular lattice. [181.0.3] This is the starting point for deriving eq. (2.1).

[181.1.1] Equation (2.5) can now be reformulated by first writing it in a more symmetric form. [181.1.2] Using eq. (2.4) we can rewrite eq. (2.5) as

\frac{\mathrm{d}}{\mathrm{d}t}P(i,j,t)=(\omega _{b}-\omega)P(j,i,t)+\omega P(j,t)-\gamma P(i,j,t) (2.6)

where \gamma=\omega _{b}+(z-1)\omega, and z denotes the coordination number of the lattice. [181.1.3] Note that eq. (2.6) reduces to the master equation for a random walk on a regular lattice if one sets b=1 and sums over all sites j which are nearest neighbours of site i. [181.1.4] Next we differentiate eq. (2.6) and sum over j. [181.1.5] We then employ it for i and j interchanged to eliminate the term \frac{\mathrm{d}}{\mathrm{d}t}P(j,i,t) and find

\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(i,t)+\gamma\frac{\mathrm{d}}{\mathrm{d}t}P(i,t)=(\omega _{b}-\omega)\gamma P(i,t)+\omega\sum _{{j\{ i\}}}\frac{\mathrm{d}}{\mathrm{d}t}P(j,t)-(\omega _{b}-\omega)\gamma\sum _{{j\{ i\}}}P(j,i,t). (2.7)

[181.1.6] Solving eq. (2.6) for P(j,i,t) and inserting the result into eq. (2.7) one obtains a closed second order equation for P(i,t)

\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(i,t)+(\gamma+\omega _{b}-\omega)\frac{\mathrm{d}}{\mathrm{d}t}P(i,t)=\omega\sum _{{j\{ i\}}}\frac{\mathrm{d}}{\mathrm{d}t}\big[P(j,t)-P(i,t)\big]+\omega\gamma\sum _{{j\{ i\}}}\big[P(j,t)-P(i,t)\big] (2.8)

where the summations, as before, run over all nearest neighbour sites j of site i. [page 182, §0]    [182.0.1] Eq. (2.8) contains the same information as eq. (2.5) but no longer involves the directional quantities P(i,j,t). [182.0.2] This form can now be used to introduce disorder and it leads directly to eq. (2.1). [182.0.3] We now turn to the introduction of a time dependent network.

2.2 Correlated Hopping in a Dynamic Percolation Network

[182.1.1] Consider a system where the configuration of accessible sites fluctuates in time. [182.1.2] We are interested in the case where the B-particles perform a random walk. [182.1.3] Because we are dealing with bond percolation this random walk occurs on the dual lattice. [182.1.4] In an elementary step a blocking bond swings around either one of its end points through an angle \pm 2\pi/z where z is the coordination number of the underlying lattice. [182.1.5] It then occupies the new bond position if it is vacant. [182.1.6] This process is repeated on the average after a time \tau _{{\mathrm{B}}} which is the characteristic time scale for the blocker motion. [182.1.7] In Figure 1 we depict the possible rotations of a B-particle for the case of a hexagonal lattice. [182.1.8] This model has been termed “dynamic bond percolation” model[22]. [182.1.9] The characteristic hopping time for a single B-particle is called \tau _{{\mathrm{B}}}. [182.1.10] The ratio \tau=\tau _{{\mathrm{B}}}/\tau _{{\mathrm{A}}} between the typical hopping time of the blockers and the walker will be the main variable characterizing the dynamics of the environment.

[182.2.1] Equation (2.1) must be generalized to allow for time dependent transition rates. [182.2.2] Therefore we have to consider an equation of the form

\displaystyle\frac{\mathrm{d}^{2}}{\mathrm{d}t^{2}}P(i,t)+(\gamma _{i}+\omega _{b}-\omega)\frac{\mathrm{d}}{\mathrm{d}t}P(i,t)=\omega\sum _{{j\{ i\}}}A_{{ij}}(t)\frac{\mathrm{d}}{\mathrm{d}t}\big[P(j,t)-P(i,t)\big]+\omega\gamma _{i}\sum _{{j\{ i\}}}A_{{ij}}(t)\big[P(j,t)-P(i,t)\big] (2.9a)

where now the coefficients A are time dependent,

\displaystyle A_{{ij}}(t)=\begin{cases}1&\text{if the bond }[ij]\text{ is vacant at time }t,\\ 0&\text{if the bond is occupied by a blocker at time }t.\end{cases} (2.10a)
Figure 1: Blocker motion on the hexagonal lattice. Three possible elementary rotations of a blocking bond around either one of its endpoints are indicated by arrows. Full lines represent blocked bonds, dashed lines represent open bonds.

[page 183, §0]    [183.0.1] The time dependence of these coefficients could in principle be determined from the many particle master equation for all blockers. [183.0.2] However, because that equation is much too complicated we will approximate the true time dependence by a simple renewal model in Section 4. [183.0.3] Equation (2.9) completes the formulation of the model. [183.0.4] We remark here that other forms of a two step memory are possible and may be useful for applications. [183.0.5] For example one can consider enhanced or reduced transitions continuing in the same direction as the last step. [183.0.6] Such correlations lead to more complicated equations, but they can be treated by the same general approach presented here.

[183.1.1] We conclude this section with the formulas that will be used to calculate the frequency dependent conductivity \sigma(\omega) from P(i,t). [183.1.2] This is done via a generalized Einstein relation which reads

\sigma(\omega)=\frac{\rho e^{2}}{k_{{\relax\mathrm{B}}}T}D(\omega) (2.11a)

where \rho is the carrier density, e their electric charge, k_{{\relax\mathrm{B}}}, the Boltzmann constant, T the absolute tempreature, and D(\omega) the generalized frequency dependent diffusion coefficient. [page 184, §0]    [184.0.1] D(\omega) will be calculated in standard fashion from[23]

D(\omega)=-\frac{\omega^{2}}{z}\int^{\infty}_{0}\sum _{{{\vec{r}}_{i},{\vec{r}}_{0}}}({\vec{r}}_{i}-{\vec{r}}_{0})^{2}\mathrm{e}^{{-\mathrm{i}\omega t}}P\left({\vec{r}}_{i},t\middle|{\vec{r}}_{0},0\right)\mathrm{d}t (2.12a)

where P\left({\vec{r}}_{i},t\middle|{\vec{r}}_{0},0\right) is the solution to eq. (2.9) or eq. (2.1) for the frozen case. [184.0.2] The latter will be determined in the next section.