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# 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 . [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 , 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 , be the switching frequency between the two states, i. e.  is an effective renewal time. [188.2.6] Then the probability to find the bond occupied by a blocker at time is easily seen to relax as , where , is either or depending upon whether at time 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 -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 where 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

 (4.1)

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

 (4.2)

[189.0.2] The proportionality constant 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 . [189.0.5] The substitution rule can also be obtained from a simple probabilistic argument[22]. [189.0.6] Consider the inverse Laplace transform of which was calculated in eq. (3.13). [189.0.7] As is well known, 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 Aâparticle are independent, the transition rate for the dynamic problem at time is the product of the corresponding rate for the frozen case and the probability that there was no renewal up to time , i. e.

 (4.3)

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

 (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 is not normalized. [190.0.5] Consequently the distinction between the case above and below persisted, and it was necessary to utilize the substitution rule of eq. (4.3) to treat the case above . [190.0.6] In addition our previous approach was limited to the case . [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 and . [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.