6 Discussion and Conclusions
[198.2.1] We begin with the discussion of the crossover in σω
as a result of the mobility of the blockers.
[198.2.2] As a general fact we note that the blocker motion
introduces a crossover from a low frequency regime
dominated by the blocker rearrangement to a high frequency regime
in which the disorder appears frozen.
[198.2.3] For small frequencies Reσ∼1/τ while in the high frequency regime
σ behaves as for the frozen case.
[198.2.4] This is independent of the correlations introduced by b.
[page 199, §1]
[199.1.1] In the last section we had found from Figure 4
that the crossover frequency ωτ between the two regimes
behaves as ωτ∼τ-1/2.
[199.1.2] This can be understood from the model of independently fluctuating bonds.
[199.1.3] In this model each bond switches independently between its two states,
closed and open, with a relaxation time τ.
[199.1.4] Let the system be in a stationary state at t=0,
i. e. when the walker starts.
[199.1.5] The number of bonds that have remained in the same state
since t=0 decreases exponentially with time.
[199.1.6] We define the crossover time as the average time
after which the walker first encounters a bond
that has switched at least once since t=0.
[199.1.7] The walker can cross a bond repeatedly
as long as the bond has not switched since the start of the walk.
[199.1.8] If the walker makes n steps then the average length
of the time interval during which all of the it crossed bonds
remain in their original state is τ/n.
[199.1.9] The crossover occurs when this time equals the number of steps,
i. e. when n⋅1∼τ/n, where the charcteristic hopping time of the walker
is again assumed to be 1.
[199.1.10] This immediately gives the observed τ-1/2 behaviour.
[199.2.1] Next we discuss the effect of correlations, i. e. the case b≠1.
[199.2.2] At high frequencies the conductivity is determined mostly
by the conductance (i. e. transition rate) of the last bond that was passed.
[199.2.3] This conductance is increased resp. decreased per definitionem by a factor b.
[199.2.4] Thus the limiting high frequency value of σω
will be increased for enhanced reversals, b>1,
resp. decreased for reduced reversals, b<1.
[199.3.1] At low frequencies the walker explores a much larger region,
and it will begin to feel the existence of the percolation threshold.
[199.3.2] Assume for the moment that the blocker conï¬guration is frozen, i. e. τ=∞.
[199.3.3] Let us consider ï¬rst the case 1-p-pc,
i. e. when there is an infinite network of vacancies.
[199.3.4] To estimate whether the d. c. conductivity σ0 is enhanced or not,
consider the limiting cases b→1 and b→∞
for a lattice of low coordination, e. g. the linear chain.
[199.3.5] In the limit b→∞ the particle moves deterministically
to the right or to the left depending upon its initial conditions.
[199.3.6] For b→∞ the particle will oscillate between two sites.
[199.3.7] Therefore we expect that σ0 will be decreased for b>1,
and increased for b<1.
[page 200, §1]
[200.1.1] A different situation arises for 1-p>pc (still with τ=∞)
because now σ0=0.
[200.1.2] For reduced reversals (b<1) the particle has a higher probability of fully
exploring all the dead ends of the percolating network
than for the enhanced reversals (b>1).
The exploration of the dead ends will however not contribute to the d. c. conductivity,
and we therefore expect σ0 to be decreased for b<1, and increased for b>1.
[200.2.1] From these qualitative arguments one expects by continuity that for 1-p>pc
there will be at least one point of intersection between ReD(ω,b=1,τ=∞)
and ReD(ω,b≠1,τ=∞).
[200.2.2] For 1-p<pc there may be no point of intersection or an even number of them.
[200.2.3] This is confirmed by the calculations.
[200.2.4] For 1-p<pc and τ=∞ there are two points of intersection.
[200.2.5] If b<1 the conductivity is decreased in the very low frequency regime,
increased at intermediate frequencies, and again decreased at high frequencies
relative to its value for b=1.
[200.2.6] For small enough b a maximum can arise at a finite frequency.
[200.2.7] On the other hand for b>1 the conductivity is increased at very low frequencies
then decreased, and finally again increased at high frequencies.
[200.2.8] For the case 1-p>pc and τ=∞ we find exactly one point of intersection.
[200.2.9] For b<1 the conductivity is increased at low frequencies,
and decreased at high frequencies as compared to its value for b=1.
[200.2.10] The reverse is true for b>1.
[200.3.1] The maximum for b→0 arises from the competing effects
of disorder induced correlations and memory induced correlations.
[200.3.2] The memory correlations tend to enhance the low frequency conductivity
over the values at high frequencies just as they do for the regular lattices[17, 18, 19, 20].
[200.3.3] On the other hand the disorder has the opposite effect.
[200.3.4] At high frequencies i. e. short time scales the memory correlations prevail,
while at very low frequencies (i. e. on long time scales) the disorder is dominant.
If now one allows also blocker motion, i. e. for finite τ,
an additional crossover arises for 1-p<pc.
[200.4.1] At very low frequencies the diffusion is controlled by the slowly moving vacancies
which can occupy every site in the underlying regular lattice,
and σ0 will be nonzero.
[200.4.2] One therefore expects that σ0 is increased for b<1,
and decreased for b>1, as for a regular lattice.
[page 201, §0]
[201.0.1] Thus in this should be three points of intersection with the curve for b=1.
[201.0.2] This is indeed borne out by the numerical solution
although it is a small effect as seen in Figure 10
for the case of the hexagonal lattice.
[201.1.1] In summary, in this paper we have analyzed the correlated hopping of an A-particle
in a background of mobile B-particles.
[201.1.2] We have calculated the corresponding frequency dependent diffusion coefficient
using the substitution rule and a generalized effective medium theory
for correlated diffusion.
[201.1.3] We have found a rich variety of new features especially below the vacancy percolation threshold.
[201.1.4] The simultaneous presence of several crossover frequencies
gives rise to a complicated structure of the frequency dependent response.
[201.1.5] In particular we find the possibility of a maximum in the real part at finite frequency,
or apparent power law behaviour over more than a decade in frequency
as a consequence of disorder and correlations.
[201.1.6] Our model has a wide range of applicability.
[201.1.7] As an example we refer back to the introduction
and point out that our approach can be used to
model the simulation results of Refs. [10] or [11] for lattice gases
with short range attractive or repulsive interactions
by employing the correlation parameter b.
[201.1.8] Here we have applied our approach in the d. c. limit
to existing Monte-Carlo simulation data of A-B-lattice gases
for the uncorrelated case (b=1).
[201.1.9] We have found remarkably good agreement.
