4 Dynamic Disorder: Renewal Approach

[188.1.1] So far we have discussed equation (2.1). [188.1.2] We now turn to equation (2.9), i. e. we consider dynamic disorder. [188.1.3] The difficulty lies not so much in solving eq. (2.9) but in specifying the stochastic coefficients $A(t)$ . [188.1.4] They are determined by the random motion of all blockers. [188.1.5] Our general strategy will be to assume a simpler form for the random process $A(t)$ , and then to find a solution for eq. (2.9) which makes use of the solution for the frozen case, i. e. eq. (2.1).

[188.2.1] The basic idea is to approximate the actual correlated dynamics of the environment by a simple exponential renewal process. [188.2.2] Consider a single bond. [188.2.3] It is either occupied or vacant, and switches randomly between these two states. [188.2.4] This can be modelled by a two state Markov chain as suggested by Harrison and Zwanzig[30]. [188.2.5] Let $1/\tau _{r}$ , be the switching frequency between the two states, i. e. $\tau _{r}$ is an effective renewal time. [188.2.6] Then the probability to find the bond occupied by a blocker at time $t$ is easily seen to relax as $p+p_{0}\exp(-t/\tau _{r})$ , where $p_{0}$ , is either $1-p$ or $-p$ depending upon whether at time $0$ the bond was occupied or not. [188.2.7] Thus, in this model, the bonds flip randomly and independently between the two states. [188.2.8] Using a $1$ -bond effective medium approximation, as the one described above, Harrison and Zwanzig[30] have shown that for this model the generalized diffusion coefficient is given by $D_{0}(u+1/\tau _{r})$ where $D_{0}(u)$ is the diffusion coefficient for the corresponding frozen problem (e. g. eq. (3.13)). [188.2.9] The same result, to which we will refer as the substitution rule, had been obtained by Druger, Ratner and Nitzan[31] for a model in which full configurations are renewed instead of single bonds. [188.2.10] This is not surprising because in the 1-bond-approximation the behaviour for the full lattice is calculated by considering only the possible configurations of a single bond. [188.2.11] Let us therefore approximate the dynamics of the environment by an exponential renewal process for full lattice configurations with renewal density

$\psi _{r}(t)=\frac{1}{\tau _{r}}\mathrm{e}^{{-t/\tau _{r}}}.$

(4.1)

[page 189, §0] [189.0.1] The mean renewal time $\tau _{r}$ is now an effective renewal time which should be proportional to $\tau$ , the ratio of jump rates between B- and A-particles, i. e.

$\tau _{r}=c\tau.$

(4.2)

[189.0.2] The proportionality constant $c$ contains the effects from correlations in the blocker dynamics that are not taken into account by the renewal approach. [189.0.3] It is, however, not an adjustable parameter. [189.0.4] It will be determined below by comparison with well known exact results for the limit $p\to 1~(\tau=1,\, b=1)$ . [189.0.5] The substitution rule can also be obtained from a simple probabilistic argument[22]. [189.0.6] Consider the inverse Laplace transform $D_{0}(t)$ of $D_{0}(u)$ which was calculated in eq. (3.13). [189.0.7] As is well known, $D_{0}(t)$ is the kernel of a generalized master equation[32]. [189.0.8] Therefore it can be interpreted as a generalized time dependent transition rate, i. e. transition probability per unit time. [189.0.9] It describes the frozen problem in a mean field picture and is the proper kernel to use between renewal events. [189.0.10] Because the renewal process and the random walk of the Aâparticle are independent, the transition rate for the dynamic problem at time $t$ is the product of the corresponding rate for the frozen case and the probability that there was no renewal up to time $t$ , i. e.

$D(t)=D_{0}(t)\left[1-\int^{t}_{0}\psi _{r}(t^{\prime})\mathrm{d}t^{\prime}\right]=D_{0}(t)\mathrm{e}^{{-t/\tau _{r}}}.$

(4.3)

[189.0.11] From this one recovers the substitution rule by Laplace transformation. [189.0.12] Using eq. (4.2) we get

$D(u)=D_{0}\left(u+\tfrac{1}{c\tau}\right)$

(4.4)

as our final result.

[189.1.1] To proceed we have to specify a particular lattice of interest, and solve eq. (3.10). [189.1.2] Before doing so we comment briefly on the relation to our previous continuous time random walk approach to the same problem[22]. [page 190, §0] [190.0.1] Here, as in our previous work we have attempted to find a framework in which correlation effects resulting purely from the dynamics of the environment can be incorporated. [190.0.2] In the present paper we have attacked the problem by considering random walks with memory which has the additional advantage that other types of correlations can be considered. [190.0.3] In our previous CTRW-approach we have attempted to incorporate correlation effects into the waiting time distribution by way of crude probabilistic and physical arguments about the nature of the deblocking mechanism. [190.0.4] This was based on the fact that the effective medium waiting time distribution below $p_{c}$ is not normalized. [190.0.5] Consequently the distinction between the case above $p_{c}$ and below persisted, and it was necessary to utilize the substitution rule of eq. (4.3) to treat the case above $p_{c}$ . [190.0.6] In addition our previous approach was limited to the case $\tau\ll 1$ . [190.0.7] Its main advantage was to exhibit the theoretical possibility of a sequential deblocking mechanism resulting in a nonmonotonous waiting time density and interesting consequences for the conductivity. [190.0.8] On the other hand our present approach applies for all $\tau$ and $p$ . [190.0.9] It allows for other sources of correlations, and gives relatively good quantitative results. [190.0.10] This will be demonstrated in the next section.

Author:	R. Hilfer and R. Orbach
originally published in:	Dynamical Processes in Condensed Molecular Systems, J. Klafter et al. (eds.), World Scientific, Singapore, 1989, pages 175-202, ISBN 981-02-3457-0
originally published:	1989