PHYSICAL REVIEW E 89, 012115 (2014)

Scaled Brownian motion as a mean-field model for continuous-time random walks

Felix Thiel* and Igor M. Sokolov†

Institut fur Physik, Humboldt-Universitat zu Berlin, Newtonstrasse 15, D-12489 Berlin, Germany(Received 9 October 2013; published 13 January 2014)

We consider scaled Brownian motion (sBm), a random process described by a diffusion equation with explicitlytime-dependent diffusion coefficient D(t) = αD0t

α−1 (Batchelor’s equation) which, for α < 1, is often used forfitting experimental data for subdiffusion of unclear genesis. We show that this process is a close relative ofsubdiffusive continuous-time random walks and describes the motion of the rescaled mean position of a cloud ofindependent walkers. It shares with subdiffusive continuous-time random walks its nonstationary and nonergodicproperties. The nonergodicity of sBm does not however go hand in hand with strong difference between itsdifferent realizations: its heterogeneity (“ergodicity breaking”) parameter tends to zero for long trajectories.

DOI: 10.1103/PhysRevE.89.012115 PACS number(s): 05.40.Fb, 05.10.Gg

Anomalous diffusion is a generic name for a class oftransport processes which are close to diffusion in their origin(i.e., can be represented via generalized random-walk schemesor Langevin equations) but do not lead to the mean-squareddisplacement growing as the first power of time

〈r2(t)〉 = 2dDt (1)

(with D being the diffusion coefficient and d the dimension ofspace), as predicted by Fick’s laws. Within the random-walkschemes such deviations from the normal diffusion picturecan arise either due to broad distributions of the waiting timesbetween the steps [continuous-time random-walk (CTRW)models], or due to slow decay of correlations between steps,or both, see [1] for a review, leading to the change of the powerlaw in the time dependence of the mean-squared displacement,

〈r2(t)〉 ∝ tα.

The processes with α < 1 are called subdiffusion; the oneswith α > 1 are termed superdiffusion. In the first case theformal diffusion coefficient D in Eq. (1) vanishes in the long-time limit; in the second case it diverges.

The single trajectory dynamics in normal diffusion isdescribed by the Langevin equation

r =√

2Dξ (t)

with white, δ-correlated Gaussian noise ξ (t), 〈ξ (t)〉 = 0,〈ξ (t)ξ (t ′)〉 = δ(t − t ′); the time dependence of the probabilitydensity function (PDF) of the process or the one of its transitionprobabilities is given by Fick’s second law (diffusion equation)

∂

∂tp(r,t) = D

∂2

∂r2p(r,t) (2)

(both equations given here in one dimension). The descriptionof anomalous diffusion of different origins often follows bymodification of one of the equations above.

In experiments, many processes of anomalous diffusion ofunknown origin, i.e., when the observable of interest cannot befitted to the solutions of Eq. (2), are fitted to the results obtainedfor the so-called scaled Brownian motion (sBm) [2], a diffusion

*thiel@physik.hu-berlin.de†igor.sokolov@physik.hu-berlin.de

process with explicitly time-dependent diffusion coefficientD(t) = αD0t

α−1. Numerical simulations of Ref. [3] show that,at least for the case of FRAP (fluorescence recovery after photobleaching), the fits may be astonishingly good, independent onthe true nature of the simulated process (percolation, CTRW,etc.). This means that the form of FRAP recovery curves, ifthey hint onto anomalous diffusion, hardly depends on theorigin of the corresponding anomaly. Using the sBm modelfor calculating other properties may however be dangerous, aslong as the nature of the sBm model itself and the one of theprocess under investigation are not well understood [1].

The PDF P (x,t) and the transition probabilities of the sBmprocess X(t) are given by the Batchelor’s equation [4],

∂

∂tP (x,t) = αD0t

α−1 ∂2

∂x2P (x,t), (3)

leading to the mean-squared displacement 〈X2(t)〉 = 2D0tα

(the original one was for α = 3). Initially, Eq. (3) wasproposed for description of superdiffusive turbulent disper-sion, as an alternative to Richardson’s diffusion equationwith the distance-dependent diffusion coefficient. Some ofits shortcomings for description of the turbulent dispersionwere clear to Batchelor himself; see Ref. [5] for more detaileddiscussion. The equations of Batchelor’s type are typicallypostulated and do not follow from any explicit physical model,one of the seldom exclusions being [6].

The Langevin description of the corresponding process isgiven by

X =√

2D0αtα−1ξ (t), (4)

with white, δ-correlated Gaussian noise ξ (t), 〈ξ (t)〉 = 0, and〈ξ (t)ξ (t ′)〉 = δ(t − t ′). Both the Langevin and the diffusionequation for sBm can be reduced to the ones for the normalone by the time rescaling t → u = tα . As a consequence,and in contrast to CTRW, the sBm X(t) is Gaussian. Itcan be represented via a Brownian motion B(t) as X(t) =√

2D0B(tα), as we will show below. CTRW, on the other handcan be approximated on long length scales via B[N (t)], whereN (t) is the stochastic process, that counts the steps. Since〈N (t)〉 ∝ tα , it is valid to interpret sBm as a CTRW with apreaveraged counting process. Thus sBm is a random processwhich is subordinated to a Brownian motion (Wiener process)under the deterministic time change given above and as such is

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FELIX THIEL AND IGOR M. SOKOLOV PHYSICAL REVIEW E 89, 012115 (2014)

a relative of a Gaussian CTRW with the subtle (but important)difference that in CTRW the time change is stochastic.

The qualitative discussion of the relation between thesBm and the CTRW was given in [1] without proofs andcalculations. Here we close this gap and provide deeperanalysis of sBm. Thus we show that sBm can be consideredas a homogenized (mean-field) approximation to CTRWand shares its property of aging and ergodicity breaking,but this ergodicity breaking does not go hand in handwith inhomogeneity (nonconvergence in distribution of themean-squared displacement as obtained by the moving timeaverage). This stresses that the absence of ergodicity (dueto nonstationarity) does not imply the nonzero value ofthe “ergodicity breaking parameter,” which characterizes theheterogeneity of realizations. In sBm, although a close relativeof CTRW, this heterogeneity is removed by the preaveragingprocedure. In what follows we concentrate on the subdiffusivecase α < 1, as considered in [3].

Let us first discuss a general situation and consider arandom process Z(t) in continuous time t . Let us assumethat the process possesses all necessary single point and crossmoments.

The process will be taken to possess zero mean and varianceσ 2(t). We now create m i.i.d. copies Zj , j ∈ {1,2, . . . ,m}, andassociate each of them with the coordinate of some walker of acloud of m random walkers. Let us now consider the behaviorof the center of mass of this cloud [i.e., the mean coordinateof m independent random processes Zj (t), j = 1,2, . . . ,m]Zc.m.(t) = (1/m)

∑mj=1 Zj (t). Note that 〈Zj (t)〉 = 0 due

to the symmetry of the process, 〈Zj (t)Zl(t)〉 = 0 (forj = l) due to independence of different realizations, and〈Zj (t)Zj (t)〉 = σ 2(t), so that 〈Z2

c.m.(t)〉 = σ 2(t)/m. In orderto prove the corresponding limit theorems, we need to definethe mean position in such a way that its second moment doesnot depend on m. We define the “rescaled mean position” as

Z(m)(t) = 1√m

m∑j=1

Zj (t).

The random process Z(m)(t) giving the rescaled mean positionretains the correlation function of the initial Zj (t) process.Note that for symmetric processes with correlation func-tion 〈Zj (t)Zl(t ′)〉 = C(t,t ′)δlj , Z(m)(t) has the same corre-lation function: 〈Z(m)(t)Z(m)(t ′)〉 = 1

m

∑mj,l=1〈Zj (t)Zl(t ′)〉 =

1m

∑mj,l=1 C(t,t ′)δjl = 1

m

∑mj=1 C(t,t ′) = C(t,t ′). The mean-

squared change in Z(m), given by C(t,t), does not dependon m. The limit process, X(t) = limm→∞ Z(m)(t), willcorrespond to the “mean-field coordinate” (MF position) ofthe walker. For finite m, the position of the cloud’s center ofmass is obtained by inverse rescaling. The limit X(t) will turnout to be sBm, if the process Z(t) is chosen to be a CTRW.

We now show that the position X(t) and all possible vectors[X(t1),X(t2), . . . ,X(tn)] comprising the walker’s positions atdifferent times converge to Gaussian, so that the processZ(m)(t) converges to a Gaussian process for m → ∞.

Let us consider a sequence of times t1,t2, . . . ,tn and acorresponding vector Zj = [Zj (t1),Zj (t2), . . . ,Zj (tn)]. Thisvector (stemming from probing the position of the j th walkerat time instants t1,t2, . . . ,tn) is unbiased, i.e., 〈Zj 〉 = 0 and

possesses a covariance matrix between its components withfinite elements Cjk = C(tj ,tk). According to the central limittheorem for multivariate (vector) distributions the correspond-ing mean Z(m) = 1√

m

∑mi=1 Zj converges in distribution to a

multivariate Gaussian with zero mean and the same covariationmatrix; see, e.g., Refs. [7] and [8, Sec. 3.5]. Since this happensfor any sequence of observation times, the limit process, i.e.,the one describing the rescaled mean position of a cloud ofwalkers, is a Gaussian process with the correlation functioninherited from the single realization Zj (t). This process canbe considered as a kind of “mean-field approximation” for ourinitial process Zj (t).

We have seen that pooling (superimposing many statisticalcopies of the initial process) leads us to a Gaussian processwith the correlation function inherited from a single copy.Now we turn to Z(t) being a CTRW with the power-lawwaiting time density ψ(t) � τα

0 t−1−α and the mean-squareddisplacement σ 2(t) � 2D0t

α , with D0 being the combinationof mean-squared displacement per step a2 and typical waitingtime τ0 [9]. Let us subdivide the time axis in short intervalsof duration dt , and resample each of the CTRW processesas a simple random walk with the step duration dt and withthree possible step lengths si = 0 or si = ±1 (i = [t/dt] is thelargest integer smaller than t/dt ; the double steps during thedt intervals can be neglected provided dt is small enough).The steps of this process are not independent (if the leadingprocess is not a Poissonian one, i.e., if the waiting timedistributions are not exponential), but always uncorrelated,because of the symmetry; hence 〈sisi ′ 〉 = δi,i ′ . The gener-alization to correlated steps is neither straightforward noreasy to treat and will not be considered in this paper. Thedisplacements Z(t1,t2) and Z(t ′1,t

′2) of a walker during two

nonintersecting time intervals (t1,t2) and (t ′1,t′2) are given by

Z(t1,t2) = ∑[t2/dt]−1i=[t1/dt] si , and the analogous expression for

Z(t ′1,t′2). Since the intervals are nonoverlapping, all indices

i of the sum Z(t1,t2) are different from all the indices i ′corresponding to Z(t ′1,t

′2). We conclude that the increments

are also uncorrelated, i.e., 〈Z(t1,t2)Z(t ′1,t′2)〉 = 0, for two

nonintersecting intervals (t1,t2) and (t ′1,t′2). This observation

allows us to get the position-position correlation functionC(t,t ′) = 〈Z(t)Z(t ′)〉 for the CTRW process Z(t). Takingt ′ > t one can put C(t,t ′) = 〈Z(t)[Z(t) + Z(t,t ′)]〉 =〈Z2(t)〉 + 〈Z(0,t)Z(t,t ′)〉. The mean 〈Z(0,t)Z(t,t ′)〉vanishes as discussed above. Therefore, the correlation func-tion C(t,t ′) for the CTRW Z(t) is given by

C(t,t ′) = 〈Z2[min(t,t ′)]〉 = 2D0[min(t,t ′)]α. (5)

We now consider the superposition of m independentCTRWs, that is the same limit as discussed above. Whenthe number m of independent pooled (superimposed) randomprocesses Zj tends to infinity, so does also the number of eventswithin each interval of a fixed length, and the displacement X

of a pooled process during this interval tends to a Gaussian,since it is the normalized sum of independent random variableswith finite variance. The displacements at different intervalsare noncorrelated Gaussian random variables and are thereforeindependent [7]: the dependence present in the values ofZj (t) gets “dissolved” when the number of processes tendsto infinity. As pointed out above, X(t) shares its correlation

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SCALED BROWNIAN MOTION AS A MEAN-FIELD . . . PHYSICAL REVIEW E 89, 012115 (2014)

function with Zj (t), Eq. (5). As also already mentioned, X(t)can be represented in terms of a Brownian motion B(t) asX(t) = √

2D0B(tα), due to its Gaussianity and the specialform of its correlation function. The difference between X(t)and the CTRWs Zj (t) it is composed of is Gaussianity. Noneof the Zj are Gaussian processes.

We proceed with investigation of the ergodic propertiesof X(t). The mean-squared displacement of X(t) during theinterval (t,t + τ ) is given by

〈X2(t,t + τ )〉 = 2D0(t + τ )α − 2D0tα � 2αD0t

α−1τ. (6)

We then can write X(t + t) = X(t) + X(t) and interpret itas a finite-difference approximation to a Langevin equation

X = η(t),

i.e., as the result of integrating this equation over the finitetime interval (t,t + t). Since the distribution of X(t) isGaussian, the values of X at different time intervals areuncorrelated, and 〈X2〉 = 2D0αtα−1t , the properties ofthe noise follow: this noise is Gaussian with 〈η(t)〉 = 0 and〈η(t)η(t ′)〉 = 2D0αtα−1δ(t − t ′), i.e., exactly as in Eq. (4).This diffusion process with explicitly time-dependent diffu-sion coefficient is the sBm, and the Batchelor’s equation,Eq. (3), appears as a Fokker-Planck equation for the Langevinequation above. The mean-field process shares with theoriginal CTRW Zj (t) the same time dependence of the mean-squared displacement: 〈X2(t)〉 ∝ tα in the ensemble aver-age. Let X2(τ )T = 1

T −τ

∫ T −τ

0 dt X2(t,t + τ ) be the movingtime-averaged mean-squared displacement. Then the doubleaverage, ensemble, and moving time average goes as

〈X2(τ )T 〉 = 1

T − τ

∫ T −τ

0dt 〈X2(t,t + τ )〉

= 2D0

T − τ

∫ T −τ

0dt [(t + τ )α − tα] ≈ 2D0T

α−1τ

for τ T , just like in CTRW [10], showing ergodicitybreaking (a trivial one, due to the nonstationarity of theprocess). At difference with CTRW, the “ergodicity breakingparameter” for the mean-field process vanishes for T → ∞,showing that its different realizations are extremely similar.

In CTRW, X2(τ )T is a random variable: its distribution doesnot narrow for T → ∞ in the sense that the heterogeneity(“ergodicity breaking”) parameter J [11]

J (τ,T ) = 〈X2(τ )2

T 〉 − 〈X2(τ )T 〉2

〈X2(τ )T 〉2(7)

tends to a finite limit J = 2�2(1 + α)/�(1 + 2α) − 1 forT → ∞. Expressing J via correlation functions we see thatvanishing of this parameter for a stationary process leadsto ergodicity (i.e., equality of the ensemble mean and themoving time average) as a consequence of Birkhoff-Khinchintheorem, stressing that ergodicity implies stationarity andsome amount of “homogeneity.” Scaled Brownian motion(Batchelor’s process) is nonstationary (since it is explicitlytime inhomogeneous), and nonergodic, but shows a highamount of homogeneity among its trajectories, as we proceedto show by explicit calculation of J and showing that itvanishes for T → ∞.

The only thing not yet calculated is 〈X2(τ )2

T 〉:

〈X2(τ )2〉 = 1

(T − τ )2

⟨ ∫ T −τ

0dt1 X2(t1,t1 + τ )

×∫ T −τ

0dt2 X2(t2,t2 + τ )

⟩

= 2

(T − τ )2

⟨ ∫ T −τ

0dτd

∫ T −τ−τd

0dtm

×X2(tm,tm + τ )X2(tm + τd,tm + τd + τ )

⟩,

(8)

where we changed the variables of integration to tm =min(t1,t2), τd = |t2 − t1| and used the symmetry of the ex-pression. The order of integration and ensemble averaging canbe interchanged and the mathematical expectation is readilyevaluated by using the Gaussian property of X(t). Let us denoteby C(t1,t2) = 〈X(t1,t1 + τ )X(t2,t2 + τ )〉 the correlationfunction of the increments. Expressed in variables of Eq. (8),C reads C(tm,tm + τd ) = 2D0{(tm + τ )α − [min(tm + τ,

tm + τd )]α} = 2D0[(tm + τ )α − (tm + τd )α]I[0,τ )(τd ), whereIA(x) is an indicator function that equals unity if x ∈ A

and vanishes otherwise. Note that the increment process isalso Gaussian, so that its higher correlators decompose intoproducts of C(t1,t2). Thus the integrand in the last expressionof Eq. (8) reads

〈X2(tm,tm + τ )X2(tm + τd,tm + τd + τ )〉= 2C2(tm,tm + τd ) + C(tm,tm)C(tm + τd,tm + τd )

= 8D20[(tm + τ )α − (tm + τd )α]2I[0,τ )(τd )

+ C(tm,tm)C(tm + τd,tm + τd ).

The last summand can be identified with the 〈X2(τ )T 〉2

term in the numerator of Eq. (7) and therefore cancels out.The remaining summand vanishes, if τd > τ . This is themain difference to usual CTRW, where the existence of suchhigher-order correlations at longer times is responsible fornonvanishing J parameter.

Now let us assume T to be sufficiently large, i.e.,T − τ,� τ , and proceed:

〈X2(τ )2

T 〉 − 〈X2(τ )T 〉2 = 16D20

(T − τ )2

∫ τ

0dτd

∫ T −τ−τd

0dtm

× [(tm + τ )α − (tm + τd )α]2. (9)

For large tm the integrand is asymptotically equal toα2t2α−2

m (τ − τd )2. For α > 1/2 this asymptotic form can beimmediately plugged into Eq. (9) and the integral readilyevaluated. Its limit for T � τ is

〈X2(τ )2

T 〉 − 〈X2(τ )T 〉2 ≈ 16α2D20τ

3T 2α−3

3(2α − 1).

Inserting the corresponding expression into Eq. (7) shows thatJ approaches zero as T −1.

If α = 1/2, the double integral Eq. (9) can be calculated ex-plicitly, revealing logarithmic corrections, i.e., J ∼ T −1 ln T .

For α < 1/2 the substitution of the integrand by itsasymptotic expression for large tm leads to the divergence

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FELIX THIEL AND IGOR M. SOKOLOV PHYSICAL REVIEW E 89, 012115 (2014)

of the inner integral in Eq. (9) at the lower limit ofintegration. Let us split the integral into two parts atsome intermediate time ti > 0 and use the asymptoticsubstitution discussed above only in the second part:∫ T −τ−τd

0 dtm [(tm + τ )α − (tm + τd )α]2 = ∫ ti0 dtm[(tm + τ )α −

(tm + τd )α]2 + α2∫ T −τ−τd

tidtm (τ − τd )2t2α−2

m . The firstsummand is bounded from above by ti[τα − τα

d ]2, and thesecond one converges on the upper limit, which thus can betaken to go to infinity. Therefore, the inner integral [and thusthe whole double integral in Eq. (9)] tends for large T to afunction independent on T . We thus conclude that

J (τ,T ) �

⎧⎪⎪⎨⎪⎪⎩

4Kα

(τT

)2α, α < 1

2 ,

13

τT

ln(

Tτ

), α = 1

2 ,

4α2

3(2α−1)τT

, α > 12 ,

(10)

with Kα = ∫ 10 dy

∫ ∞0 dx[(x + 1)α − (x + y)α]2.

We see that, for any positive value of α, J will vanish,and shows a crossover between two types of T dependence atα = 1/2. Figure 1 shows simulated values of J for differentvalues of α and T on a double logarithmic plot together withthe predictions of Eq. (10).

In conclusion, scaled Brownian motion (Batchelor’s pro-cess), described by a diffusion equation with explicitly time-dependent diffusion coefficient and often used in fitting datafor experimental situations showing anomalous diffusion ofunclear origin, is, in the subdiffusive case α < 1, a closerelative of CTRW and describes the motion of the rescaledmean position of a cloud of independent continuous-time

10-5

10-4

10-3

10-2

10-1

100

102 103 104 105

Het

ero

gen

eity

Par

amet

erJ

Observation Time T

FIG. 1. (Color online) Scaling behavior of J . X(t) = B(tα) wassimulated for different values of α and T . α values range from 0.2(+), 0.4 (×), 0.6 (�), 0.8 (�), and 1.0 (�). Each point is calculatedfrom 5000 trajectories. The given power laws are (from top to bottomline) T −0.4, T −0.8, and T −1.0.

random walkers. The model shows the same kind of ergodicitybreaking due to nonstationarity as the corresponding CTRWprocess. This one, however, does not go hand in handwith strong heterogeneity of its different realizations: thecorresponding “ergodicity breaking” parameter vanishes in thelimit of long trajectories.

The authors acknowledge financial support by DFG withinIRTG 1740 research and training group project.

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