Eur. Phys. J. B (2013) 86: 331DOI: 10.1140/epjb/e2013-40436-1

Regular Article

THE EUROPEANPHYSICAL JOURNAL B

Random time averaged diffusivities for Levy walks

D. Froemberga and E. Barkai

Department of Physics, Bar Ilan University, 52900 Ramat Gan, Israel

Received 25 April 2013 / Received in final form 2nd June 2013Published online 24 July 2013 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2013

Abstract. We investigate a Levy walk alternating between velocities ±v0 with opposite sign. The sojourntime probability distribution at large times is a power law lacking its mean or second moment. The first casecorresponds to a ballistic regime where the ensemble averaged mean squared displacement (MSD) at largetimes is

⟨x2⟩ ∝ t2, the latter to enhanced diffusion with

⟨x2⟩ ∝ tν , 1 < ν < 2. The correlation function

and the time averaged MSD are calculated. In the ballistic case, the deviations of the time averaged MSDfrom a purely ballistic behavior are shown to be distributed according to a Mittag-Leffler density function.In the enhanced diffusion regime, the fluctuations of the time averages MSD vanish at large times, yetvery slowly. In both cases we quantify the discrepancy between the time averaged and ensemble averagedMSDs.

1 Introduction

Dispersion of Brownian particles is described on themacroscopic level by Fick’s second law which states thatthe local change in particle density is proportional to thenegative gradient of the local particle flux, where the diffu-sion constant D is the proportionality factor. Accordingly,the density of the particles is a spreading Gaussian withthe mean squared displacement (MSD) going linearly withtime, 〈x2〉 = 2Dt. Einstein [1] established a relationshipbetween the macroscopic quantity D = 〈(δx)2〉/2〈τ〉 andthe underlying stochastic process with jump lengths (δx)of variance 〈(δx)2〉 and the average time passing betweentwo jumps 〈τ〉. For a single Brownian particle, the timeaveraged MSD (TAMSD) δ2 has to be considered,

δ2 =1

t−Δ

∫ t−Δ

0

[x(t′ +Δ) − x(t′)]2 dt′. (1)

Here the averaging was performed over all displacementsalong the single trajectory occuring during a fixed lag timeΔ within the measurement time t. Due to the station-ary increments, for a Brownian motion the time averagedMSD will attain the limit δ2 = 2DΔ for large times t.Brownian motion is therefore ergodic, ensemble and timeaverages are equal, 〈x2〉 = δ2. It is important to note thatin practice an estimation of ensemble averaged diffusionconstants from single particle trajectories even for nor-mal diffusion is a subtle issue due to the lack of statistics.However, one can exploit the ergodic property of such pro-cesses in order to find suitable estimators [2,3].

Unlike in normal diffusion, in strongly disordered me-dia the mean squared displacement often does not grow

a e-mail: daniela.froemberg@web.de

linearly with time, but according to a power law 〈x2〉 ∝ tν .The case 0 < ν < 1 corresponds to subdiffusion, 1 < ν < 2indicates enhanced diffusion or superdiffusion. In disor-dered and complex systems time averages can differ con-siderably from the corresponding ensemble averages. Theincreasing employment of single particle tracking tech-niques makes it indispensable to understand these differ-ences in order to interpret the experiments and gain in-sight to the mechanisms underlying anomalous transport.

Examples for enhanced diffusion are experimentson active transport of microspheres [4], of polymericparticles [5] or of pigment organelles [6] in living cells.Enhanced diffusion in living cells is promoted by molec-ular motors that move along microtubules or the cy-toskeleton [7,8]. In vivo experiments usually examine thetrajectories x(t) of single particles and hence assess theTAMSD equation (1) instead of ensemble averages. Mea-surements often find this quantity to be a random variable,so that ensemble average and (single trajectory) time av-erage differ. For this behavior several reasons come intoquestion, either variations in the probed environments orcells, ergodicity breaking or too short measurement times.In references [4,6] the observed enhanced diffusion wasdescribed within the framework of generalized Langevinequations (GLE), which is Gaussian and ergodic [9–11].Using this theory, the experimentally observed exponentscharacterizing the anomalous transport were reproduced.However, this Gaussian approach cannot explain the mul-tiscaling of moments found for the enhanced motion ofpolymeric particles in living cells [5], a feature that seemsto be more consistent with a Levy walk scheme.

Levy flights describe enhanced diffusion in terms ofrandom walk processes where the distribution of the parti-cle displacements lack the second or even the first moment

Page 2 of 13 Eur. Phys. J. B (2013) 86: 331

(i.e. give rise to a Levy statistics). Levy flights have beenused in the past to describe phenomena as diverse as thedispersal of bank notes [12], tracer diffusion in systems ofbreakable elongated micelles [13] or animal foraging pat-terns [14]. However, Levy flight models can lead to unphys-ical behavior with regard to velocities since Levy flightsare characterized by extremely large jump lengths, cor-responding to the heavy tails in the jump length distri-butions. Levy walks address finite velocities by either pe-nalizing long instantaneous jumps with long resting times(jump models), or by letting the particles move at a cer-tain velocity for a certain time or displacement, and choos-ing a new direction (or velocity) and sojourn time accord-ing to given probabilities (velocity models) [15,16].

The theory of Levy walks finds a wide range of ap-plications. Experiments with passive tracer particles ina laminar flow have shown that the flight times andhence displacements within the resultant chaotic trajec-tories of the tracer particles can exhibit power-law dis-tributions [17]. Likewise, the motion of tracers in turbu-lent flows can be described as Levy walks [18]. Anotherexample is the stochastic description of on-off-times inblinking quantum dots, where the intensity correspondsto the velocity of a particle alternating between states ofzero and constant nonzero velocity. Nonergodic behaviorwas found in the correlation functions for sojourn timedistributions lacking their mean [19,20]. Another exampleis the dynamics of cold atoms in optical traps [21] andthe related Brownian motion in shallow logarithmic po-tentials [22], or perturbation spreading in many-particlesystems [23]. Moreover, also deterministic systems suchas certain classes of iterated nonlinear maps may showenhanced diffusion. The chaotic behavior of resistivelyshunted Josephson junctions manifests itself in an anoma-lous (deterministic) phase diffusion, which can, therefore,be modelled by means of such maps [24]. In turn, theenhanced diffusion behavior emerging from such iteratedmaps can be modelled stochastically using the Levy walkapproach (in particular velocity models) [15].

In this article, we study Levy walks in one dimensionwhere the persistence times in the positive or negative ve-locity state are drawn according to a probability densityfunction ψ(τ) with either first or second moment lack-ing. The first case is referred to as the ballistic case, thelatter as the subballistic or enhanced case. We note thatcontrary to the normal diffusion case this model for Levywalks on the basis of persistence times is equivalent tothe Levy walk model where flight times are distributed asψ(τfl), but positive or negative velocity are drawn ran-domly with equal probability (see Appendix B). Thus thederived properties and relations we are interested in willhold also for that Levy walk model. Recently, the dynam-ics emerging from a nonlinear map similar to the Levywalk was investigated [25]. However, this work did not ad-dress the fluctuations of the TAMSD. Fluctuations wereconsidered in a very recent numerical study for the spe-cial case of Levy walks exhibiting enhanced diffusion [26],and numerically and analytically for enhanced and ballis-tic case in a brief publication of the authors [27].

The article is divided into four parts. The first one isdedicated to the ballistic case of a Levy walk, the secondone to a Levy flight with a step size distribution lackingthe second moment, and the third one to the enhancedLevy walk case. For both the ballistic and the enhancedcase we first review briefly occupation times, propagatorsand ensemble averaged mean squared displacements. Thenwe turn to the ensemble averaged quantities such as cor-relation functions and ensemble averages of the TAMSDs.Finally, we investigate the distributions and properties ofthe fluctuations of the TAMSDs. In the second part, weinvestigate for comparison the Levy flight correspondingto the enhanced case. We also provide the fluctuationsof time averages of Levy flights, using simple arguments,thus adding to the work in reference [28] who addressedthis problem rigorously and more generally.

2 Ballistic regime

We consider a one-dimensional motion of a particle witha two-state-velocity ±v0, where the sojourn times inthe states are drawn from a probability density func-tion (PDF) ψ(τ). Hence the particle has a velocity +v0for period τ1 drawn from ψ(τ), after that switches tovelocity −v0 and remains in this state for another pe-riod τ2 also drawn from ψ(τ). This process is then re-newed. In particular, this PDF is chosen such that it lacksits first moment with a power-law decay at large times,ψ(τ) ∼ A/Γ (−α)τ−1−α with 0 < α < 1. Particularly inthe simulations we will use

ψ(τ) ={ατ−1−α τ ≥ 10 else. (2)

The Laplace transform of equation (2) in the small-u-limit is

ψ(u) � 1 −Auα (3)

with A = Γ (1 − α) and u being the Laplace conju-gate of τ . This relation can easily be derived via Laplacetransformation of

∫ τ0 ψ(τ ′)dτ ′ � 1 − τ−α, which yields

1u ψ(u) � 1

u (1 − Γ (1 − α)uα) by using convolution andTauberian theorems.

In the limit of long times t, the particle moves ballis-tically so that the mean-squared displacement is

⟨x2⟩

=(1−α)t2 [29,30]. Recently, a similar system had been gen-erated in the context of deterministic superdiffusion andthe ensemble average of the time averaged mean squareddisplacement 〈δ2〉 was derived [25]. Processes whose tem-poral dynamics is governed by heavy-tailed PDFs lackingtheir mean exhibit ageing [31], which manifests itself ina deviation of the ensemble averaged mean-squared dis-placement from the TAMSDs which are random them-selves. Therefore, the fluctuations of TAMSDs are an im-portant signature of this process.

Eur. Phys. J. B (2013) 86: 331 Page 3 of 13

�1.0 �0.5 0.0 0.5 1.00

2

4

6

8

z

p�z�

�1.0 �0.5 0.0 0.5 1.00.0

0.5

1.0

1.5

2.0

2.5

zp�z�

Fig. 1. Distribution of the position of particles at t = 107 forα = 0.5 (left panel) and α = 0.7 (right panel) in terms of thescaling variable z (v0 = 1). Sample size 104, ψ(τ ) is given byequation (2).

2.1 Occupation fraction and particle position

The distribution for the fraction of time z± = t±/t spentin one state (positive or negative velocity) after a largetime t is given by Lamperti’s law [32],

pocc(z±) =sinπαπ

× zα−1± (1 − z±)α−1

z2α± + (1 − z±)2α + 2 cosπαzα± (1 − z±)α,

(4)

a generalization of the arcsine law which is reproduced forthe case α = 1/2. The particle position x(t) is given by theintegral over the velocities

∫ t0v(t′) dt′, and the temporal

mean of the velocity is x/t = v0(t+ − t−)/t. The distribu-tion of this quantity in the limit of large times had beencalculated in the context of the mean magnetization of atwo-state system [33]. A related problem is the calculationof the integral over the intensity of emitted light in blink-ing quantum dots. In that case one state corresponds tothe “on”-state, i.e. the emitting state, and the other stateto the “off”-state where no light is emitted [19].

The probability to find the particle at position x attime t for large times can be obtained in terms of thescaling variable z = x/(v0t) by using equation (4) andchange of variables. We have x

v0t= 2z+−1, hence ∂z+

∂z = 12

and p(z) =∣∣∣∂z+∂z

∣∣∣ pocc(z+), which finally results in

p(z) =2 sinπα

(1 − z2

)α−1

π((1 + z)2α + (1 − z)2α + 2 cosπα (1 − z2)α

) .

(5)Figure 1 shows this distribution of the scaled particle po-sition for two different values of α.

2.2 Ensemble average of δ2

First we will analyze the ensemble averaged TAMSD:

〈δ2〉 =1

t−Δ

⟨∫ t−Δ

0

[x(t′ +Δ) − x(t′)]2 dt′⟩

. (6)

Fig. 2. Sketch of the forward recurrence time τf . Ticks sym-bolize renewal events.

Changing the order of integration and ensemble averagingin equation (6), we get

〈δ2〉=∫ t−Δ

0

〈x2(t′ +Δ)〉 + 〈x2(t′)〉 − 2〈x(t′)(t′ +Δ)〉t−Δ

dt′.

(7)In order to find 〈δ2〉, we first derive the Levy walk cor-relation function 〈x(t1)x(t2)〉, which turns out to exhibitageing. The position correlation function 〈x(t1)x(t2)〉 isrelated to the velocity correlation function 〈v(s1)v(s2)〉via

〈x(t1)x(t2)〉 =⟨∫ t1

0

v(s1) ds1∫ t2

0

v(s2) ds2

⟩

=∫ t1

0

ds1

∫ t2

s1

〈v(s1)v(s2)〉 ds2

+∫ t1

0

ds2

∫ s2

0

〈v(s1)v(s2)〉 ds1, (8)

where we took into account that t2 > t1. Using the ap-proach of reference [33], we obtain for the velocity corre-lation function 〈v(t1)v(t2)〉:

〈v(t1)v(t2)〉 = v20

∞∑

n=0

(−1)npn(t1, t2), (9)

where pn(t1, t2) is the probability of the velocity to switchits sign n times within the time interval [t1, t2], i.e. foreven n between t1 and t2 we have v(t1)v(t2) = v2

0 , and forodd n, v(t1)v(t2) = −v2

0 .In the scaling limit where t1 and t2 are large and

equation (3) applies, the particle gets stuck in the + or− state for times of the order of the measurement timedue to the lacking first moment of the sojourn time dis-tribution ψ(τ). Therefore, only the first term n = 0 isrelevant in equation (9). The corresponding probabilityp0(t1, t2) of the velocity not switching its sign from agiven t1 on up to t2 is called the persistence probability,to which the velocity correlation function is proportional,〈v(t1)v(t2)〉 = v2

0p0(t1, t2). Let us denote the first waitingtime from an arbitrary time t1 up to the next switchingevent by τf (see Fig. 2). This first waiting time is calledthe forward recurrence time, and its PDF ψf,t1(τf ) differsfrom ψ(τ) since t1 does not necessarily coincide with arenewal event.

With t2 = t1 +Δ, we express the persistence probabil-ity as [33]:

p0(t1, t1 +Δ) =∫ ∞

Δ

ψf,t1(τf ) dτf . (10)

Page 4 of 13 Eur. Phys. J. B (2013) 86: 331

In terms of the scaling variable θ = Δ/t1, the PDF of theforward recurrence time to take a value of Δ at time t1reads in the scaling limit where also Δ is large:

limt1→∞ψf (θ) =

sinπαπ

1θα(1 + θ)

. (11)

The limit theorem for forward recurrence times ψf,t1 equa-tion (11) is due to Dynkin [34]. Hence,

〈v(t1)v(t1 +Δ)〉 = v20

∫ ∞

Δt1

sinπαπ

1θα(1 + θ)

dθ

= v20

∫ t1t2

0

sinπαπ

1ξ1−α(1 − ξ)α

dξ

= v20

sinπαπ

B

(t1t2

;α, 1 − α

), (12)

where B(y; a, b) =∫ y0 duu

a−1(1 − u)b−1 denotes the in-complete Beta-function [35]. Note that this expressionyields only real values for t2 ≥ t1. In the case of t1 > t2,the t1 and t2 in equation (12) have to be interchanged.Inserting equation (12) into equation (8) and using inte-gration by parts we find

〈x(t1)x(t2)〉 = v20

sinπαπ

[t1t2B

(t1t2

;α, 1 − α

)

−12t22B

(t1t2

; 1 + α, 1 − α

)

−12t21B

(t1t2

;−1 + α, 1 − α

)]

−αv20

2t21. (13)

In particular, for t1 = t2 we find the MSD

〈x2(t1)〉 = (1 − α)v20t

21, (14)

in agreement with [29,30,33]. The theoretical autocorrela-tion functions increase in this case with increasing timedifference. For normal diffusion, we have

〈x(t1)x(t2)〉 = 2Dmin(t1, t2),

so that〈x(t1)x(t2)〉/〈x2(t1)〉 = 1

for t2 ≥ t1. In contrast, the behavior of the Levy walkis governed by long periods of ballistic motion. Thus,it exhibits strong correlations compared to normal dif-fusion which are due to the long sticking times in thepositive or negative velocity states. In the limiting caseα → 0 the particle remains in state +v0 or −v0 through-out the measurement so that x(t) = ±v0t with probabil-ity 1/2 for either sign. Therefore we expect the purelyballistic, deterministic behavior 〈x(t1)x(t2)〉 = v2

0t1t2for the position-position correlation function, and hence〈x(t1)x(t2)〉/〈x2(t1)〉 = t2/t1. To see this, note that for

0 2� 107 4� 107 6� 107 8� 107 1� 1080

1� 1015

2� 1015

3� 1015

4� 1015

5� 1015

t1

�x 1x 2�

Fig. 3. Simulational results for 〈x(t1)x(t2)〉 for α = 0.5 (uppergraph) and α = 0.7 (lower graph), and the respective theoret-ical predictions (solid lines) (Eq. (13)). t2 was fixed at 108,v0 = 1; sample size 104.

α → 0, B(t1t2

;α, 1 − α)

diverges and the first term inequation (13) is the only term that remains:

v20t1t2 sinπα

πB

(t1t2

;α, 1 − α

)=

v20t1t2 sinπα

π

(t1t2

)α ∞∑

i=0

Γ (α+ i)Γ (α)

1(α+ i)i!

(t1t2

)i,

recalling that t1/t2 < 1 and taking the limit α → 0, weare left with

limα→0

〈x(t1)x(t2)〉 = v20t1t2

where we used de l’Hospital’s rule. Simulations of the sys-tem for moderate α show a good agreement with theoryequation (13), (see Fig. 3).

The asymptotic behavior of the position-autocorrelation function equation (13) is:

〈x(t1)x(t1 +Δ)〉 =

⎧⎪⎨

⎪⎩

v20 sin παπα(1−α2) t

21

(1 + Δ

t1

)1−α, t1 Δ

v20(1 − α)

(t21 +Δt1

), t1 Δ.

(15)Note that we made again the transformation from (t1, t2)to t1, Δ = t2 − t1 and t1 ≤ t2. Inserting the above re-sults for the correlation function equation (13) and meansquared displacement equation (14) into equation (7), in-tegrating by parts and using again the integral definitionof the incomplete Beta function, we obtain the ensembleaveraged TAMSD. In the limit Δ/t 1 we get

〈δ2〉 ≈ v20

[

Δ2 − sinπαπα

2Δ2(Δt

)1−α

6 − 11α+ 6α2 − α3

]

. (16)

Note that in fact the short time behavior of the correla-tion function is not negligible in the integral equation (7)since it affects the long-time behavior of the ensemble-averaged TAMSD 〈δ2〉. It is important to point out that

Eur. Phys. J. B (2013) 86: 331 Page 5 of 13

100 200 500 1000 2000 5000 1� 1041

100

104

106

�

�v0��2�Δ2

Fig. 4. Deviations from ballistic motion of TAMSDs v20Δ

2−δ2versus Δ for ten different trajectories; α = 0.5, v0 = 1, t = 108.Points belonging to the same trajectory are connected by astraight line.

even the first term of 〈δ2(Δ)〉 (Eq. (16)), differs from〈x2(t)〉 = v2

0(1 − α)t2 by a factor. This term was alsofound recently in a different context of deterministic mapsby reference [25].

2.3 Fluctuations of the time averages

More important are the fluctuations of the TAMSDs, sincethese allow conclusions to be drawn with respect to the er-godic properties of the system. Our simulations revealedthat these fluctuations are quite small compared to thevalue of the TAMSDs, and become even smaller with timerelative to the ballistic contribution (v0Δ)2. The fluctu-ations are more pronounced if one looks at the shiftedTAMSD v2

0Δ2 − δ2, which is the natural random variable

of this process as will turn out soon. In Figure 4 we plotv20Δ

2 − δ2 versus the lag time Δ for ten different trajec-tories. v2

0Δ2 − δ2 remains visibly random.

Small Δ limit

In order to specify the distribution of v20Δ

2 − δ2, we con-struct the following special case. Consider again the so-journ time PDF equation (2) with 0 < α < 1. Moreover,let us for now only adhere to small Δ < 1, so that atmaximum one change of direction takes place during theinterval Δ. In this case, the TAMSD equation (6) can beexplicitly calculated. Clearly, if there were no changes ofdirection, the TAMSD would be δ2 = v2

0Δ2. For a single

switching event at time ts within a time interval (t, t+Δ),the integrand in equation (6) becomes

[x(t+Δ) − x(t)]2 ={v20Δ

2 for t ≤ (ts −Δ) and ts ≤ t

(2v0ts − v0Δ− 2v0t)2 for (ts −Δ) ≤ t ≤ ts.(17)

Hence, using equation (1), one change of direction withinthe observation time t reduces the TAMSD to a value of

δ2 = v20Δ

2 − 23v20t Δ

3 for t Δ (t−Δ ≈ t), as shows inte-gration of equation (17). Two changes of direction result inδ2 = v2

0Δ2 − 2 2

3v20t Δ

3 and so forth. Altogether we find foran amount nt of switching events within the observationtime t,

δ2 � v20

t

[Δ2t− 2

3Δ3nt

], (18)

where the amount of direction changes nt within (0, t) isa random variable. The probability pn(t) of the number ofevents n within (0, t) is determined by the convolution ofn sojourn time PDFs with the probability of no event afterthe nth one [34], and is well investigated. Taking ψ(u) �1−Auα in Laplace domain and using the convolution theo-rem results in pn(u) = Auα−1 exp [n ln(1 − Auα)]. Laplaceinversion yields:

pn(t) =1α

t

A1/αn1+1αlα,1

[t

A1/αn1/α

]. (19)

lα,1(t) denotes the one-sided Levy density whose Laplacetransform is given by exp[−uα] [34]. Hence, with 〈nt〉 ∼tα(AΓ (1 + α)) the ensemble average of equation (18)becomes

〈δ2〉 = v20Δ

2 − 2v20

3AΓ (1 + α)tα−1Δ3, (20)

which clearly differs from equation (16). Equation (18)and, hence, equation (20) describes a special case of thesojourn time distribution equations (2) and (3) fulfill-ing the relations t1−α Δ/(AΓ (1 + α)) and Δ ≤ 1so that the first ballistic term in equation (16) remainslarger than second term. In contrast, equation (16) re-quires Δ 1. From equation (18) we find that the quan-tity (v2

0Δ2−δ2) and the amount of switching events within

t are proportional,

(v20Δ

2 − δ2) =23Δ3

tnt. (21)

Therefore, in terms of a new variable

ξ =v20Δ

2 − δ2

v20Δ

2 −⟨δ2⟩ =

nt〈nt〉 (22)

and using equation (19) the rescaled distribution of theTAMSD becomes

p(ξ) =Γ 1/α(1 + α)αξ1+1/α

lα,1

[Γ 1/α(1 + α)

ξ1/α

]. (23)

This PDF is the density of the Mittag-Leffler distribu-tion, a distribution already encountered in the context ofTAMSD fluctuations in the subdiffusive continuous timerandom walk [36–38]. Figure 5 shows the PDF (23) andthe respective results for simulations of the Levy walk fortwo different values of α, but for large Δ. It is interest-ing to note that 〈v2

0Δ2 − δ2〉 differs for small and large

Δ regimes, however the distribution of the rescaled vari-able (22) does not depend on Δ.

Page 6 of 13 Eur. Phys. J. B (2013) 86: 331

0 1 2 3 4 50.00.10.20.30.40.50.60.7

Ξ

p�Ξ�

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.7

Ξp�Ξ�

Fig. 5. PDF of ξ = ((v0Δ)2 − δ2)/((v0Δ)2 − 〈δ2〉) at t = 106

for α = 0.5 (left panel) and α = 0.7 (right panel); Δ = 103.The histogram shows the result of the simulations, with ψ(τ )equation (2), the solid blue curve the theory equation (23);sample size 104, v0 = 1.

Crossover to the large Δ regime

The ensemble average of the fluctuations of (v20Δ

2− δ2) isgiven by equation (16) for large 1 Δ t. In the presentcase where ψ(τ) = 0 at short times τ < 1, the behaviorof the ensemble averaged TAMSD at Δ 1 is given byequation (20). This behavior at small Δ constitutes thelower bound for more general ψ(τ) with arbitrary shapeat small τ . However, the fluctuations of the time averagesequation (23) are governed only by the tail of the persis-tence PDF ψ(τ) and are therefore the same for small andlarge Δ. Hence we can write

v20Δ

2 − δ2 = χ2nt,

χ2 =

⎧⎨

⎩

2v20Δ3

3t Δ 1

2v20 sinπαΔ3−α

πα(6−11α+6α2−α3)AΓ (1+α)t Δ 1. (24)

Here χ2 gives the deterministic part that governs the en-semble mean of the shifted TAMSD, while the full fluctu-ations enter via nt, compare equations (22) and (23). InFigure 6 we plot t(v2

0Δ2 − ⟨δ2⟩) versus Δ. Simulational

results match the theoretical short time as well as longtime behaviors, equations (20) and (16), respectively. Thecrossover takes place in the region of the cutoff of the so-journ time PDF ψ(τ) at small times, i.e. at Δcr ≈ 1.

Finally we demonstrate numerically that the above dis-tributions are indeed the limiting distributions at largetimes. For this purpose, we calculate the ergodicity break-ing (EB) parameter [36] for the shifted TAMSD ξ:

EB = limt→∞

⟨ξ2⟩− 〈ξ〉2〈ξ〉2 =

2Γ 2(1 + α)Γ (1 + 2α)

− 1, (25)

where we used equation (23). Numerics for α = 0.5 showthat the EB-parameter for ξ tends indeed to the predictedfinite value EB = 0.571 (Fig. 7), i.e. the variable ξ remainsdistributed according to equation (23).

Note that, however, the EB-parameter for the original(not shifted) TAMSD δ2 slowly tends to zero for nonzeroΔ as t−2(1−α). Hence, for the ballistic Levy walk, non-ergodicity in the sense of the distribution of time averagesdoes not find its expression in the (decaying) fluctuationsof the TAMSDs δ2 themselves, but in the (persisting) fluc-tuations of the shifted and rescaled variable ξ.

0.1 10 1000 1051

104

108

1012

1016

1020

�

t��v 0��2��Δ2��

Fig. 6. t(v20Δ

2 − ⟨δ2⟩) versus Δ, α = 0.5. The small–Δ regionis sensitive to the shape of ψ(τ ). Lines indicate theory equa-tion (20) (dashed) matching numerical data (dots) at small Δ,and equation (16) (solid) for large Δ. Note that the crossovertakes place in the region of the small-time cutoff of the sojourntime PDF equation (2), Δcr ≈ 1; sample size 104, v0 = 1,t = 107.

�

���� � ���� �

���� � ����

� ����

����

�

����� ���� � ����

1000 104 105 106 107 108

0.56

0.58

0.60

0.62

0.64

0.66

0.68

0.70

t

EB

Fig. 7. EB-parameter of ξ for α = 0.5. Red circles indicateΔ = 0.5, blue diamonds Δ = 100, green triangles Δ = 5000.The grey solid line indicates the theoretical value EB = 0.571(Eq. (25)); sample size 5000.

3 Levy flight TAMSD

Before we turn to the behavior of the TAMSD of the Levywalk in the enhanced diffusion regime, let us illustrate thesituation in the related Levy flight. We present a ratherillustrative than rigorous argument, to avoid complicatedmath. A more general and rigorous treatment can be foundin reference [28]. The Levy flight is a random walk processwhere at each renewal the displacement x of the walker isdrawn according to a jump PDF λ(x), but in contrast tothe normal random walk the jump PDF lacks the secondmoment. With a unit time span passing between consecu-tive renewal events, the number of jumps acts as the (dis-crete) time variable t. Hence consider the coordinate of aLevy flight after t steps as a sum of independent identically

Eur. Phys. J. B (2013) 86: 331 Page 7 of 13

distributed (i.i.d.) random variables or displacements xi

Xt =t∑

i=1

xi.

Let the {xi} be distributed according to a two-sided sym-metric distribution λ(x) = λ(−x) falling off as a powerlaw for large |x|,

λ(x) ∝ A0

2|x|−1−α , 1 < α < 2. (26)

In particular, for our simulations we used equation (26)with A0 = α for |x| ≥ 1, and λ(x) = 0 for |x| < 1 (in theAppendix it is shown that this approach is equivalent tothe alternating one for the Levy flight). The distributionof the sum Xt will yield a two-sided Levy law for large t,Lα,0(Xt/(tA0Γ (1 − α)/α)1/α), according to the general-ized central limit theorem [31,34]. The function L0,α(x) isdefined as the inverse Fourier transform of

Lα,0(k) = exp [−|k|α] (27)

with k being the Fourier variable [34]. This Green functionof the Levy flight has a similar behavior as the central partof the Levy walk Green function when 1 < α < 2, as willbecome obvious in the next section.

The time-averaged mean squared displacement(TAMSD) is defined as [28]:

δ2 =1

t−Δ

t−Δ∑

i=1

[Xi+Δ −Xi]2

=1

t−Δ

t−Δ∑

i=1

(i+Δ∑

k=i

xk

)2

, (28)

where the integer Δ is the lag time. We have

δ2 =1

t−Δ

t−Δ∑

i=1

⎛

⎝i+Δ∑

k=i

x2k +

i+Δ∑

k=i

i+Δ∑

j �=kxkxj

⎞

⎠ . (29)

The mixed terms∑i+Δ

k=i

∑i+Δj �=k xkxj on average cancel out

for large enough Δ, hence we omit them. Moreover weassume 1 Δ t so that

δ2 � 1t−Δ

t−Δ∑

i=1

i+Δ∑

k=i

x2k

� 1t

t∑

i=1

i+Δ∑

k=i

x2k

d≈ Δ

t

t∑

k=1

x2k . (30)

We find for the distribution of the y = x2

p(x2) = p(x)∣∣∣∣dx

dy

∣∣∣∣ ∝

A0

2y−1−α

2 . (31)

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

Ζ�t2�Α

p�Ζ�t2�Α�

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

Ζ�T2�Α

p�Ζ�T2�Α�

Fig. 8. Levy flight simulation results for the distribution ofζ

t2α

for α = 1.5 and KLFα = Γ (1− α

2), Δ = 1, t = 105 (left) and

Δ = 100, t = 106 (right), and theoretical predictions accordingto equation (33) (solid curves).

Note that the transition from x to the positive valuedy results in a factor 2 in the normalization. The large(y = x2) asymptotics can be obtained in Laplace domain,using the Tauberian theorem:

p(uy) � 1 − A0

αΓ(1 − α

2

)u

α2y . (32)

Hence, the sum over these x2k = yk in equation (30) is

a sum over positive i.i.d. random variables distributedaccording to a PDF with a power law tail of exponent−1 − α/2. Hence, due to the generalized central limittheorem we find in the large t limit [31,34] the PDF ofζ = tδ2/Δ:

p (ζ) =1

(KLFα t)

2α

Lα2 ,1

{ζ

(KLFα t)

2α

}

, (33)

where KLFα = A0

α Γ (1 − α2 ) and Lα/2,1(·) is the one-sided

Levy PDF given by exp[−uα/2] in Laplace domain [34].

A similar result was obtained in reference [28] though aslightly different scaling was reported. Figure 8 shows thePDF equation (33) obtained with Mathematica, and thecorresponding simulational results which perfectly matchthe theory.

Hence, this example illustrates that the TAMSD ofLevy flights with jump length distributions without sec-ond moment is a one-sided Levy density lacking the firstmoment. As will be shown later, the TAMSD distributionfor the corresponding Levy walk is completely different de-spite the similarity in the central part of the propagator(see Fig. 9).

4 Enhanced diffusion regime

In this section, we consider the regime of sojourn timesdistributed according to a PDF with existing mean 〈τ〉,but diverging second moment, i.e. equation (2) with 1 <α < 2. The expansion in Laplace domain is hence

ψ(u) � 1 − 〈τ〉u+Auα. (34)

In our case equation (2) the average sojourn time is 〈τ〉 =α/(α− 1) and A = |Γ (1 − α)|.

Page 8 of 13 Eur. Phys. J. B (2013) 86: 331

�4 �2 0 2 40.00

0.05

0.10

0.15

0.20

0.25

0.30

z

p�z�

Fig. 9. For the Levy walk with 1 < α < 2 the central part ofthe particle distribution (vs. the scaling variable z = x/t1/α)is a symmetric Levy distribution equation (37) (solid line).Simulation (histogram) for α = 3/2, t = 106, v0 = 1, cα (seeEq. (38)), sample size 105.

4.1 Particle position distribution

Again, the particle position is given by the integral overthe velocities v0(t+ − t−). The MSD is well-known [33]:

⟨x2(t)

⟩ � 2v20A

〈τ〉(α− 1)Γ (4 − α)

t3−α, (35)

with the generalized diffusion constant

Kα =v20A(α − 1)

〈τ〉Γ (4 − α). (36)

The propagator is given by a two-sided Levy-distributionfor the central part where |x| v0t,

p(x, t) � 1

(cαt)1/α

Lα,0

(x

(cαt)1/α

)

, (37)

whereLα,0(k) = exp [−kα]

and [33]cα = −A cos(πα/2)/〈τ〉. (38)

Due to the finite velocity the Levy walk propagator ex-hibits a cutoff so that p(x, t) = 0 at |x| > v0t, similarlyto other Levy walk models [29,39]. Moreover, we expectthose particles that have never changed their directionof motion to form a delta-peak at the edge of the cut-off [40]. In the following we will calculate the time aver-aged mean squared displacement (TAMSD) of the Levywalk described above.

4.2 Ensemble average of δ2

It is important to note that also in the subballistic regime1 < α < 2, the velocity correlation is governed by thepersistence probability p0(t) to stay in one state for atime t: for sojourn time PDFs ψ(τ) with existing mean

the corresponding forward recurrence time PDF ψf,t1(τf )reaches stationarity at large t1 and obeys the limitingdistribution [33]

limt1→∞ψf,t1(τf ) =

1〈τ〉∫ ∞

τf

ψ(τ)dτ � A

〈τ〉 |Γ (1 − α)|τ−αf .

This first waiting time in turn has no mean and thereforep0 dominates 〈v(t1)v(t1 +Δ)〉. Again, with equations (9)and (10), 〈v(t1)v(t1 +Δ)〉 = v2

0p0(t1, t1 +Δ) holds for thevelocity correlation function for t1 and Δ both large. Thep0 for the present process is well-known, so that following,e.g. the procedure presented in reference [33] we have

〈v(t1)v(t1 +Δ)〉 =v20A

(α − 1) 〈τ〉 |Γ (1 − α)| t1−α1

×(

t1t1 +Δ

)α−1((

1 − t1t1 +Δ

)1−α− 1

)

. (39)

In the equilibrated regime (or stationary state) t1 Δ,〈v(t1)v(t1 +Δ)〉 becomes independent of time t1 so that

〈v(t1)v(t1 +Δ)〉eq � v20A

〈τ〉 (α− 1)|Γ (1 − α)|Δ1−α. (40)

By integrating equation (39) as in equation (8) we obtainthe position autocorrelation

〈x(t1)x(t1 +Δ)〉 =Av2

0

〈τ〉Γ (4 − α)t3−α1

×[− (y)3−α + (1 + y)3−α + (α− 3) (1 + y)2−α + α

]

(41)

where y = Δ/t1. Our simulations have shown that thisestimation reproduces the large t1 behavior of the positioncorrelation 〈x(t1)x(t2)〉 quite well. ForΔ→ 0 the behaviorof the MSD equation (35) is reproduced.

Note that in the subballistic case the MSD for a processstarting with the beginning of the measurement differsfrom the MSD for a process that started a long time beforethe beginning of the measurement time t0, as was foundearlier in the context of a stochastic collision model [41].This behavior is due to the predominant role of the per-sistence probability p0 in the correlation functions dis-cussed above. In what follows the MSD for the processthat started long before t0 will be called the equilibriumMSD

⟨x2⟩eq

, alluding to the fact that this process has nomemory of its starting time. The situation is sketched inFigure 10. It is important to introduce the equilibriumMSD at this point for the following reason: since the timeaveraging procedure comprises averaging over all contin-uously shifted time lags, and not only over those start-ing at a switching event, there is an inherent averagingover disorder. The equilibrium MSD accounts for this av-eraging over disorder and is therefore the natural ensem-ble averaged quantity to later compare the time averagedMSD to. Note also that such a definition of an equilibrium

Eur. Phys. J. B (2013) 86: 331 Page 9 of 13

Fig. 10. Sketch of the process starting at the beginning ofthe measurement t0 (upper panel) and the equilibrium processstarting at t = −tini with tini � 〈τ 〉. The measurement be-gins at a time t0 between two renewal events (lower panel).Renewal events are indicated by black ticks, the start of themeasurement t0 is marked by a cross.

MSD is not possible in the ballistic case since there a sta-tionary state does not exist – the respective MSD wouldnever become independent of the time difference betweenstart of the process in the past and the actual start of themeasurement.

Using equation (40) and

⟨x2⟩eq

= 2∫ t

0

dt1

∫ t

0

dt2〈v(t1)v(t2)〉eq,

we obtain the known equilibrium MSD [41]

〈x2〉eq = v20

A

〈τ〉2

Γ (4 − α)t3−α. (42)

For the TAMSD, we use again the definition equation (1)and write equation (7) for the respective ensemble average.To obtain a description of the ensemble averaged TAMSD,we insert the autocorrelation function equation (41) andthe MSD equation (35) into equation (7). Thus, we have

〈δ2〉 � v20

(t−Δ)2A

〈τ〉Γ (5 − α)

×[− (t−Δ)4−α + t4−α −Δ4−α

+ (4 − α)tΔ3−α − (4 − α)t3−αΔ], (43)

which becomes in leading orders by expansion for Δ/tsmall:

〈δ2〉 � 2Av20

〈τ〉Γ (4 − α)Δ3−α − Av2

0

〈τ〉Γ (3 − α)Δ2t1−α. (44)

Note that this result for 〈δ2〉 complies with the timedependence of the equilibrium ensemble average

⟨x2⟩eq

(Eq. (42)), and hence differs from the MSD (35) by lack-ing a factor:

limt→∞

δ2

〈x2〉 =1

α− 1. (45)

Numerical simulations also indicate that these two aver-ages appear to differ by a factor (Fig. 11). Further nu-merical evidence for this behavior was found in refer-ences [25,26]. Especially, for small α the convergence of

100 200 500 1000 2000 5000 1� 1041

2

3

4

5

�

�Δ2���x2�

Fig. 11. Ratio⟨δ2⟩/⟨x2⟩

for α = 1.25 (triangles), α = 1.5

(diamonds) and α = 1.7 (circles); t = 107. The theory predicts4, 2 and 1.43 for this ratio (solid lines).

100 200 500 1000 2000 5000 1� 1041000

104

105

106

107

�

Δ2

Fig. 12. TAMSDs of particle trajectories in dependence on thelag time Δ, t = 105; α = 5/4. For finite measurement times tfluctuations in δ2 are observed. Larger times t and smaller lagtimes Δ result in smaller fluctuations.

〈δ2〉 is extremely slow. Moreover, in our simulations thelarge time behavior of the

⟨x2⟩

is not represented veryaccurately at the corresponding relatively small times (upto 104).

4.3 Fluctuations of the TAMSD

The TAMSDs of trajectories measured up to a certain ob-servation time appear to be distributed. Figure 12 showsthe TAMSD evolution of some sample trajectories. Thefluctuations of the TAMSDs decrease with increasing to-tal measurement time. However, since in experiments theobservation time is always finite, these fluctuations mayplay a role in practice. In our simulations, for example,we find large fluctuations among the δ2 for α = 1.25 andΔ = 100, t = 105 (see Fig. 12).

Although the propagators of the Levy walk and flightlook very similar in the central part |x| < v0t, unlike the

Page 10 of 13 Eur. Phys. J. B (2013) 86: 331

� ������������������

��������� ���������

� � �������� � ��������� �������� � ��������

1000 104 105 106 107

0.01

0.1

1

10

t

EB

� ��������� ��������� ��������� ��������� ����

� ��������� ��������� ��������� ��������� ����

1000 104 105 106 107

1.000.50

5.00

0.10

10.00

0.05

0.01

tE

B

Fig. 13. EB-parameter equation (46), for α = 3/2 (left) and5/4 (right). Blue diamonds indicate Δ = 100, green trianglesΔ = 5000, the grey solid line indicates the slope (1 − α).

flight case simulations suggest that the TAMSD distribu-tion for the Levy walk cannot be expressed by a Levy dis-tribution of stability index α/2. For the ergodicity break-ing (EB) parameter of the subballistic Levy walk regime,

EB = limt→∞

⟨(δ2)2⟩−⟨δ2⟩2

⟨δ2⟩2 , (46)

we find a steady decay with t, which is yet very slow withan exponent of roughly (1−α), to the value zero indicatingthat the width of the distribution tends to zero (Fig. 13,for α = 1.5, 1.25). Hence, the TAMSDs do not remaindistributed in the limit of very long times in the subbal-listic regime. Such a very slow decay to ergodic behavioris not a distinct feature of the enhanced phase of the Levywalk model but can also be found for completely differentergodic systems such as relaxation of confined fractionalBrownian motion [42].

In contrast to the flight case, the width of the δ2–distribution for the Levy walk at finite times always existsdue to finite velocity. Simulations suggest that it increaseswith the lag time Δ as Δ4. The width of the TAMSD dis-tribution hence appears to have the same Δ-dependenceas a ballistic motion. However, it is not quite clear whetherthis dependence represents the large time behavior of thedistribution of the TAMSDs, or whether it is an artefact ofthe ballistic peaks of the propagator due to the extremelylong transients.

5 Summary

In this article, we have shown that the shifted time av-eraged MSD of the ballistic Levy walk is described bythe Mittag-Leffler distribution, similar to the distributionof the TAMSD in the sub-diffusive continuous time ran-dom walk (CTRW) [36,43]. This distribution describes thefluctuations of the time averages and is universal. TheTAMSD averaged over an ensemble of trajectories 〈δ2〉 isnot equal to the ensemble average 〈x2〉 as already pointedout by Akimoto [25]. Interestingly 〈δ2〉 − (v0Δ2) exhibitstwo behaviors valid for Δ < 1 and Δ > 1, and it wouldbe interesting to see if a similar cross-over takes place in

other models such as the sub-diffusive CTRW. For Levyflights the TAMSDs are random with a PDF given by theone sided Levy PDF, which is in agreement with rigorousresults (though note that our coefficients are different thanthose reported in reference [28], possibly due to a typo). Inthe enhanced diffusion regime, the PDF of the particle po-sition of the Levy walk is similar to the Levy flight case, atleast in its center. However, the TAMSD of the two modelsis vastly different, and for Levy walks no fluctuations arefound for δ2. This indicates that the TAMSD is controlledby rare events, since the tails of the mentioned distribu-tions are where one finds differences between the models.Thus taking into consideration finite velocity (like in theLevy walk model) is crucial for our understanding of theergodic properties of these processes. In the future it mightbe worth while checking the time averages of lower ordermoments, since they might exhibit behavior very differ-ent the second moment considered here. We note that forfinite times the fluctuations of TAMSDs are large. Con-sequently, in the laboratory where experiments are madefor finite time the process may seem non ergodic, but thisis only a finite time effect. Moreover, in this sub-ballisticcase 〈δ2〉 is equal to the equilibrium MSD 〈x2〉eq, but notto 〈x2〉. If one wishes to compare time and ensemble av-erages, the conclusion on equality of these two averageswill depend on how the ensemble is prepared. To attainergodicity we need to start the ensemble in a stationarystate, which is not so surprising. The point is that fornormal processes, e.g. the case where all moments of thewaiting time PDF ψ(τ) exists, it does not matter how westart the process, in the long time limit the time and en-semble average procedures are all identical. In that sensethe Levy walk, in the enhanced regime, is unique. A sim-ilar effect might be found also for sub-diffusive CTRW,for the case of finite average waiting time, but with aninfinite variance, but that is left for future work. Furthertime averaged drifts, when a bias is present, also exhibitinteresting ergodic features and Einstein relations, as wediscussed recently in reference [27].

This work was supported by the Israel Science Foundation.

Appendix A: Correlation functions

For the derivation of the velocity correlation function wefollow [33]. Hence, for the time variables t1, Δ with t2 −t1 = Δ and going to Laplace domain with respect to Δwe have

pn(t1, uΔ) =

⎧⎪⎨

⎪⎩

ψf,t1(uΔ)ψn−1(uΔ)1−ψ(uΔ)uΔ

n ≥ 1

1−ψf,t1 (uΔ)

uΔn = 0,

(A.1)where ψf,t1 is the forward recurrence time PDF, i.e. thePDF of the time it takes to encounter the next event aftera given time t1, which in double-Laplace domain reads

˜ψf,u1(uΔ) =

11 − ψ(u1)

ψ(uΔ) − ψ(u1)u1 − uΔ

. (A.2)

Eur. Phys. J. B (2013) 86: 331 Page 11 of 13

Here uΔ and u1 are the Laplace variables conjugate toΔ and t1, respectively. Inserting (A.1) and (A.2) backinto (9) we get

LΔ {〈v(t1)v(t1 +Δ)〉|uΔ} = v20

1 − ψf,t1(uΔ)uΔ

+ v20

∞∑

n=1

(−1)nψf,t1(t1, uΔ)ψn−1(uΔ)1 − ψ(uΔ)

uΔ

= v20

⎡

⎣1 − ψf,t1(, uΔ)uΔ

− ψf,t1(uΔ)1 − ψ(uΔ)

uΔ

(1 + ψ(uΔ)

)

⎤

⎦ ,

Lt1,Δ {〈v(t1)v(t1 +Δ)〉|u1, uΔ} =

v20

[1

uΔu1− 1

1 − ψ(u1)ψ(uΔ) − ψ(u1)(u1 − uΔ)uΔ

− 11 − ψ(u1)

ψ(uΔ) − ψ(u1)u1 − uΔ

1 − ψ(uΔ)

uΔ

(1 + ψ(uΔ)

)

⎤

⎦

= v20

1uΔ

[1u1

− ˜ψf,u1(uΔ)

21 + ψ(uΔ)

](A.3)

where we denote the Laplace transformation by L.

A.1 Ballistic phase

Let us now specify the sojourn time distribution at largetimes, in Laplace domain ψ(u) � 1 − Auα, 0 < 1 < αwhich yields

Lt1,Δ {〈v(t1)v(t1 +Δ)〉|u1, uΔ} =

v20

[1

uΔu1− 1

1 − ψ(u1)ψ(uΔ) − ψ(u1)(u1 − uΔ)uΔ

]

= v20

[1

uΔu1− 1uΔ

˜ψf,u1(uΔ)]

(A.4)

and after Laplace inversion

〈v(t1)v(t1 +Δ)〉 = v20

(

1 −∫ Δ

0

ψf,t1(Δ′) dΔ′

)

= v20

∫ ∞

Δ

ψf,t1(Δ′) dΔ′, (A.5)

which we will use in the following since we know the scal-ing form of ψf,t1(Δ) for large t1 and Δ due to Dynkin’stheorem equation (11), leading to (12).

Inserting equation (12) into equation (8) and usingthe definition of the incomplete Beta function B(y; a, b) =∫ y0duua−1(1 − u)b−1 and repeated integration by parts we

find

〈x(t1)x(t2)〉 = v20

sinπαπ

×[∫ t1

0

ds1

∫ t2

s1

B

(t1t2

;α, 1 − α

)ds2

+∫ t1

0

ds2

∫ s2

0

B

(t1t2

;α, 1 − α

)ds1

]

= v20

sinπαπ

[∫ t1

0

ds1

[t2B

(s1t2, α, 1 − α

)

− s1B (1, α, 1−α)−s1B(s1t2,−1+α, 1−α

)

+ s1B (1,−1 + α, 1 − α)]

+∫ t1

0

ds2

[s2(1 − α)

π

sinπα

]]

= v20

sinπαπ

[t1t2B

(t1t2, α, 1 − α

)

− t22B

(t1t2, 1 + α, 1 − α

)

− 12t21B (1, α, 1 − α)

− 12t21B

(t1t2,−1 + α, 1 − α

)

+12t22B

(t1t2, 1 + α, 1 − α

)

+12t21B (1,−1 + α, 1 − α)

]+v20

2(1 − α)t21

(A.6)

which with B(1,−1 + α, 1 − α) = 0 for 0 < α < 1 finallyyields equation (13).

A.2 Subballistic phase

Here the Laplace transform of the sojourn time of theparticle in a velocity state for small u reads ψ(u) =1 − 〈τ〉u + Auα with 〈τ〉 = α/α− 1 and A = |Γ (1 − α)|.For t1, Δ large, i.e. u1, uΔ → 0 in Laplace domain, equa-tion (A.3) becomes again

Lt1,Δ {〈v(t1)v(t1 +Δ)〉|u1, uΔ} =

v20

[1

uΔu1− 1

1 − ψ(u1)ψ(uΔ) − ψ(u1)(u1 − uΔ)uΔ

]

= v20

[1

uΔu1− 1uΔ

˜ψf,u1(uΔ)

]

= v20˜p0(u1, uΔ) = v2

0

A

〈τ〉uα−1

1 − uα−1Δ

u1(u1 − uΔ). (A.7)

Page 12 of 13 Eur. Phys. J. B (2013) 86: 331

Laplace inversion yields

〈v(t1)v(t1 +Δ)〉= v2

0

A

〈τ〉1

(α− 1)|Γ (1 − α)|[Δ1−α − (t1 +Δ)1−α

]

= v20

A

〈τ〉1

(α− 1)|Γ (1 − α)|[(t2 − t1)1−α − t1−α2

].

Inserting this again into equation (8) gives

〈x(t1)x(t2)〉 = v20

A

〈τ〉1

Γ (4 − α)

× [−(t2 − t1)3−α + t3−α2 − (3 − α)t2−α2 t1 + αt3−α1

]

(A.8)

which finally yields equation (41).

Appendix B: Connection to other Levy walkmodels

Here, we have chosen an alternating model of a Levywalk: after each renewal the particle moves in the direc-tion opposite to the previous one with velocity v0. How-ever the derived results remain the same for a second classof Levy walks where the particle chooses to move to theleft or right with equal probability at each renewal, ir-respective of the direction it took before. The resultantbehavior of the position autocorrelation 〈x(t1)x(t2)〉 andthe TAMSD is rooted in the behavior of the velocity auto-correlation 〈v(t1)v(t2)〉. Now it is easy to argue that alsoin the second class of models the velocities after the i’thand j’th renewal are uncorrelated, 〈vivj〉 = δi,j and, thus,the velocity autocorrelation 〈v(t1)v(t2)〉 is proportional tothe probability p0(t1, t2) not to encounter a renewal be-tween t1 and t2 (only then i = j). Thus, in this model〈v(t1)v(t2)〉 = v2

0p0(t1, t2) follows even more trivially thanin the alternating one (where this is only true for sojourntime distributions in the + and − state that are lacking atleast the second moment), and as a consequence the de-rived quantities will be also the same for the two differentmodels in the long time limit, although the convergenceto the asymptotic behavior could be very different in bothmodels. Note that specific coefficients, e.g. the one charac-terizing the central part of the propagator in the subbal-listic case equation (38), may differ in both models [15].

Appendix C: Levy flights

In the Levy flight the alternating (antipersistent) modeland the one with random choice between jump directionswith equal probability yield asymptotically equivalent re-sults. Jumps take place every unit time interval, we denotethe amount of performed jumps byN . Let us briefly sketchthis fact by comparing the respective jump PDFs of thetwo models in Fourier domain.

First, for random choice of direction we have the sym-metric, two sided jump PDF λ(x) = α/2|x|−1−αΘ(|x|−1),1 < α < 2 (Θ is the Heaviside step function) and, thus, inFourier domain and in the continuum limit k → 0

λ(k) = 2∫ ∞

1

cos(kx)α

2x−1−αdx

= 1 − α|k|α∫ ∞

0

(cos(y) − 1) y−1−αdy, (C.1)

where we used the substitution y = kx, so that finally

λ(k) = 1 −A∗|k|α (C.2)

A∗ = −|Γ (1 − α)| cos(πα

2

). (C.3)

The sum of N 1 such variables∑i = 1Nxi is according

to the generalized central limit theorem

p(x,N) =1

(A∗N)1/αLα,0

(x

(A∗N)1/α

). (C.4)

In the corresponding antipersistent case the jumps are al-ternating between the positive and negative jump PDFsλ+ = αx−1−αΘ(x − 1) and λ− = α|x|−1−αΘ(1 − x). InFourier-domain we have thus for k → 0

λ+(k) = α

∫ ∞

−∞exp (−ikx)x−1−αΘ(x − 1)dx

= 1 + 〈x〉(ik) + |Γ (1 − α)|(ik)α

λ−(k) = α

∫ ∞

−∞exp (−ikx) |x|−1−αΘ(1 − x)dx

= 1 + 〈x〉(−ik) + |Γ (1 − α)|(−ik)α. (C.5)

If we now merge two successive jumps to one effectivejump, λeff(k) = λ+(k)λ−(k), we have

λeff(k) = 1 + |Γ (1 − α)|(ik)α + |Γ (1 − α)|(−ik)α

= 1 + |Γ (1 − α)||k|α[exp(iπα

2

)+ exp

(−iπα

2

)]

= 1 + |Γ (1 − α)||k|α[2 cos

(πα2

)]

= 1 −A∗eff |k|α, (C.6)

where A∗eff = −2|Γ (1 − α)| cos (πα/2). Thus, a sum of

Neff 1 such effective jumps is

p(x,Neff) =1

(A∗effNeff)1/α

Lα,0

(x

(A∗effNeff)1/α

). (C.7)

Furthermore note that Neff = N/2 and A∗eff = 2A∗ so

that this is equivalent to equation (C.4). Here, we madethe assumption that for N 1 and odd the contributionof the single excess jump is negligible in comparison to thepaired jumps.

Appendix D: Ensemble averaged TAMSD,ballistic phase

Inserting equation (13) and equation (14) into equa-tion (7) and again using integration by parts and the

Eur. Phys. J. B (2013) 86: 331 Page 13 of 13

integral definition of the incomplete Beta function yields

〈δ2〉 = v20

[1

t−Δ

[1 − α

3(t3 −Δ3)

]

− sinπαπ(t−Δ)

[(13t3 − 1

2Δt2)B

(t−Δ

t;α, 1 − α

)

−13Δ3B

(t−Δ

t;α,−2 − α

)

+12Δ3B

(t−Δ

t;α,−1 − α

)

−16t3B

(t−Δ

t; 1 + α, 1 − α

)

+16Δ3B

(t−Δ

t; 1 + α,−2 − α

)

+(−1

6t3 +

12Δt2 − 1

2Δ2t

)

×B(t−Δ

t;−1 + α, 1 − α

)

+16Δ3B

(t−Δ

t;−1 + α,−2 − α

)

−12Δ3B

(t−Δ

t;−1 + α,−1 − α

)

+12Δ3B

(t−Δ

t;−1 + α,−α

)]]

. (D.1)

For the small Δ expansion equation (16) we used theidentity

B

(t1t2, a, b

)= B (1, b, a) −B

(Δ

t2, b, a

), (D.2)

where Δ = t2 − t1 and the expansion of the incompleteBeta function for small arguments y

B (y, a, b) = ya∞∑

n=0

(1 − b)nn!(a+ n)

yn (D.3)

where (c)n = Γ (c+ n)/Γ (c) is the Pochhammer symbol.

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