**3.3 Dynamics on intermediate scales**

**3.3.1.1 Ordinary diffusion**

**3.3.1.2.3 Wells and Peaks potentials**

in the wells potential and for strongly target-dependent processes in the peaks potential
(Fig-ure 3.11 and 3.12). On the other hand for strongly source dependent processes in the peaks
potential (c = 0; 1/4 Figure 3.11) dependence on *ε* is also smooth as well as for strongly
target-dependent processes in the wells potential (Figure 3.12 *c* = 3/4; 1). However, in the
latter case, upper bands are shifted upwards. In the former case they are shifted downwards.

Thus, a difference to the ordinary diffusion consists in the upwards shift for strongly target-dependent processes, implying acceleration on corresponding scales.

In the peaks potential and strongly source dependent case, the downward shift of upper
bands for superdiffusive processes is more pronounced as compared to the ordinary
diffu-sion (Figure 3.11 and Figure 3.5). Thus, in the peaks potential source-dependent processes
are strongly attenuated on intermediate scales in the superdiffusive case. In the only
target-dependent case of the peaks potential (c = 1, Figure 3.11), a smooth behavior of the lower
bands with changing*ε, and a complicated structure of the upper ones, indicates that *
superdif-fusion on large scales is not affected by inhomogeneity to such an extent as on intermediate
scales. The complex band structure is present on intermediate scales as opposed to all scales of
analogous ordinary diffusion case (Figure 3.5). The sharp downward shift can also be found in
upper bands with the same implication on corresponding scales. Compared with the ordinary
diffusive case, this dense gap structure (Figure 3.11 and Figure 3.5) persists for larger*c*values.

A similar picture, though with broader bands on all scales and a collapse of upper bands for
large*ε* values, appears also in the wells potential in the case of the mostly source-dependent
process (c = 0Figure 3.6), indicating attenuation on corresponding scales. Compared with
the ordinary diffusion, the attenuation arises for larger*c* values (cf. *c* = 0 Figure 3.12 and
Figure 3.6). Broader bands (without additional shift) generally imply less attenuation as
com-pared with thinner bands (see discussion on page 31). For processes with small*c*and*ε*values,

0 1 2

Figure 3.10: Band structures obtained by the Kronig-Penney method in the only source-dependent
case*c* = 0. Two right panels correspond to ordinary diffusive and superdiffusive *µ* = 1 processes
in symmetric piecewise potential*η* = *λ* (see fig.3.9). Two left panels correspond to superdiffusion
(µ = 1) in asymmetrical piecewise potential with opposite bias. Note the amazing similarity to the
band structures of peaks (η/λ = 0.1) and wells (η/λ= 0.9) potentials (c= 0Figure 3.11 and Figure
3.12). Only upper and bottom boundaries for each band are plotted.

gaps emerge in lower bands, indicating a slow-down on large scales. In an intermediate range
of*ε*values, a fast downward shift of upper bands and band collapse can be observed. Thus, on
corresponding scales, the attenuation is enhanced by a downward shift and band shrinkage (for
*c*= 0). What concerns the dynamics of strongly target dependent processes on large scales, if
we take the present upward shift into account, we conclude that the most acceleration occurs
in the wells potential as compared to the cosine and piecewise potentials. The peaks potential
accelerates superdiffusive processes to the least extent. These findings are also confirmed in
Section 3.4.

Let us summarize our qualitative analysis of the band structure. In contrast to ordinary
diffu-sion, the first band becomes broader with source-target impact strength*c*in the superdiffusive
case. This implies a counterintuitive acceleration of such processes on large scales.
Accelera-tion happens after some critical value of the target impact*c >*1/2. Thus, for a superdiffusive
process to be enhanced, it is not sufficient to have only some prevalence of the target over the
source impact. The prevalence should be strong enough. This observation is fully confirmed
by perturbation analysis in the next section. On intermediate scales, we also have more
pro-nounced acceleration for large*c*values as compared with ordinary diffusion. Generally, one
frequently observes the collapse of upper bands for very small and very large*c*values. This
indicates hindering of diffusion on corresponding scales. Not only superdiffusive processes
but also ordinary diffusive ones, can distinguish between potential with opposite bias if source
and target impacts are disbalanced. In general, sensitivity of the processes to a potential shape
depends on the source-target impact. Presented qualitative insights into the dynamics on large
scales will be compared with the analytical perturbation results and numerical diagonalization
of the generalized Schrödinger equation (A.2) in the case of weak potential strengths (ε *→*0)

ε

Figure 3.11: Superdiffusion with Lévy exponent *µ* = 1 in the potential with seldom peaks. The
generalized crystal momentum*κ** _{n}*(ε)is presented in dependence on effective potential strength

*ε. For*detailed description consult Figure 3.7.

in Section 3.4.

**3.3.2 Random potentials**

In the previous section we considered behaviour of the generalized superdiffusive processes in periodic potentials on intermediate scales. In the present section, we will draw our attention to the dynamics on intermediate and small scales in random potentials. According to the results of perturbative analysis (Section 3.2), the generalized diffusion coefficient is given by

*D**µ,c*(k;*ε)/D* = 1*−ε*^{2}*G**µ,c*(k).

The function*G** _{µ,c}*(k) = 2"

_{∞}0 *dqS*(q)g* _{µ,c}*(k/q)gives us insight into the dynamics on the
corre-sponding spatial scale

*x∼*1/k. We will restrict oneselves to the case of Gaussian correlation spectrum (F). Behavior in potentials with other correlation spectra reveals essentially the same features. The dependence of

*G*

*(k)on*

_{µ,c}*k, measured in the inverse units of the correlation*length of the potential

*ξ*for different

*c*values and different Lévy exponents

*µ, is presented in*Figure 3.13 for ordinary diffusion and superdiffusion

^{3}. Three regimes can be distinguished:

a small scale regime, an intermediate regime on scales approximately equal to the correlation length, and a large scale regime, which will be the topic of Section 3.4. We discuss ordinary diffusive and superdiffusive cases separately.

3Note that due to the scaling argument from Section 3.1 we can always rescale our coordinates so that*ξ* = 1
without altering the properties of the system.

ε

Figure 3.12: Superdiffusion with Lévy exponent *µ* = 1 in the potential with seldom wells. The
generalized crystal momentum*κ** _{n}*(ε)is presented in dependence on effective potential strength

*ε. For*detailed description consult Figure 3.7.

**3.3.2.1 Ordinary diffusion**

With a varying source-target impact in the ordinary diffusive case, a general course of the
dependence*G*2,c(k)changes only quantitatively; no new global or local maxima appear. Large
scale (k *→* 0) and small scale (k *→ ∞*) limits are attained from below. The overall variation
of the factor*G*2,c(k)is maximal for*c*= 1and minimal for*c*= 0.

The small scale asymptotics does not monotonously depend on*c. For the extremec*values
(strongly source- and target-dependent processes), *G*_{2,c}(k) witnesses a stronger attenuation
because it is larger, than in the balanced case*c*= 1/2, in which*G*2,c(k)*∼*0. The small scale
asymptotics is the largest and the same for*c*= 0and*c*= 1, implying that the processes have
the same relaxation properties on corresponding scales. Note, that the small scale asymptotics
does not vanish as was in the case for the balanced transfer rate*c*= 1/2, in which the support
on small scales becomes homogeneous. In the case*c* *+*= 1/2the fractional Fokker-Planck *L*
operator (3.3) depends on the potential offset for*c* *+*= 1/2: *L*[v(x) +*v** _{o}*] =

*e*

^{(2c}

^{−}^{1)εv}

^{o}*L*[v(c)].

The small scale limit of *G**µ,c*(k) can be calculated as a product of _{2π}^{1} "

*dqS*(q) = 1 and
lim*z**→∞**g**µ,c*(z):

*k*lim*→∞**G**µ,c*(k) = 1

2µ(2c*−*1)^{2}*.* (3.19)

The dependence of*G**µ,c*(k*→ ∞*)on*µ*and*c*is illustrated in Figure 3.14(a). Hence, the smaller
*µ*values lead to greater attenuation on small scales. On the other hand, the more balanced the
source-target impact is, the more *µ-independent the small scale asymptotics becomes. For*
*c*= 1/2, it is completely*µ-independent and equals zero.*

In the intermediate regime1/k *∼ξ,G*2,c(k)is positive for small*c*values; with growing*c*it
becomes smaller. After some threshold value*c*^{"}*<*1/2,*G*_{2,c}(k)becomes negative and attains

−4 −2 0 2 4

Figure 3.13: Dependence of the
factor *G** _{µ,c}*(k) determining the
im-pact of the random phase potential
with a Gaussian correlation
spec-trum on diffusive processes with
different Lévy exponents

*µ*= 0.5; 1; 1.5and2.

its local minimum which becomes most pronounced for*c* = 1. Negative values of *G*2,c(k)
correspond to the enhancement of the process even in the diffusive case, but only for some
range of*c*values.This stands in contrast to the balanced case where negative values of*G** _{µ,c}*(k),
i.e. acceleration, occurs only for superdiffusion (Brockmann, 2003). We can approximately
calculate the dependence

*c*

*(µ) from the condition*

^{"}*g*

*µ,c*(1)

*≈*0 using the definition of

*g*

*µ,c*

(3.10). Hence, we obtain

*c** ^{"}*(µ) = 1

*−√*

1*−*2^{−}^{µ}*.* (3.20)

This dependence is presented in Figure 3.14(b). The smaller the Lévy exponent*µ, the larger*
the source-target impact is needed to obtain acceleration on the scales comparable with the
characteristic scales of the inhomogeneity. The value of*c*= 1/2(balanced case)
correspond-ing to the already obtained Lévy exponent*µ**th* *≈*0.415(Brockmann, 2003) is depicted by the
red circle.

**3.3.2.2 Superdiffusion**

The dependence of*G**µ,c*(k)(0 *< µ <* 2) in the superdiffusive case, characterizing the
tran-sient dynamics, looks very different, as compared with ordinary diffusion. Depending on the
*c*values, the curves look very distinct, most changes occurring for *c >* 1/2 (compare
Fig-ure 3.13 for *µ* = 0.5 with *c* = 1/2 and *c* = 1). For *c <* 1/2, starting from small values
of the argument, the curves go above their positive large scale asymptotic limit, reach the
global maximum, then go down, reach the global minimum and then attain their small scale
asymptotics from below. With increasing*c*after certain threshold value*c*crit the form of the
curves changes qualitatively^{4}. It starts now from its negative large scale asymptotic limit, goes
up, reaches its local maximum (except the strongly superdiffusive case,*µ* = 0.5), then goes
down for a while, reaches its local minimum and then goes up again, reaching their positive

4As we will see in Section 3.4, the threshold value of source-target impact is*c*_{crit}= 1/*√*2*≈*0.7071.

0 0.5 1 1.5 2 0

0.5 1 2

µ G µ,c(∞)

c={0;1}

c={0.1;0.9}

c={0.2;0.8}

c={0.3;0.7}

c={0.4;0.6}

c=0.5

### (a)

0 0.5 1 1.5 2

0 0.2 0.4 0.6 1

µ

c* (µ)

### (b)

Figure 3.14:(a): Small scale asymptote of the factor*G** _{µ,c}*(k)according to (3.20) as a function of

*µ*for different source-target impact values

*c. (b): threshold value of the source-target impactc*

*(µ)according to (3.20), above which enhancement on scales*

^{"}*k*

^{−}^{1}

*∼ξ*appears, as a function of Lévy exponent

*µ.*

small scales asymptotics from below. One of the messages from the picture is that we cannot
obtain enhancement on all scales simultaneously. We can see, that for small*c*(strong source
impact), the function*G**µ,c*(k)is always positive, and the process is most attenuated (the global
maximum) on the scales slightly above the correlation length. On scales approximately equal
to the correlation length, the process attenuation is least pronounced (global minimum). Note
that in the case*c*= 0, for an intermediate Lévy exponent (µ= 1), the long and small scales
asymptotics coincide, suggesting the special role of this exponent for superdiffusion among
traps: the only source dependent processes with*µ* = 1exhibit the same dynamics except on
the scales of the correlation length. For larger*c* values, the global maximum becomes less
pronounced. The whole long-scale range acquires similar relaxation properties and is
attenu-ated. As previously mentioned, on the scales of the correlation length, the process becomes
enhanced after*c** ^{"}*(µ) (3.20). On smaller scales, enhancement disappears and

*G*

*µ,c*attains its non-negative small scales asymptotics. For even larger values

*c > c*crit, the process is en-hanced on large scales, the enhancement slows with decreasing spatial scales. Next in the case

*µ*= 1.5 the enhancement gives place even to attenuation, then the enhancement increases again and after

*k*

^{−1}*∼ξ*diminishes towards zero. Afterwards the process becomes attenuated, reaching its most attenuation in the small scale limit. Note, that for the extreme

*c*values (c= 0 and

*c*= 1), while small scale limits are the same, as was also the case in the diffusive limit, the only source-dependent process exhibits a faster rate of convergence to the asymptotic limit and thus more pronounced attenuation.

Thus, we can conclude that the strong acceleration regime at the scales of the correlation length may be attained also in the ordinary diffusive limit. Acceleration is mostly pronounced in the superdiffusive case of high target impact and cannot be realized at all scales. A large-scale limit may be of particular interest, because the processes exhibits more universal prop-erties in this case. In the next section, we discuss large scale asymptotics in detail, utilizing the results of Section 3.2.