[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 At.
[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 At, 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/τr, be the switching frequency between the two states,
i. e. τr 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+p0exp-t/τr, where p0,
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 D0u+1/τr
where D0u 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
[page 189, §0]
[189.0.1] The mean renewal time τr is now an effective renewal time which should be
proportional to τ, the ratio of jump rates between B- and A-particles, i. e.
[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→1(τ=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 D0t of D0u
which was calculated in eq. (3.13).
[189.0.7] As is well known, D0t 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.
| Dt=D0t1-∫0tψrt′dt′=D0te-t/τr. | | (4.3) |
[189.0.11] From this one recovers the substitution rule by Laplace transformation.
[189.0.12] Using eq. (4.2) we get
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 pc is not normalized.
[190.0.5] Consequently the distinction between the case above pc and below persisted,
and it was necessary to utilize the substitution rule of eq. (4.3)
to treat the case above pc.
[190.0.6] In addition our previous approach was limited to the case τ≪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 τ 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.
