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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 . [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 , to the previously visited site, and jumps with a rate to any other of the nearest neighbour sites. [179.1.8] The ratio is a measure of the strength of the memory correlations. [179.1.9] For 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 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 , i. e.  the probability density to find the walker at site at time if it started from site at time . [179.1.12] We will show below that the problem can be formulated as a system of second order equations for the which reads

 (2.1a)

where and is the coordination number of site . [179.1.13] The symmetric quantities represent the bond disorder and are defined as

 (2.2a)

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

[180.1.1] Equation (2.1) has to be supplemented by initial conditions for and its derivative. [180.1.2] Special attention has to be paid to the condition on and its derivative. [180.1.3] The correct choice is

 (2.3a) (2.3b)

where the symbol stands for the limit from above. [180.1.4] Note that 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 to find the walker at site at time given that it arrived at via a direct transition from site . [180.2.7] Thus labels the previously occupied site or history. [page 181, §0]    [181.0.1] Then the symmetric probablity density is obtained from by a summation over all possible histories

 (2.4)

where the sum runs over all nearest neighbour sites of site . [181.0.2] The conditional probability densities obey the master equation

 (2.5)

where the sum runs over all nearest neighbours of site except for site 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

 (2.6)

where , and 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 and sums over all sites which are nearest neighbours of site . [181.1.4] Next we differentiate eq. (2.6) and sum over . [181.1.5] We then employ it for and interchanged to eliminate the term and find

 (2.7)

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

 (2.8)

where the summations, as before, run over all nearest neighbour sites of site . [page 182, §0]    [182.0.1] Eq. (2.8) contains the same information as eq. (2.5) but no longer involves the directional quantities . [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 where 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 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 . [182.1.10] The ratio 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

 (2.9a)

where now the coefficients are time dependent,

 (2.10a)

[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 from . [183.1.2] This is done via a generalized Einstein relation which reads

 (2.11a)

where is the carrier density, their electric charge, , the Boltzmann constant, the absolute tempreature, and the generalized frequency dependent diffusion coefficient. [page 184, §0]    [184.0.1]  will be calculated in standard fashion from[23]

 (2.12a)

where 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.