Wells and Peaks potentials

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3.3 Dynamics on intermediate scales Ordinary diffusion 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 changingε, 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 largercvalues.

A similar picture, though with broader bands on all scales and a collapse of upper bands for largeε 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 largerc 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 smallcandεvalues,

0 1 2

Figure 3.10: Band structures obtained by the Kronig-Penney method in the only source-dependent casec = 0. Two right panels correspond to ordinary diffusive and superdiffusive µ = 1 processes in symmetric piecewise potentialη = 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 ofεvalues, 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 strengthcin the superdiffusive case. This implies a counterintuitive acceleration of such processes on large scales. Accelera-tion happens after some critical value of the target impactc >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 largecvalues as compared with ordinary diffusion. Generally, one frequently observes the collapse of upper bands for very small and very largecvalues. 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)


Figure 3.11: Superdiffusion with Lévy exponent µ = 1 in the potential with seldom peaks. The generalized crystal momentumκn(ε)is presented in dependence on effective potential strengthε. 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

Dµ,c(k;ε)/D = 1−ε2Gµ,c(k).

The functionGµ,c(k) = 2"

0 dqS(q)gµ,c(k/q)gives us insight into the dynamics on the corre-sponding spatial scalex∼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 Gµ,c(k)on k, measured in the inverse units of the correlation length of the potentialξfor differentcvalues and different Lévy exponentsµ, is presented in Figure 3.13 for ordinary diffusion and superdiffusion3. 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 thatξ = 1 without altering the properties of the system.


Figure 3.12: Superdiffusion with Lévy exponent µ = 1 in the potential with seldom wells. The generalized crystal momentumκn(ε)is presented in dependence on effective potential strengthε. For detailed description consult Figure 3.7. Ordinary diffusion

With a varying source-target impact in the ordinary diffusive case, a general course of the dependenceG2,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 factorG2,c(k)is maximal forc= 1and minimal forc= 0.

The small scale asymptotics does not monotonously depend onc. For the extremecvalues (strongly source- and target-dependent processes), G2,c(k) witnesses a stronger attenuation because it is larger, than in the balanced casec= 1/2, in whichG2,c(k)0. The small scale asymptotics is the largest and the same forc= 0andc= 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 ratec= 1/2, in which the support on small scales becomes homogeneous. In the casec += 1/2the fractional Fokker-Planck L operator (3.3) depends on the potential offset forc += 1/2: L[v(x) +vo] = e(2c1)εvoL[v(c)].

The small scale limit of Gµ,c(k) can be calculated as a product of 1 "

dqS(q) = 1 and limz→∞gµ,c(z):

klim→∞Gµ,c(k) = 1

2µ(2c1)2. (3.19)

The dependence ofGµ,c(k→ ∞)onµandcis illustrated in Figure 3.14(a). Hence, the smaller µvalues 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 completelyµ-independent and equals zero.

In the intermediate regime1/k ∼ξ,G2,c(k)is positive for smallcvalues; with growingcit becomes smaller. After some threshold valuec" <1/2,G2,c(k)becomes negative and attains

−4 −2 0 2 4

Figure 3.13: Dependence of the factor Gµ,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 forc = 1. Negative values of G2,c(k) correspond to the enhancement of the process even in the diffusive case, but only for some range ofcvalues.This stands in contrast to the balanced case where negative values ofGµ,c(k), i.e. acceleration, occurs only for superdiffusion (Brockmann, 2003). We can approximately calculate the dependence c"(µ) from the condition gµ,c(1) 0 using the definition of gµ,c

(3.10). Hence, we obtain

c"(µ) = 1−√

12µ. (3.20)

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

The dependence ofGµ,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 cvalues, 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 increasingcafter certain threshold valueccrit the form of the curves changes qualitatively4. 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 isccrit= 1/20.7071.

0 0.5 1 1.5 2 0

0.5 1 2

µ G µ,c()








0 0.5 1 1.5 2

0 0.2 0.4 0.6 1


c* (µ)


Figure 3.14:(a): Small scale asymptote of the factorGµ,c(k)according to (3.20) as a function ofµfor different source-target impact valuesc. (b): threshold value of the source-target impactc"(µ)according to (3.20), above which enhancement on scalesk1 ∼ξappears, as a function of Lévy exponentµ.

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 smallc(strong source impact), the functionGµ,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 casec= 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 withµ = 1exhibit the same dynamics except on the scales of the correlation length. For largerc 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 afterc"(µ) (3.20). On smaller scales, enhancement disappears andGµ,c attains its non-negative small scales asymptotics. For even larger values c > ccrit, 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 afterk−1 ∼ξdiminishes towards zero. Afterwards the process becomes attenuated, reaching its most attenuation in the small scale limit. Note, that for the extremecvalues (c= 0 andc = 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.

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