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Scaled Brownian motion: a paradoxical process with a time dependent diffusivity for

the description of anomalous diffusion

Jae-Hyung Jeon,1 Aleksei V. Chechkin,2, 3, 4 and Ralf Metzler4, 1, ∗

1Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland2Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine

3Max-Planck Institute for the Physics of Complex Systems, Nothnitzer Straße 38, 01187 Dresden, Germany4Institute of Physics & Astronomy, University of Potsdam, 14776 Potsdam-Golm, Germany

Anomalous diffusion is frequently described by scaled Brownian motion (SBM), a Gaussian processwith a power-law time dependent diffusion coefficient. Its mean squared displacement is 〈x2(t)〉 ≃K (t)t with K (t) ≃ tα−1 for 0 < α < 2. SBM may provide a seemingly adequate description inthe case of unbounded diffusion, for which its probability density function coincides with that offractional Brownian motion. Here we show that free SBM is weakly non-ergodic but does not exhibita significant amplitude scatter of the time averaged mean squared displacement. More severely, wedemonstrate that under confinement, the dynamics encoded by SBM is fundamentally different fromboth fractional Brownian motion and continuous time random walks. SBM is highly non-stationaryand cannot provide a physical description for particles in a thermalised stationary system. Ourfindings have direct impact on the modelling of single particle tracking experiments, in particular,under confinement inside cellular compartments or when optical tweezers tracking methods are used.

PACS numbers: 87.10.Mn,02.50.-r,05.40.-a,05.10.Gg

The passive, thermally driven diffusion of fluorescentlylabelled molecules or optically visible submicron particlesis routinely measured inside living cells and complex liq-uids by methods such as single particle tracking [1] andfluorescence correlation spectroscopy [2]. Often the ob-served diffusion patterns deviate from the laws of Brown-ian motion and display anomalous diffusion characterisedby the mean squared displacement (MSD)

〈x2(t)〉 ≃ Kαtα (1)

with the anomalous diffusion coefficient Kα of dimen-sion cm2/secα. Depending on the magnitude of theanomalous diffusion exponent one distinguishes subdif-fusion (0 < α < 1) and superdiffusion (1 < α < 2) [3–5].In biological contexts one often invokes the concept

of apparent subdiffusion, which is in fact a transientcrossover between free normal diffusion with α = 1and the thermal plateau of the MSD in confinement [6].However, there is clear experimental evidence of long-ranged subdiffusion (1) in living biological cells [4, 7–12]and in artificially crowded or structured liquids [13–17].The physical mechanisms for such anomalous diffusionare very varied, even considering the most common ap-proaches: we mention the continuous time random walkapproach [3, 5, 18], in which subdiffusion is effected bymultiple trapping or sticking events with an emergingpower-law distribution of waiting times, as seen in thedata of Refs. [9, 10, 12]. Alternatively, subdiffusion arisesin fractal geometries with their dead ends and bottlenecksacross spatial scales [19], an additional mechanism pin-pointed in Ref. [12]. We also mention diffusion processesin which the anomaly arises from the spatial dependenceof the diffusion coefficients [20–22], or from interaction ofthe tracer particle with a structured environment [23–26].

A special role play the Gaussian models for anoma-lous diffusion which belong to two families, that can bestbe distinguished on the Langevin equation level. TheMandelbrot-van Ness fractional Brownian motion (FBM)follows the stochastic equation x(t) = ζfGn(t), whereζfGn(t) represents fractional Gaussian noise characterisedby the covariance 〈ζfGn(t1)ζfGn(t2)〉 ≃ α(α−1)|t1−t2|

α−2

which has long-ranged negative or positive correlations,respectively, for subdiffusion (0 < α < 1) or superdif-fusion (1 < α < 2) [27]. FBM and the closely relatedfractional Langevin equation motion are physically re-lated to the motion in a viscoelastic environment [28]. Invarious experiments this type of motion was identified assingle or partial component [9–11, 15, 29].Here we focus on the second family of Gaussian anoma-

lous diffusion models, namely, scaled Brownian motion(SBM) [30]. SBM is governed by the Langevin equation

x(t) =√

2K (t)× ζ(t), (2)

where ζ(t) is white Gaussian noise with normalised co-variance 〈ζ(t1)ζ(t2)〉 = δ(t1−t2). The prefactor in Eq. (2)is the power-law time dependent diffusion coefficient

K (t) = αKαtα−1, (3)

which decreases or increases in time for 0 < α < 1 and1 < α < 2, respectively. SBM is used to model anoma-lous diffusion in a wide range of systems [31–36], in par-ticular, in FRAP experiments [37]. In terms of the timedependent diffusivity, the MSD (1) can be rewritten as〈x2(t)〉 ≃ K (t)× t. Originally, the diffusion process witha time dependent diffusivity was introduced by Batchelorin the description of relative diffusion in turbulence [38].For unconfined SBM the probability density function

2

is the Gaussian [30]

P (x, t) =

√

1

4πKαtα× exp

(

−x2

4Kαtα

)

. (4)

It has exactly the same form as the free PDF of FBM [30],if only both processes are started at the origin. Despitethis deceiving similarity, we show that SBM is funda-mentally different from FBM and all other anomalousdiffusion models mentioned above. Most importantly,we show that it cannot represent a physical model foranomalous diffusion in stationary or thermalised systems.Concurrently, the remarkable properties of SBM revealedhere may have other relevant applications, as briefly dis-cussed below.

TIME AVERAGED MEAN SQUARED

DISPLACEMENT FOR UNCONFINED SBM

While we are used to think of a diffusion process interms of the MSD (1) calculated as the spatial averageof x2 over the probability density function P (x, t), singleparticle tracking experiments typically provide few butlong individual time series x(t). These are evaluated interms of the time averaged MSD [5]

δ2(∆, t) =1

t−∆

∫ t−∆

0

(

x(t′ +∆)− x(t′))2

dt′, (5)

where t is the length of the time series (measurementtime) and ∆ the lag time. In an ergodic system for suf-ficiently long t ensemble and time averages provide iden-tical information, formally, 〈x2(∆)〉 = limt→∞ δ2(∆, t),and δ2 for different trajectories are identical. For anoma-lous diffusion the behaviour of the MSD (1) and the timeaveraged MSD (5) may be fundamentally different. Si-multaneously δ2(∆, t) for different trajectories may be-come intrinsically irreproducible. This phenomenon of ir-reproducibility and disparity 〈x2(∆)〉 6= limt→∞ δ2(∆, t)is usually called weak ergodicity breaking [5] and was dis-cussed in detail for the subdiffusive CTRW [5, 39–43]. Inthe following, for simplicity we neglect the explicit de-pendence of δ2 on the measurement time t.

For SBM the mean of the time averaged MSD over

multiple trajectories i,⟨

δ2(∆)⟩

= N−1∑N

i=1 δ2i(∆) can

be derived from the Langevin equation (2), yielding [47]

⟨

δ2(∆)⟩

=2Kαt

1+α

(α+ 1)

[

1−(

∆t

)1+α

−(

1− ∆t

)1+α]

t−∆. (6)

In the limit ∆ ≪ t, we recover the behaviour [48]

⟨

δ2(∆)⟩

∼ 2Kα

∆

t1−α, (7)

10-2

10-1

100

101

102

100 101 102 103 104

<x2 (t

)>, <

– δ2 (∆)>

time t, lag time ∆

α=0.5

tα

∆

MSDaveraged TA MSDindividual TA MSD

theory

100

101

102

103

104

105

106

100 101 102 103 104

<x2 (t

)>, <

– δ2 (∆)>

time t, lag time ∆

α=1.5

tα

∆

MSDaveraged TA MSDindividual TA MSD

theory

FIG. 1: MSD and time averaged MSD for SBM with α = 1/2(top) and α = 3/2 (bottom). The simulations of the SBM-Langevin equation (2) show excellent agreement with the an-alytical results, Eqs. (1) and (6). We also show results forthe time averaged MSD for 20 individual trajectories. Apartfrom the region where the lag time ∆ approaches the lengtht of the time series and statistics worsen, there is hardly anyamplitude scatter between individual δ2.

and when the lag time ∆ approaches the measurementtime t the limiting form

⟨

δ2(∆)⟩

∼ 2Kαtα−

αKα

t1−α(t−∆)+

α(α− 1)Kα

3t2−α(t−∆)2

(8)describes a cusp around ∆ = t. Result (6) is importantto deduce the full behaviour of δ2(∆) when ∆ approachest, as shown in Fig. 5 in the Appendix.In Fig. 1 we show results from simulations of the SBM

process for both sub- and superdiffusion, observing excel-lent agreement with the analytical findings. We also seethat the scatter between individual trajectories is verysmall. Such scatter is a characteristic for anomalous dif-fusion models and can be used, e.g., to reliably distin-guish FBM from subdiffusive CTRW processes [39, 45].For CTRW subdiffusion we find pronounced scatter be-tween δ2(∆) from individual trajectories even in the limitof extremely long trajectories t → ∞ [5, 39, 40], while forFBM the scatter vanishes for longer t, a characteristic of

3

the ergodic nature of FBM [28, 39, 45, 49]. The ampli-tude scatter of individual δ2 can be characterised in termsof the dimensionless variable ξ = δ2(∆)/

⟨

δ2(∆)⟩

. If ξ

has a narrow distribution around ξ = 1 and the widthdecreases with increasing t, the process is usually con-sidered ergodic. This width is characterised in terms ofthe ergodicity breaking parameter EB = 〈ξ2〉 − 〈ξ〉2. ForSBM it was found [48]

EB =

4Iα (∆/t)2α

, 0 < α < 1/2

13 (∆/t) ln(t/∆), α = 1/2

4α2

3(2α−1) (∆/t) , α > 1/2

(9)

with the integral Iα =∫ 1

0dy

∫

∞

0dx[(x+ 1)α − (x+ y)α]2

[48]. The EB parameter for SBM thus clearly decays tozero for increasing t. For α = 1 we obtain the knownform EB = 4

3∆/t for Brownian motion. The ∆/t scalingalso characterises FBM for α < 3/2 [49]. Our analysis forSBM shows that the scatter distribution φ(ξ) is indeednarrow and of Gaussian shape, albeit it is broader thanthe Gaussian form predicted for FBM in Ref. [45] (notshown). It decreases with the ratio ∆/t and thus indi-cates a reproducible behaviour between individual trajec-tories. The coexistence of the disparity δ2(∆) 6= 〈x2(∆)〉and asymptotically vanishing ergodicity breaking param-eter, limt→∞ EB = 0 is a new class of non-ergodic pro-cesses the more detailed mathematical nature of whichremains to be examined.From the analysis so far for free motion we can see

that the SBM process has a truly split personality. Thusits PDF is identical to that for free FBM. In contrastto the ergodic behaviour δ2(∆) = 〈x2(∆)〉 of FBM forsufficiently long t [39, 49], however, SBM exhibits weakergodicity breaking, as demonstrated in Eq. (7). Thisscaling form δ2 ≃ ∆/t1−α exactly matches the result forCTRW subdiffusion [5, 39, 40] or diffusion processes withspace-dependent diffusivity [20, 21]. Unlike the weaklynon-ergodic dynamics of the latter two, for SBM the fluc-

tuations around the mean⟨

δ2(∆)⟩

measured by the dis-

tribution φ(ξ) are narrow and decrease with longer t. Wenow show that the behaviour of confined SBM is also un-conventional.

CONFINED SBM

An important physical property of a stochastic processis its response to external forces or spatial confinement.From an experimental point of view, this is of relevanceto tracer particles moving in the confines of cellular com-partments or when the particle is traced by the help ofoptical tweezers, which exert an Hookean restoring forceon the particle [9]. We study the paradigmatic case of anharmonic potential V (x) ∝ 1

2kx2, for which the motion

is governed by the Fokker-Planck equation

∂

∂tP (x, t) =

∂

∂x

(

kx+ K (t)∂

∂x

)

P (x, t), (10)

which follows directly from the Langevin equation (2)with the additional Hookean force term −kx(t). Notethat in absence of the external force (k = 0) this equationis that of Batchelor [38]. For the MSD we find an exactexpression in terms of the Kummer function M [50],

〈x2(t)〉 = 2Kαtαe−2ktM(α, 1 + α, 2kt), (11)

from which we obtain the initial free subdiffusion (1) fort ≪ 1/k and the scaling form

〈x2(t)〉 ∼ αKαk−1tα−1 (12)

in the long time limit t ≫ 1/k. For subdiffusion (0 < α <1) the MSD thus has a power-law decay to zero, whilefor superdiffusion (1 < α < 2) it grows infinitely. Thiscounterintuitive behaviour of SBM is due to the fact thatthe time dependence of the diffusivity K (t) correspondsto a time dependent temperature [51] or a time dependentviscosity. Thus SBM is a most non-stationary processthat never reaches stationarity. Fig. 2 corroborates thisanalytical result with simulations based directly on theLangevin equation (2): after the free anomalous diffusionbehaviour of the MSD for a particle starting at the vertexof the potential, we observe a turnover to a power-lawbehaviour with negative or positive scaling exponent.The result for the time averaged MSD is similarly re-

markable. As shown in Fig. 2 for SBM simulations basedon the Langevin equation (2) with the Hookean forcing,it exhibits a pronounced apparent plateau for lag times∆ ≫ 1/k for both sub- and superdiffusive SBM. Thisbehaviour is in excellent agreement with the full analyt-ical solution (16) provided in the Appendix in terms ofKummer functions. Taking the limit ∆ ≪ t we obtain

⟨

δ2(∆)⟩

∼Kα

k

[

tα −∆α

t−∆+ (t−∆)α−1

(

1− 2e−k∆)

]

,

(13)which indeed features the extended plateau and providesa good approximation for ∆ ≫ 1/k. Note that when ∆approaches the measurement time t the time averaged

MSD⟨

δ2(∆)⟩

converges to the value of the MSD 〈x2(t)〉

due to the pole in expression (5). This behaviour is anal-ysed in more detail in Fig. 5 in the Appendix.Fig. 3 analyses this behaviour in the harmonic poten-

tial further by showing the time series x(t) for subdiffu-sive SBM, Brownian motion, and superdiffusive SBM.Indeed we see that for subdiffusive and superdiffusiveconfined SBM the fluctuations continue to decrease andincrease with time, while those in the Brownian limit be-come stationary. This is the direct effect of the time de-pendent temperature or viscosity encoded in the SBM

4

10-3

10-2

10-1

100

101

100 101 102 103 104

<x2 (t

)>, <

– δ2 (∆)>

time t, lag time ∆

α=0.5

tα

∆

tα-1

MSD (k=0.01)MSD (k=0.1)

TA MSD (k=0.01)TA MSD (k=0.1)

theory

100

101

102

103

104

105

100 101 102 103 104

<x2 (t

)>, <

– δ2 (∆)>

time t, lag time ∆

α=1.5

tα

∆

tα-1

MSD (k=0.01)MSD (k=0.1)

TA MSD (k=0.01)TA MSD (k=0.1)

theory

FIG. 2: MSD 〈x2(t)〉 and time averaged MSD⟨

δ2(∆)⟩

for

α = 1/2 (top) and α = 3/2 (bottom) in an harmonic poten-tial. In each case we consider the force constants k = 0.01and k = 0.1. Convergence of the corresponding ensembleand time averages at t = 5 × 104 can be shown numerically.The shown analytical curve is based on the full solution for⟨

δ2(∆)⟩

provided in the Appendix.

0 25000 50000

-4

-2

0

2

4

=3/2=1=1/2

x(t)

time t

k=0.01 k=0.1 k=1

0 25000 50000-40

-20

0

20

40

time t0 25000 50000

-500

0

500

time t

FIG. 3: Trajectories of SBM in an harmonic potential, forα = 1/2 (left), α = 1 (centre), and α = 3/2 (right) for threedifferent confinement strengths k. The fluctuations are sta-tionary only in the Brownian case α = 1.

diffusivity K (t). If the system is stationary or ther-malised this behaviour clearly underlines the unsuitabil-ity of SBM for the description of anomalous diffusion.

The behaviour of SBM dynamics under confinement isthe central result of our study. The continued temporaldecay or increase of the MSD that we obtained for SBMis in stark contrast to the behaviour of confined CTRWsubdiffusion, for which the MSD 〈x2〉 saturates to thethermal plateau, while the time averaged MSD continuesto grow in the power-law form δ2 ≃ (∆/t)1−α up until thelag time ∆ approaches t [39, 42]. It strongly differs fromFBM, which relaxes to a plateau for both the MSD andthe time averaged MSD, and for which only a transientdisparity between 〈x2〉 and δ2 exists [15]. Finally, SBMis also at variance with heterogeneous diffusion processeswith a space dependent diffusion coefficient relax to aplateau for both 〈x2〉 and δ2 [52].

DISCUSSION

As we showed SBM is a truly paradoxical stochasticprocess. Somewhat similar to a chameleon, each timewe compare SBM with other established anomalous dif-fusion processes we find certain similarities. Looking atthe sum of its features, however, SBM is a truly indepen-dent process with a range of remarkable properties.

For free SBM the probability density function P (x, t)equals that of FBM, despite the fact that both processesare governed by different stochastic (Langevin) equa-tions. SBM’s time averaged MSD scales equally to thoseof the weakly non-ergodic CTRW subdiffusion and diffu-sion processes with space-dependent diffusivity. Despitethis weakly non-ergodic character of the mean time av-

eraged MSD⟨

δ2⟩

, the amplitude scatter between the

time averaged MSD δ2 of individual realisations is smalland the distribution has a Gaussian shape, as otherwiseobserved for the ergodic FBM or for Brownian motionof finite time t. In that sense SBM represents a newclass of non-ergodic processes. The most striking be-haviour of SBM is, however, its strongly non-stationarybehaviour under confinement. Instead of relaxing to aplateau the MSD acquires a power-law decay or growthmirroring a continuously decreasing or increasing tem-perature encoded in SBM’s time dependent diffusion co-efficient K (t).

SBM is thus at variance with the currently availableexperimental observations in complex liquids using sin-gle particle tracking by video microscopy or by opticaltweezers tracking of single submicron particles. Thus thefree anomalous diffusion data garnered so far was clas-sified into FBM-like and CTRW-like behaviour, or com-binations thereof [8, 10, 12]. Note that also for fluores-cence correlation spectroscopy experiments recent anal-ysis tools corroborated an FBM nature of the data [13].

5

For optical tweezers tracking of lipid granules in differentcomplex liquids the time averaged MSD either contin-ues to grow under confinement, reflecting the non-ergodicfeatures of CTRWs [5, 9], or it relaxes to a plateau valuemirroring an ergodic dynamics [15, 53].

How can FBM and SBM have the same distribution (4)in free space? Simply put for FBM, the viscoelastic prop-erties of the environment effecting the long-range corre-lations of FBM lead to a frequency dependent responseof the environment to a disturbance, while the materialsproperties remain unchanged in time. For free FBM thisgives rise to the subdiffusive MSD (1), in which the timedependent diffusivity effectively encodes the frequencydependent response of the viscoelastic environment. Thedistribution (4) for FBM and its description in terms ofa Fokker-Planck equation of the form (10) is treacher-ous, however. This can already be seen when we usethe PDF (4) or the dynamics equation (10) to calcu-late the first passage behaviour. This procedure leads tothe wrong result [54], and the full analytical descriptionof FBM in the presence of boundaries remains elusive,a difficulty imposed by the highly correlated fractionalGaussian noise driving its Langevin equation. In con-trast, SBM, according to its Langevin description (2), isdriven by uncorrelated noise but the environment itselfis changing as function of time, effecting an extremelynon-stationary process. The equivalence of the PDF (4)of both processes is thus simply due to the fact that aGaussian PDF is completely defined by its second mo-ment, the MSD (1).

With its interesting behaviour that is so different fromthe other conventional anomalous diffusion models, SBMmay indeed have relevant applications to weakly coupledor fully adiabatic systems, as well as for active systems,in which the existence of a temperature is not mean-ingful. In particular, in the superdiffusive case SBM oranalogous dynamics with other increasing effective diffu-sivities may represent an alternative approach to activeBrownian motion [55].

The difference of SBM to other processes can also beseen in Fig. 4. For the subdiffusive case a sample trajec-tory of SBM is compared to that of FBM and the noisyCTRW [46], in which the pure subdiffusive CTRW is su-perimposed with Ornstein-Uhlenbeck noise to accommo-date the thermal noise of the environment observed in ex-periments [25]. We see that SBM with its Gaussian prob-ability density function and uncorrelated driving noiseappears more similar to the noisy CTRW motion, albeitit has a more pronounced tendency to reach larger am-plitudes than the CTRW with its waiting time periods.The fluctuations of SBM are dramatically less than thoseof the anticorrelated FBM, which also shows the largestamplitudes in x(t). For superdiffusion, we compare SBMwith a noisy version of the Levy walk process [56] andagain with FBM. Note the vastly different size of thewindow shown on the ordinate. This time, the persis-

-30

-20

-10

0

10

20

30

0 2000 4000 6000 8000 10000

x(t)

time t

α=0.5

FBM

SBMnCTRW

-6000

-4000

-2000

0

2000

4000

6000

0 2000 4000 6000 8000 10000

x(t)

time t

α=1.5

FBM

SBM

nCTRW

FIG. 4: Individual trajectories of anomalous stochastic pro-cesses: FBM, SBM, and noisy CTRW (nCTRW), for subdif-fusion with α = 1/2 (top) and superdiffusion with α = 3/2(right).

tent FBM shows pronouncedly larger excursions and adistinct reduction of the noise compared to SBM. Theshape of x(t) of the latter does not appear to be quali-tatively different from the subdiffusive case. The noisyCTRW is fundamentally different from both SBM andFBM. Despite their disparate physical nature and theirdissimilarity when setting one against the other in a di-rect comparison, we note that generally it is difficult toidentify a stochastic process solely from the appearanceof the recorded trajectory.

From this discussion it is obvious that one thing is tohave at our disposal a handy and easy-to-use descriptionfor anomalous diffusion processes. SBM with its Gaus-sian and uncorrelated nature appears deceivingly simpleand is therefore easy to implement in numerical analy-ses and descriptions such as diffusion limited reactions.However, the other face of the medal is about the phys-ical relevance of a model process. Using SBM with itstime-dependent diffusion coefficient violates the physi-cal setting in typical experiments, in which the systemis held at approximately constant temperature and itspredictions are at odds with actual observations. The

6

unphysical nature for the kind of processes we have inmind is most obvious for confined motion. In contrast,FBM and fractional Langevin equation motion are er-godic processes with the physical background of an ef-fective particle motion in a viscoelastic multi-body envi-ronment. FBM and fractional Langevin equation motionexhibit a transient disparity between the MSD and thetime averaged MSD under confinement [15]. Weakly non-ergodic CTRW processes emerge due to immobilisationperiods imprinted on the dynamics by the structure ofthe environment of binding events to the environment.Finally, diffusion processes with space dependent diffu-sivities arise naturally in non-homogeneous systems suchas biological cells or subsurface aquifers. To extract phys-ically meaningful information from anomalous diffusiondata one needs to have some physical insight into the ob-served process and properly analyse the data using com-plementary tools [5, 10–12, 15, 46, 57–59] before settlingfor the appropriate physical model.We finally note that it will be interesting to compare

the predictions of the SBM model with that of activeprocesses in viscoelastic environments [60, 61].We acknowledge funding from the Academy of Finland

within the FiDiPro scheme.

Appendix

The covariance of the position for SBM in an harmonicpotential follows from the Langevin equation (2). Ourresult is

〈x(t1)x(t2)〉 = 2Kαtα

1 e−k(t1+t2)M(α, 1 + α, 2kt1) (14)

for t1 < t2. In absence of the confinement (k = 0) thecovariance reduces to the MSD (1) and in the Brownianlimit α = 1 we recover the familiar covariance

〈x(t1)x(t2)〉 =K1

k

(

e−k(t2−t1) − e−k(t1+t2))

. (15)

From Eq. (14) we derive the exact result for the timeaveraged MSD (5),

⟨

δ2(∆)⟩

=2Kα

1 + α(t−∆)αe−2k(t−∆)

×M(1 + α, 2 + α, 2k[t−∆])

+2Kα

(1 + α)(t−∆)

[

t1+αe−2ktM(1 + α, 2 + α, 2kt)

−∆1+αe−2k∆M(1 + α, 2 + α, 2k∆)]

−4Kα

1 + α(t−∆)αe−2kt+k∆

×M(1 + α, 2 + α, 2k[t−∆]). (16)

To derive the limit (13) of the time averaged MSD forconfined SBM in the long time limit t → ∞ we use the

2.5

2.6

2.7

2.8

4.8 4.9 5log 1

0 <

x2 (∆)>

, lo

g 10

<δ- 2

(∆)>

log10 ∆

α=1/2

4.8 4.9 5 7.5

7.6

7.7

7.8

log10 ∆

α=3/2

-1.6

-1.2

-0.8

-0.4

0

0.4

4.999 4.9995 5

log 1

0 <

x2 (∆)>

, log

10 <

δ- 2(∆

)>

log10 ∆

α=1/2

4.7 4.8 4.9 5 3.5

3.6

3.7

3.8

log10 ∆

α=3/2

FIG. 5: Convergence of the time averaged MSD⟨

δ2(∆)⟩

to

the MSD 〈x2(∆)〉 at ∆ → t for α = 1/2 and α = 3/2. Themeasurement time t = 105. Top: Free SBM. For the su-perdiffusive case the dashed black line of unity slope aids in

demonstrating the non-linear behaviour of⟨

δ2(∆)⟩

. Bottom:

SBM in an harmonic potential with k = 0.1. Note the differ-ent ranges of the abscissa in the two cases. For the extremelysmall window for the subdiffusive case needed to illustrate thecusp at ∆ → t and the relatively large variation of

⟨

δ2(∆)⟩

the MSD 〈x2(∆)〉 appears almost constant.

property

M(α, 1 + α, x) ∼Γ(1 + α)

Γ(α)×

ex

x(17)

of the Kummer function [50].In Fig. 5 we illustrate the convergence of the time av-

eraged MSD⟨

δ2(∆)⟩

to the value of the MSD 〈x2(t)〉 in

the limit ∆ → t.

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7

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