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Aging Scaled Brownian Motion Hadiseh Safdari,1,2 Aleksei V. Chechkin,3,2,4 Gholamreza R. Jafari,1 and Ralf Metzler2,5,∗ 1Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839, Iran 2Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam-Golm, Germany 3Institute for Theoretical Physics, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine 4Max-Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany 5Department of Physics, Tampere University of Technology, FI-33101 Tampere, Finland 5 (Dated: January 21, 2015) 1 0 Scaled Brownian motion (SBM) is widely used to model anomalous diffusion of passive tracers 2 in complex and biological systems. It is a highly non-stationary process governed by the Langevin equation for Brownian motion, however, with a power-law time dependence of the noise strength. n Here we study the aging properties of SBM for both unconfined and confined motion. Specifically, a we derive theensemble and time averaged mean squared displacements and analyze their behavior J intheregimesofweak,intermediate,andstrongaging. Averyrichbehaviorisrevealedforconfined 0 agingSBMdependingondifferentagingtimesandwhethertheprocessissub-orsuperdiffusive. We 2 demonstrate that the information on theaging factorizes with respect to the lag time and exhibits afunctionalform, thatisidenticaltotheaging behaviorofscale freecontinuoustimerandom walk ] h processes. WhileSBM exhibits adisparity between ensemble and time averaged observables and is c thus weakly non-ergodic, strong aging is shown to effect a convergence of the ensemble and time e averaged mean squared displacement. Finally, we derive the density of first passage times in the m semi-infinitedomain that features a crossover defined bytheaging time. - t PACSnumbers: 05.40.-a a t s . I. INTRODUCTION fusion studies is reported in artificially crowdedenviron- t a ments mimicking aspects of the superdense state of the m cellularfluid[17,18]. Withinandalonglipidmembranes Deviations from normal Brownian motion were re- - anomalousdiffusionwasfoundfromexperimentandsim- ported already in the work of Richardsonon the spread- d ulations [16]. n ing of tracers in turbulent flows [1], and deviations from o theBrownianlawarediscussedbyFreundlichandKru¨ger c [2]. Today anomalous diffusion is typically defined in Brownian motion is intimately connected with the [ terms of the power-law form Gaussianprobabilitydensityfunctiondescribingthespa- tial spreading of a test particle as function of time. This 1 v hx2(t)i∼2K∗tα (1) Gaussianis effected a forteriori by the centrallimit the- α 0 orem,asBrownianmotioniswelldescribedonastochas- 1 of the mean squared displacement (MSD) [3, 4]. De- tic level by the Wiener process. Anomalous diffusion 8 pendingonthevalueoftheanomalousdiffusionexponent loses this universal character, and instead different sce- 4 we distinguish the regimes of subdiffusion (0 < α < 1) narios corresponding to the physical setting need to be 0 andsuperdiffusion(α>1),including the specialcasesof considered. Among the most popular models we men- . 1 Brownianmotion(α=1)andballistictransport(α=2). tion the Scher-Montroll continuous time random walk 0 ThegeneralizeddiffusioncoefficientK∗ inEq.(1)hasthe (CTRW), in which individual jumps are separated by 5 α physical dimension cm2/secα. independent, random waiting times [5, 19]. If the dis- 1 : Anomalousdiffusionisobservedinawiderangeofsys- tributionofthesewaitingtimes isscalefree,subdiffusion v tems, including fields as diverse as charge carriermotion emerges[20]. FractionalBrownianmotionandtheclosely Xi in amorphous and polymeric semiconductors [5, 6], dis- relatedfractionalLangevinequationmotionarestochas- persionof chemicals in groundwateraquifers [7], particle ticprocessesfueledbyGaussianyetpower-lawcorrelated r a dispersionincolloidalglasses[8],orthemotionoftracers noise [21, 22]. Anomalous diffusion emerges when a con- in weakly chaotic systems [9]. With the rise of experi- ventional random walker is confined to move on a ma- mental techniques such as fluorescence correlation spec- trix with a fractal dimension [11, 23, 24]. Stochastic troscopy or advanced single particle tracking methods, processes with multiplicative noise, corresponding to a the discovery of anomalous diffusion has gone through space-dependent diffusion coefficient, also effect anoma- a sharp rise for the motion of endogenous and artificial lous diffusion [25, 26]. A contemporary summary of dif- tracers in living biological cells [10–15]. Concurrent to ferentanomalousdiffusionprocessesexceeding the scope thisdevelopmentanincreasingamountofanomalousdif- of this introduction is provided in Ref. [27]. Here we deal with the remaining of these popular anomalous diffusion models, namely Scaled Brownian ∗Electronicaddress: [email protected] Motion (SBM). SBM is a highly non-stationary process 2 defined in terms of the stochastic equation II. AGEING EFFECT ON UNCONFINED SBM dx(t) = 2K (t)×ξ(t), (2) The position autocorrelation function (covariance) for dt SBM in the conventional (ensemble) sense reads [30] which is driven by whpite Gaussian noise of zero mean hξ(t)i=0 and with autocorrelationhξ(t1)ξ(t2)i=δ(t1− hx(t )x(t )i=2K∗min(t ,t )α. (4) 1 2 α 1 2 t ) . The explicitly time dependent diffusion coefficient 2 is taken as Foranagedprocess,inwhichwemeasuretheMSDstart- K (t)=αK∗tα−1, (3) ing fromthe agingtime ta until time t, the resultfor the α MSD thus becomes We allow α to range in the interval (0,2), such that the process describes both subdiffusion and sub-ballistic su- hx2(t)ia = h[x(ta+t)−x(ta)]2i perdiffusion. The idea of a power-law time dependent = 2K∗[(t+t )α−tα]. (5) α a a diffusion coefficient is essentially dating back to Batche- lor (albeit he used α=3) in his approachto Richardson For a non-aged process with ta =0 the standard scaling turbulent diffusion [28]. SBM, especially in its subdiffu- (1) of the MSD is recovered, as it should. In the aged sive form, is widely used to describe anomalousdiffusion process, the MSD (5) is reduced by the amount accu- [29]. SBMwasstudiedsystematicallyinRefs.[26,30–32]. mulated until time ta, at which the measurement starts. In stationary systems correlations measured between The limiting cases of expression (5) interestingly reveal two times t and t are typically solely functions of the the crossoverbehavior 1 2 time difference, f(|t1 −t2|). In non-stationary systems 2αK∗tα−1t, t ≫t this functional dependence is generally more involved, hx2(t)i = α a a . (6) e.g.,it canacquirethe formf(t2/t1) [33]. Insucha non- a ( 2Kα∗tα, t≫ta stationary setting the origin of time can no longer be While for weak aging (t ≪ t) the aged MSD (5) be- chosenarbitrarily. Thisraisesthequestionofaging,that a comes identical to the non-aged form (1), for strong ag- is,the explicitdependence ofphysicalobservablesonthe ing (t ≫t) the scaling with the process time t is linear time spant betweenthe originalpreparationofthe sys- a a and thus, deceivingly, identical to that of normal Brow- temandthestartoftherecordingofdata. Traditionally, nian diffusion. However, the presence of the power tα−1 agingisconsideredakeypropertyofglassysystems[34]. a is reminiscent of the anomaly α of the process. We note Theagingtimet canbeadjusteddeliberatelyincertain a that the behavior (5) and thus (6) is identical to the re- experiments, such as for the time of flight measurements sult for the subdiffusive CTRW [35, 36] as well as aged of charge carriers in polymeric semiconductors in which heterogeneousdiffusion processes with a power-law form the systemis preparedby knocking outthe chargecarri- of the position dependent diffusivity [37]. ers by a light flash [6]. Similarly, aging could be checked In single particle tracking experiments [38–41] one directly in blinking quantum dot systems, in which the measures the time series x(t) of the position of a labeled initiationtime is givenby the firstexposureofthe quan- particle,whichisthentypicallyevaluatedintermsofthe tum dot to the laser light source. In other systems, for time averaged MSD. For an aged process originally ini- instance, the motion of tracers in living biological cells, tiated at t = 0 and measured from t for the duration the aging time is not always precisely defined. In such a (measurementtime)t thistime averagedMSDis defined cases it is therefore important to have cognisance of the in the form [36] functional effects of aging as developed here. In the following, we will analyze in detail the aging 1 t+ta−∆ 2 properties encoded in the SBM dynamics in both un- δ2(∆)= x(t′+∆)−x(t′) dt′, (7) a t−∆ confined and confined settings. For free aging SBM in Zta h i Section II we show that the result for the time averaged as a function of the lag time ∆ and the aging time t . a MSDfactorizesintoatermcontainingalltheinformation Averaging over an ensemble of N individual trajectories on the aging time t and another capturing the physi- a in the form cally relevant dependence on the lag time ∆. This fac- torization is identical to that of heterogeneous diffusion 1 N δ2(∆) = δ2 (∆), (8) processes and scale-free, subdiffusive CTRW processes. a N a,i In Section III we explore the aging dynamics of confined D E Xi=1 SBM.Forincreasingagingtimetaitisdemonstratedthat the structure function h[x(t′+∆)−x(t′)]2i in the inte- the non-stationary features of SBM under confinement gral of expression (7) can be evaluated in terms of the are progressively washed out, a feature, which is impor- covariance (4). The exact result reads tant for the evaluation of measured time series. Section IVreportsthefirstpassagetimedensityonasemi-infinite 2K∗ δ2(∆) = α (t+t )α+1−(t +∆)α+1 domainforagedSBMwhichincludesacrossoverbetween a (α+1)(t−∆) a a twoscalingregimesasaresultoftheadditionaltimescale D E h introduced by ta. Finally, Sec. V concludes this paper. −(t+ta−∆)α+1+tαa+1 . (9) i 3 In the absence of aging,we recoverthe knownresult[26, 103 31, 32] > a 102 α=0.5 δ2(∆) ∼2K∗ ∆ . (10) (t) t1−α 2x 101 D E < Its linear lag time dependence contrasts the power-law >, 100 formoftheensembleaveragedMSD(1)andthusdemon- ) strates that this process is weakly non-ergodic in the ∆( 10-1 2 individual TA MSD sense of the disparity [27, 42, 43] δ<a10-2 averaged TAth MeoSrDy δ2(∆) 6=hx2(∆)i. (11) 10-3 MSD 100 101 102 103 104 105 D E The equivalence and therefore ergodicity in the Boltz- t, ∆ mann sense is only restored in the Brownian case α=1. In the presence of aging, expansion of expression (9) in 103 the limit t,ta ≫∆ of short lag times yields > a 102 α=0.5 ) (t δa2(∆) ∼Λα(ta/t)× δ2(∆) , (12) 2x 101 < in which weDdefinedEthe so-called agDing depEression as >, 100 )  Λα(z)=(1+z)α−(z)α. (13) ∆2( 10-1 individual TA MSD In this experimentally relevant limit all the information δ<a10-2 averaged TAth MeoSrDy on the age of the process is thus contained in the aging MSD 10-3 depressionΛα,andthe physicallyimportantdependence 100 101 102 103 104 105 on the lag time ∆ factorizes, such that Eq. (12) con- t, ∆ tains the non-aged form (10). Result (12) is identical to the behavior of aged subdiffusive CTRW [36] and het- 103 erogeneous diffusion processes [25]. In the limit ta ≫ t a 102 α=0.5 > of strong aging, the time averaged MSD (9) remarkably ) reduces to the form 2 (t 101 x δa2(∆) =2αKα∗taα−1∆. (14) >, < 100 ∆ D E ∆) 10-1 Inthislimit,thetimeaveragedMSDthusbecomesequiv- 2( 10-2 individual TA MSD ahxle2n(t∆t)oia,thaesaegveiddenencesedmbbylecaovmepraagriesdonMwSiDth, hrδea2s(u∆lt)i(5=). δ<a10-3 averaged TAth MeoSrDy In this limit, that is, ergodicity is restored, as already 10-4 MSD observed for aged CTRW processes [36]. 100 101 102 103 104 105 Figure 1 shows the behavior of the ensemble and time t, ∆ averaged MSD for unconfined SBM at different aging times in the subdiffusive case with α = 1/2. The thin FIG. 1: Ensemble and time averaged MSD for SBM with lines depict the simulations results for the time averaged α = 1/2. Thin lines: time averaged MSD for 20 individual MSD for 20 individual trajectories. The first observa- trajectories from simulations of the SBM Langevin equation tion is that the amplitude spread between these 20 time (2) with trajectory length t = 105. Circles: averages over traces is fairly small. Note that the larger scatter for those 20 trajectories. Black thin line: theory result (12). longerlag times ∆ is due to worseningstatistics when ∆ Thick green line: ensemble averaged MSD (5). Three different aging times were considered (top to bottom): (a) non- approachesthetracelengtht. ThecirclesinFig.1corre- spond to the averageover the 20 different results for the agedcaseta =0,(b)weakagingcaseta=103,and(c)strong timeaveragedMSD.Thelattercompareverynicelywith aging case ta =106. In all simulations Kα∗ =1/2. the theoretical expectation(12). Finally, the thick green lineisthetheoreticalresult(5)fortheensembleaveraged MSD. The detailed behavior in the three different aging (ii) In the weak aging case (t = 102, middle panel a regimes is as follows: of Fig. 1) a major change is visible in the behavior of (i) In the non-aged case (ta = 0, top panel of Fig. 1) the MSD, namely, we see the crossover from the aging- the power-lawgrowthhx2(t)i≃tα of the MSD contrasts dominated linear scaling hx2(t)i ≃ tα−1t to the anoma- a the linear form hδ2(∆)i ≃ ∆, this disparity being at the lous scaling hx2(t)i ≃ tα, encoded in Eqs. (6). The be- heart of the weak ergodicity breaking [26, 31, 32]. haviorofthe time averagedMSDislargelyunchangedin 4 A. Ensemble averaged MSD of confined SBM >a 108 α=1.5 The ensemble averaged MSD for aging SBM, ) 2 (t 107 hx2(t)ia =h[x(ta+t)−x(ta)]2i becomes x >, < 106 ∆ hx2(t)ia =2M1(ta+t)+2M1(ta)−4e−ktM1(ta), (17) ) ∆( 105 individual TA MSD where we used the abbreviation δ 2a averaged TA MSD M (t)=K∗tαexp(−2kt)M(α,α+1,2kt). (18)  104 theory 1 α < MSD In the limit k → 0 of vanishing confinement, Eq. (5) for 103 100 101 102 103 104 105 freeSBMisreadilyrecoveredfromthepropertyM(α,α+ t, ∆ 1,0)=1. We now discuss the result (17) in the three limits of the non-aged, weakly aged, and strongly aged processes. FIG. 2: Ensemble andtime averaged MSD for SBM (2) with α=3/2 in the strong aging case, ta =106. The observation The analysis reveals a rich behavior depending on the time is t = 105. The spread of the 20 single trajectory time values of the aging time ta and the anomalous diffusion averages is fairly small. As before, the ensemble and time exponent α. For sub- and superdiffusion, respectively, averaged MSDs coincide, apparently restoring ergodicity. the various crossoversare displayed in Figs. 3 and 4. (i)Inthe absenceofaging(t =0),wegetbackto the a result comparison to case (i). hx2(t)i=2M (t) (19) (iii) In the strong aging case (t = 106, bottom panel 1 a of Fig. 1) we see the apparent restoration of ergodicity: reported in Ref. [32]. For t ≪ 1/k this reduces to the ensemble and time averagedMSDs coincide, as given by non-agedfreeSBMresult(1),whileinthelongtimelimit Eq. (14). t≫1/k we use the expansion The convergence of the ensemble and time averaged exp(z) MSDsinthestrongagingcaseforthesuperdiffusivecase M(α,α+1,z)∼α (20) with α=3/2 is nicely corroboratedin Fig. 2. z of the Kummer function to obtain [32] ∗ αK III. AGEING EFFECT ON CONFINED SBM hx(t)2i∼ αtα−1. (21) k Thisresultunderlinestheinherentlynon-stationarychar- In many cases an observed particle cannot be consid- acter of SBM: for subdiffusion the MSD hx(t)2i progres- ered free during the observation. Examples contain par- sively decays, while for superdiffusion it increases. This ticles moving in confined space, for instance, within the propertyreflectsthetimedependenceofthetemperature confines of living biological cells [14, 44]. Similarly, par- encodedin the diffusivity (3) [32]. This non-agedbehav- ticles measured in optical tweezers setups experience a ior is shown in Figs. 3 and 4 as the grey lines for two confining Hookean force [12, 18, 45]. As a generic ex- different strengths k of the external confining potential. ample for confined SBM we consider the linear restoring How does aging modify this behavior? force −kx(t) with force constant k. The corresponding (ii) We first consider the case t ≪1/k. If in addition stochastic equation for this confined SBM reads [32] a t ≪ 1/k, this is but the above result (5) for free aging SBM. However, care needs to be taken when t ≫ 1/k. dx(t) =−kx(t)+ 2αK∗t(α−1)×ξ(t), (15) FromEq.(17),wefindthatthefirsttwoterms(thethird dt α q oneisexponentiallysmallintandcanbeneglected)lead to the asymptotic behavior where, as before, ξ(t) represents white Gaussian noise of zeromean. The covarianceinthis confinedcaseyieldsin αK∗ the form [32] hx2(t)ia ∼ kαtα−1+2Kα∗tαa. (22) hx(t )x(t )i=2K∗tαe−k(t1+t2)M(α,α+1,2kt ) (16) This implies that for subdiffusion (0 < α < 1) the first 1 2 α 1 1 termtendstozeroandtheleadingbehavioristheplateau fort1 <t2 intermsoftheconfluenthypergeometricfunc- hx2(t)ia ∼2Kα∗tαa. (23) tion of the first kind, also referred to as the Kummer function [32, 47]. Based on this result we now present Even for very weak aging, the ensemble averaged MSD the ensemble and time averagedMSDs. hx2(t)i becomest -dependent. Whenexperimentaldata a a 5 1 10 1/2 0 10 -1 10 > (t) 10-2 2 x 1 0 < -3 10 g o l 1 -4 10 2 4 <x (t)> -5 10 -6 10 -2 0 2 4 6 8 10 10 10 10 10 10 10 10 log t FIG.3: EnsembleaveragedMSDhx2(t)iforconfinedagingSBMinthesubdiffusivecasewith α=0.5at differentagingtimes: (i) non-aged (ta =0) denoted by the grey lines; (ii) weakly aged (ta = 0.1) denoted by the blue lines; and (iii) strongly aged (ta = 106) denoted by the orange lines. In all cases, the full lines correspond to the force constant k =0.1, while the dashed lines stand for k =0.01. The green line at the bottom of the graph is a blowup (hx2(t)i4 of the case ta =106 and k =0.1) in which thecrossover between thetwo plateaux is better visible. are evaluated and the exact equivalence t = 0 is not according to Eq. (25) again differs between sub- and su- a guaranteed, the erroneous conclusion could be drawn perdiffusive motion. For 0<α<1 the plateau thattheprocessisstationary. Note,however,thatresult ∗ αK (23)isindependentofthe strengthk oftheconfiningpo∗- hx2(t)ia ∼ kαtαa−1 (27) tential and only depends on the diffusion coefficient K α emerges. Note, however, that in comparison to Eq. (26) and the aging time t , mirroring the fact that this term a we now have half the amplitude. In the superdiffusive stems from the initial free motion during the aging pe- caseα>1werecoverresult(24). Thisintricatebehavior riod. Conversely, for superdiffusion (α > 1) the leading is shown in Figs. 3 and 4 as the orange lines. In Fig. 3 order term indeed shows the growth wepronouncethe crossoverbetweenthe twoplateauxby ∗ hx2(t)i ∼ αKαtα−1 (24) plotting the fourth power of the ensemble MSD as the a k green line. of the ensemble averagedMSD. The weakly agedbehav- ior is shown in Figs. 3 and 4 as the blue lines. B. Time averaged MSD of confined SBM (iii) With the asymptotic expansion (20) of the Kum- mer function, we find that in the strong aging regime The time averagedMSD for confined SBM can be de- t ≫1/k the ensemble averagedMSD becomes a rived by substituting the above covariance (16) into the hx2(t)i ∼αk−1K∗ (t +t)α−1+tα−1 1−2e−kt . integral (7). By help of the relation [46] a α a a (25) x (cid:2) (cid:0) (cid:1)(cid:3) yαe−yM(α,1+α,y)dy = At short times t≪1/k,this leads us back to the unconfined result hx2(t)i ∼ 2αK∗tα−1t of Eq. (6). At long Z a α a 1 time t ≫ 1/k, however, we have to distinguish two dif- x1+αe−xM(1+α,2+α,x) (28) 1+α ferent regimes. First, for t ≫ t ≫ 1/k we obtain the a plateau this procedure yields the general result ∗ hx2(t)ia ∼ 2αkKα∗taα−1, (26) δa2(∆) = (t−∆2K)(1α+α) M2(t+ta)−M2(ta+∆) D E +M (t+t −∆h)−M (t ) 2 a 2 a which differs from the above result (24) by the factor of two. Second, for t ≫ ta ≫ 1/k the leading order −2e−k∆ M2(t+ta−∆)−M2(ta) ,(29) (cid:16) (cid:17)i 6 8 10 6 10 104 3/2 > t) 1/2 ( 2 2 x 10 < g o 0 l 10 1 -2 10 3/2 -4 10 -2 0 2 4 6 8 10 10 10 10 10 10 10 10 log t FIG. 4: Ensemble averaged MSD hx2(t)i for confined aging SBM in the superdiffusive case with α = 1.5 at different aging times: (i) non-aged (ta =0) denoted by thegrey lines; (ii) weakly aged (ta =0.1) denoted by theblue lines; and (iii) strongly aged (ta = 106) denoted by the orange lines. In all cases, the full lines correspond to the force constant k = 0.1, while the dashed lines stand for k=0.01. In all cases theterminal scaling ≃tα−1 is reached. where we used the abbreviation (iii) The second, more interesting case corresponds to long aging times compared to the relaxation time scale, M2(t)=t1+αe−2ktM(1+α,2+α,2kt). (30) ta ≫ 1/k. When also t ≫ 1/k, the result is independent of the specific magnitude of the lag time. From the (i) In the limit k → 0 we recover the result (9) of generalexpression(29)byhelpofrelation(20)weobtain unconfined aging SBM, while the complete absence of ∗ K aging restores the result from Ref. [32]. In the presence δ2(∆) ∼ α (t+t )α−(∆+t )α of confinement, we distinguish the following regimes. a k(t−∆) a a D E h (ii) We now consider the case when the aging time +(1−2e−k∆)[(t+t −∆)α−tα] .(33) a a is short compared to the relaxation time of the system, i ta ≪1/k. Fromthe generalexpression(29) wethen find If we now consider the regime in which the lag time is thefollowingbehaviors: (a)wheninadditionthelagtime short, t,t ≫ 1/k ≫ ∆, we obtain result (12) with the a is short (t ≫ 1/k ≫ ∆ & ta) we recover the non-aged agingdepression(13)fromunconfinedagingSBM.Inthe result (10) with its linear scaling in the lag time ∆. (b) oppositelimitt,t ≫∆≫1/k whenthe lagtime islong a When the lag time is long (t ≫∆≫1/k≫ta), we find compared to the relaxation time, we find the plateau δ2(∆) ∼Λ (t /t) δ2(∆) , (34) ∗ a a a 2K δ2(∆) ∼ αtα−1 (31) D E D E a k where δ2(∆) is equal to expression (31) and Λ (z) is D E a knownfromthenon-agedcase[27]. (c)Finally,whenthe againtDheaginEgdepression(13). Inthestrongaginglimit lag time approaches the length t of the time series, the t,ta ≫∆≫1/k,thatis,the agedtimeaveragedMSDis time averagedMSD generally given by δ2(∆) ∼ Λ (t /t) δ2(∆) for any a a a αK∗ lag time. Similar tDo subdiEffusive CTRWD proceEsses [36], δ2(∆) ∼ αtα−1 (32) the occurrence of the factor Λ appears like a general a k a feature for the aging dynamics of SBM. D E becomes equivalent to the ensemble averaged MSD, Figure 5 shows the behavior of the ensemble and time Eq. (24). In contrast to the ensemble averaged MSD, averaged MSD for confined SBM at different degrees of we thus find that the time averagedMSD is not affected aging. The graphs represent the full behavior according by short aging times as compared to the relaxationtime to Eqs. (17) and (29). In the absence of aging, the ini- scale, t ≪1/k. tiallineargrowthhδ2(∆)i≃∆ofthetimeaveragedMSD a a 7 102 ×103 TA MSD k=0.1 TA MSD k=0.01 2>, <x (t)>a110001 ~ tα MMSSDD kk==00..10 1 2>, <x (t)>a100 α~= t1, .∆5 δ∆ 2<() a1100--21 ~ ∆ α=0.5 δ∆ 2<() a 10 TTMMAASS MMDD,,SS DD ,, kkkk====0000....1001 11 1 100 101 102 103 104 100 101 102 103 104 t, ∆ t, ∆ 101 α FIG.6: Ensembleandtimeaveraged MSDfor confinedSBM a ~ t α=0.5 with α = 3/2 in the strong aging case with ta = 106. The )> observation time is t=5×104. 2 (t 100 x < >, temperature) [32]. When aging effects come into play, ∆) 10-1 ~ ∆ theensembleaveragedMSDdisplaysnotabledifferences. ( In the case of weak aging displayed in the middle panel δ 2<a TTMAAS MMD SS DD kkk===000...101 1 obfecFoimg.e5adpepvaiaretniotnfsorfrolomngtehretpimowese,r-tla≫wtdec≫ay1o/fkh.xE2(vte)ina- 10-2 MSD k=0.01 a tuallytheconvergencetoacommonvalueindependentof 100 101 102 103 104 the force constantsis observed,as predictedby Eq.(23). t, ∆ Finally, in the strong aging limit, the ensemble and time 10-1 averagedMSDareequivalentandergodicityisseemingly α=0.5 restored: hδ2(∆)i=hx2(∆)i ,ascanbe witnessedinthe a a a bottom panel of Fig. 5. The apparent equivalence of en- > (t) ~ t, ∆ sembleandtimeaveragedMSDsinthestrongaginglimit 2x is also proven in the superdiffusive case for α = 3/2 in < >, 10-2 Fig. 6. The slight discrepancy remaining between time ) and ensemble averagedMSD in the latter strong (but fi-  ∆ nite) aging case is shown in Fig. (7). This Figure also ( TA MSD k=0.1 δ 2a TMAS MD S D kk==00..10 1 dereamgoednsMtraStDess.the convergence of ensemble and time av- < MSD k=0.01 10-3 100 101 102 103 104 t, ∆ IV. FIRST PASSAGE TIME DENSITY FIG.5: Ensembleandtimeaveraged MSD forconfinedSBM ApartfromtheMSDthefirstpassagebehaviorisasig- for α = 1/2. From top to bottom, the panels represent the nature quantity of a stochastic process. We here study non-aged (ta = 0) case, the case of weak aging (ta = 10−1), how aging changes the first passage statistic of SBM in andthecaseofstrongaging(ta =106),wheretheobservation the semi-infinite domain. The probability density func- time is chosen as t = 5×104. The lines represent Eqs. (17) tion (PDF) of first passage is found by solving the SBM and (29). The force constants k are indicated in the panels. diffusion equation with the time dependent coefficient Note that the time averaged MSD indeed converges to the K (t), ensembleMSDin thelimit ∆→t,comparethediscussion in Ref. [32]. ∂ ∂2 P(x,t)=K (t) P(x,t), (35) ∂t ∂x2 however,with the aged initial condition crossesovertoanapparentplateau,contrastingthefunc- tional behavior of the ensemble average: at short times, P (x,t )= 1 e−x2/(4Kα∗tαa). (36) we observe the power-law growth hx2(t)ia ≃ tα of un- 0 a 4πK∗tα α a confined SBM, while after engaging with the confining potential, the monotonic decrease hx2(t)i ≃ tα−1 re- This aged initial condiption emerges from a δ(x) peak for a flects the temporal decay of the noise strength (i.e., the a system initialized some aging time t before. In this a 8 ×10-3 ×105 9.8 1.49 2x(t)>a kα==00..15 ETAA MMSSDD 2x (t)>a 1.48 kα==01..15 < < >, 9.7 >, 1.47 ∆) ∆) δ(< 2a δ 2<(a 1.46 ETAA MMSSDD 9.6 1.45 101 102 103 104 101 102 103 104 t, ∆ t, ∆ ×10-3 ×105 98 >a kα==00..051 ETAA MMSSDD >a 1.48 2 (t) 2 (t) x x < < >, 97 >, 1.47 ) ) ∆ ∆ δ 2(a δ 2(a 1.46 k=0.01 EA MSD < < α=1.5 TA MSD 96 1.45 103 104 103 104 t, ∆ t, ∆ FIG. 7: Full behavior of aging confined SBM for ta = 106 with α = 1/2 (left) and α = 3/2 (right), demonstrating the convergence of the time averaged MSD to the ensemble averaged MSD in the limit ∆ → t, shown for two different potential strengths, as indicated in thepanels. setuptmeasuresthetimespanfromtheagedinitialcon- Forα=1(Brownianmotion)andintheabsenceofaging dition(36). ToobtainthefirstpassagePDFforthesemi- (t =0) we recover the well known Lévy-Smirnov distri- a infinitedomainwesolvetheSBMdiffusionequation(35) bution. Result (40) exhibits a crossover relative to the forunconfinedmotionwiththeagedinitialcondition(36) aging time, and then use the method of images. For the PDF of the aged process we obtain t−3α/2tα−1, t ≫t,(x2/[4K∗])1/α αx a a 0 α 0 ℘≃ × . P(x,t)= 4πKα∗1(tαa +tα)e−x2/4Kα∗(tαa+tα). (37) p4πKα∗  t−1−α/2, t≫ta,(x20/[4Kα∗])1/(α41) Inthestrongaginglimitthescalingexponentis−(1−α), Inthepresenpceofanabsorbingboundaryattheorigin, andwe observethe explicitpresence ofthe agingtime t a the survival probability for a process initiated originally withexponentα. Forweakaging,thescalingexponentof in x >0 is therefore given by 0 t is −(1+α/2), as known from subdiffusive CTRW pro- ∞ cesses. However,thedetailedcrossoverbehaviorisdiffer- S(t)= [P(x−x0,t)−P(x+x0,t)]dx. (38) ent, compare Ref. [48]. We also note that for fractional Z0 Brownianmotiontheanomalousdiffusionexponentαen- ters oppositely [30, 49]. Fig. (8) shows the crossover of Substituting the aged PDF (36) into this expression the first passage density for aging SBM. yields x S(t)=erf 0 (39) V. CONCLUSIONS 4Kα∗(tαa +tα)! p SBM is possibly the simplest anomalous diffusion in terms of the error function. The first passage PDF follows from the relation ℘(t)=−dS(t)/dt, model, and it is therefore widely used in literature. De- spite its apparentsimplicity SBM exhibits weak ergodic- αx tα−1 x2 ity breaking in the sense that we observe a distinct dis- ℘(t)= 4πKα∗0(tαa +tα)3 exp(cid:18)−4Kα∗(tαa0+tα)(cid:19). pofartihtyisbpertowceeesns [t2h6e, e2n7s,e3m1b,l3e2a].ndIttiimsethaevreerfoargeedaMnaStDus- q (40) ralquestionto explore the aging effects ofSBM, i.e., the 9 10-4 uously decaying (subdiffusion) or increasing (superdiffusion). As shownhere the functionalbehavior ofconfined 10-6 aging SBM is remarkably rich. Concurrently, the time 10-8 -1/2 averaged MSD exhibits an intermediate plateau. In the 10-10 presence of aging, we observe a deviation from the non- g p(t) 10-12 aagteldonpgoewretrim-laews abnedhaavuionrivoefrtshaelceonnsveemrbgelencaevetroaagepdlaMteSaDu o l 10-14 value. For strong aging, we again observe the convergence of ensemble and time averaged MSDs. We note 10-16 -5/4 that the behavior of confined CTRW is opposite: the 10-18 time averaged MSD exhibits a power-law growth with 10-20 exponent 1−α, while the ensemble averaged MSD con- 10-2 100 102 104 106 108 1010 1012 1014 verges to the thermal plateau value [33]. log t In addition to these quantities we considered the first passagetimedensityinthesemi-infinitedomain. Incon- FIG. 8: First passage time density ℘(t) for α = 1/2, with x0 = 1, and aging time ta = 106. As shown by the dashed trast to the non-aging fractional Brownian motion we line, the crossover between the aging dominated slope −1/2 foundacrossoverbetweentwocharacteristicscalinglaws to theslope −5/4 is distinct. dependingonthecompetitionbetweenagingandprocess time, t and t. a When using SBM as a stochastic model cognizance explicit dependence of physical observables on the time should be taken of the fact that it is a highly non- span t between the original system preparation. We a stationary process. The time dependence of its diffusiv- here showed how the ensemble and time averagedMSDs ity corresponds to a time dependent temperature (noise dependont forboththeunconfinedandconfinedcases. a strength), and is therefore physically meaningless as de- For unconfined aging SBM we obtained the exact de- scription for a system coupled to a thermostat. There pendence on the aging time t and observed a strik- a exist, however, cases, in which SBM may turn out to be ingsimilarityto bothsubdiffusive CTRWwithscale-free a physically meaningful approach. For instance, it was waiting time distributions and heterogeneous diffusion demonstratedthat SBMprovides a useful meanfield de- processeswith power-lawpositiondependence ofthe dif- scriptionforthemotionofataggedparticleinagranular fusivity. In particular, for short lag times the time av- gas with a sub-unity restitution coefficient in the homo- eraged MSD factorizes into the non-aged expression and geneous cooling phase [51]. theagingdepressionΛ ,whichindeedhasthesamefunc- a tionalformasforthesubdiffusiveCTRWandthehetero- A number of aging features are quite similar between geneousdiffusionprocess. Inthelimitofstrongaging,we subdiffusive CTRWs [36], heterogeneous diffusion pro- alsoshowedthatergodicityisseeminglyrestoredandthe cesses [37], and aging SBM, as shown here. While frac- disparity between ensemble and time averages becomes tional Brownian motion is ergodic, transient deviations increasingly marginal. fromergodicity and transientaging occur under confine- Confined aging SBM, in contrast, is qualitatively a ment [18, 50]. To reliably distinguish these models from quite unique process. Due to the time dependence of another, it is therefore imperative to employ other diag- the diffusivity K (t) there is no thermal plateau for the nostic stochastic quantities with characteristicbehaviors ensemble averagedMSD. Insteadthis quantity is contin- for the respective processes [27, 52, 53]. [1] L. F. Richardson, Proc. Roy.Soc. A 110, 709 (1926). Berkowitz, Geophys. Res. Lett. 29, 1061 (2002). [2] H. Freundlich and D. Kru¨ger, Trans. Faraday Soc. 31, [8] E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, 906 (1935). and D. A. Weitz, Science 287, 627 (2000); J. Mattsson, [3] J.-P. Bouchaud and A. Georges, Phys. Rep. 195, 127 H. M. Wyss, A. Fernandez-Nieves, K. Miyazaki, Z. Hu, (1990). D.R.Reichman,andD.A.Weitz,Nature462,83(2009). [4] R.Metzler and J. Klafter, Phys. Rep.339, 1 (2000). [9] T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys. [5] H. Scher and E. W. Montroll, Phys. Rev. B. 12, 2455 Rev.Lett.71,3975(1993);E.R.WeeksandH.L.Swin- (1975). ney,Phys. Rev.E 57, 4915 (1998). [6] M. Schubert, E. Preis, J. C. Blakesley, P. Pingel, U. [10] F.HöflingandT.Franosch,Rep.Progr.Phys.76,046602 Scherf, and D.Neher, Phys. Rev.B 87, 024203 (2013). (2013). [7] M. Dogan, R. L. Van Dam, G. C. Bohling, and J. J. [11] M. J. Saxton and K. Jacobsen, Ann. Rev. Biophys. ButlerJr.,Geophys.Rev.Lett.38,L06405(2011);J.W. Biomol. Struct. 26, 373 (1997); M. Saxton, Biophys. J. Kirchner,X.Feng,andC.Neal,Nature403,524(2000); 103, 2411 (2012); 72, 1744 (1997). H. Scher, G. Margolin, R. Metzler, J. Klafter, and B. [12] J.-H.Jeon,V.Tejedor,S.Burov,E.Barkai,C.Selhuber- 10 Unkel,K.Berg-Sørensen,L.Oddershede,andR.Metzler, [29] M. J. Saxton, Biophys. J. 81, 2226 (2001); T. J. Feder, Phys.Rev.Lett. 106, 048103 (2011). I. Brust-Mascher, J. P. Slattery, B. Baird, and W. W. [13] S. M. A. Tabei, S. Burov, H. Y. Kim, A. Kuznetsov, T. Webb, Biophys. J. 70, 2767 (1996); N. Periasmy and A. Huynh,J.Jureller,L.H.Philipson,A.R.Dinner,andN. S. Verkman, Biophys. J. 75, 557 (1998); M. Weiss, M. F.Scherer,Proc.Natl.Acad.Sci.USA110,4911(2013); Elsner, F. Kartberg, and T. Nilsson, Biophysical J. 87, D. Robert, T. H. Nguyen, F. Gallet, and C. Wilhelm, 3518 (2004); J. Wu and K. M. Berland, Biophys. J. 95, PLoS ONE4, e10046 (2010). 2049 (2008). [14] I. Golding and E. C. Cox, Phys. Rev. Lett. 96, 098102 [30] S.C.LimandS.V.Muniandy,Phys.Rev.E66,021114 (2006); S.C.Weber,A.J.Spakowitz,andJ.A.Theriot, (2002). Phys.Rev.Lett. 104, 238102 (2010). [31] F. Thiel and I. M. Sokolov, Phys. Rev. E 89, 012115 [15] K. Burnecki, E. Kepten, J. Janczura, I. Bronshtein, Y. (2014). Garini, and A. Weron, Biophys. J. 103, 1839 (2012); I. [32] J.-H.Jeon,A.V.Chechkin,andR.Metzler,Phys.Chem. Bronstein, Y. Israel, E. Kepten, S. Mai, Y. Shav-Tal, Chem. Phys.16, 15811 (2014). E. Barkai, and Y. Garini, Phys. Rev. Lett. 103, 018102 [33] S.Burov,R.Metzler,andE.Barkai,Proc.Natl.Acadm. (2009). Sci. USA107, 13228 (2010). [16] E. Yamamoto, T. Akimoto, M. Yasui, and K. Ya- [34] E.-J. Donth, The Glass Transition (Springer, Berlin, suoka, Scientif. Reports 4, 4720 (2014); G. R. Kneller, 2001); W. Götze, Complex Dynamics of Glass-Forming K. Baczynski, and M. Pasienkewicz-Gierula, J. Chem. Liquids (Oxford University Press, Oxford, UK,2009). Phys.135,141105(2011);J.-H.Jeon,H.Martinez-Seara [35] E.Barkai,Phys.Rev.Lett.90,104101(2003); E.Barkai Monne,M.Javanainen,andR.Metzler,Phys.Rev.Lett. and Y.C. Cheng, J. Chem. Phys.118, 6167 (2003). 109, 188103 (2012). [36] J. H. P. Schulz, E. Barkai, and R. Metzler, Phys. Rev. [17] W. Pan, L. Filobelo, N. D. Q. Pham, O. Galkin, V. V. Lett.110,020602(2013);Phys.Rev.X4,011028(2014). Uzunova,andP.G.VekilovPhys.Rev.Lett.102,058101 [37] A.G.Cherstvy,A.V.Chechkin,andR.Metzler,J.Phys. (2009); J. Szymanski and M. Weiss, Phys. Rev. Lett. A 47, 485002 (2014). 103, 038102 (2009); G. Guigas, C. Kalla, and M. Weiss, [38] W. E. Moerner and M. Orrit, Science238, 1670 (1999). Biophys. J. 93, 316 (2007). [39] M. J. Saxton and K. Jacobsen, Ann. Rev. Biophys. [18] J.-H.Jeon,N.Leijnse,L.B.Oddershede,andR.Metzler, Biomol. Struct.26, 373 (1997). NewJ. Phys.15, 045011 (2013). [40] C. Bräuchle, D. C. Lamb, and J. Michaelis, Single [19] E.W.MontrollandG.H.Weiss,J.Math.Phys.10,753 Particle Tracking and Single Molecule Energy Transfer (1969). (Wiley-VCH, Weinheim,Germany, 2012). [20] J. Klafter, A. Blumen,and M. F. Shlesinger, Phys.Rev. [41] X. S. Xie, P. J. Choi, G.-W. Li, N. K. Lee, and G. Lia, A 35, 3081 (1987). Annu.Rev.Biophys. 37, 417 (2008). [21] B. B. Mandelbrot and J. W. van Ness, SIAM Rev. 10, [42] J.-P. Bouchaud, J. Phys. (Paris) I 2, 1705 (1992); C. 422 (1968); A. N. Kolmogorov, Dokl. Akad. Nauk SSSR MonthusandJ.-P.Bouchaud,J.Phys.A29,3847(1996). 26, 6 (1940). [43] E.Barkai,Y.Garini,andR.Metzler,Phys.Today65(8), [22] I.Goychuk,Phys.Rev.E80,046125(2009);Adv.Chem. 29 (2012). Phys.150,187(2012);E.Lutz,Phys.Rev.E64,051106 [44] Y. He, S. Burov, R. Metzler, and E. Barkai, Phys. Rev. (2001); G. Kneller, J. Chem Phys. 141, 041105 (2014); Lett. 101, 058101 (2008). P.Hänggi, Zeit.PhysikB31,407 (1978); P.Hänggiand [45] M.A.Taylor,J.Janousek,V.Daria,J.Knittel,B.Hage, F. Mojtabai, Phys. Rev. E 26, 1168 (1982); S. C. Kou, H.-A. Bachor, and W. P. Bowen, Nature Phot. 7, 229 Ann.Appl.Stat.2, 501 (2008). (2013). [23] S. Havlin and D. Ben-Avraham, Adv. Phys. 51, [46] A. B. Prudnikov and Yu. A. Brychkov, Integrals and 187(2002);Y.Meroz,I.M.Sokolov,andJ.Klafter,Phys. Series, Volume 3 (Gordon and Breach, New York, NY, Rev.E81,010101(R)(2010);M.Spanner,F.Höfling,G. 1986). E. Schröder-Turk, K. Mecke, and T. Franosch, J. Phys. [47] M. Abramowitz and I. A. Stegun, Handbook of mathe- Cond. Mat. 23, 234120 (2011). matical functions (DoverPubl., New York,NY,1965). [24] A. Klemm and R. Kimmich, Phys. Rev. E 55, 4413 [48] H.Kru¨semann, A.Godec, and R.Metzler, Phys.Rev.E (1997); A. Klemm, R. Metzler, and R. Kimmich, Phys. 89, 040101(R) (2014). Rev.E 65, 021112 (2002). [49] J.-H. Jeon, A. V. Chechkin, and R. Metzler, Europhys. [25] A.G.Cherstvy,A.V.Chechkin,andR.Metzler,NewJ. Lett. 94, 20008 (2011). Phys. 15, 083039 (2013); Soft Matter 10, 1591 (2014); [50] J. Kursawe, J. H. P. Schulz, and R. Metzler, Phys. Rev. A.G.CherstvyandR.Metzler,Phys.Rev.E90,012134 E88,062124(2013);J.-H.JeonandR.Metzler,ibid.85, (2014); Phys. Chem. Chem. Phys. 15, 20220 (2013); P. 021147 (2012). Massignan,C.Manzo,J.A.Torreno-Pina,M.F.Garc´ıa- [51] A. Bodrowa, A. G. Cherstvy, A. V. Chechkin, and R. Parako, M. Lewenstein, and G. L. Lapeyre, Jr., Phys. Metzler, E-print arXiv:1501.04173. Rev.Lett. 112, 150603 (2014). [52] V. Tejedor, O. Benichou, R.Voituriez, R.Jungmann, F. [26] A. Fulin´ski, J. Chem. Phys. 138, 021101 (2013); Phys. Simmel, C. Selhuber-Unkel, L. Oddershede and R. Met- Rev.E 83, 061140 (2011). zler, Biophys. J. 98, 1364 (2010). [27] R. Metzler, J.-H. Jeon, A. G. Cherstvy, and E. Barkai, [53] M. Magdziarz, A. Weron, K. Burnecki and J. Klafter, Phys.Chem. Chem. Phys.16, 24128 (2014). Phys. Rev.Lett. 103, 180602 (2009). [28] G.K.Batchelor,Math.Proc.CambridgePhilos.Soc.48, 345 (1952).