Scaling Green-Kubo Relation and Application to Three Aging Systems

A. Dechant,1 E. Lutz,1,2 D. A. Kessler,3 and E. Barkai31Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany2Institute for Theoretical Physics II, University of Erlangen-Nürnberg, 91085 Erlangen, Germany

3Department of Physics, Institute for Nanotechnology and Advanced Materials, Bar Ilan University,Ramat-Gan 52900, Israel

(Received 29 October 2013; published 24 February 2014)

The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorre-lation function. We derive a generalization of the Green-Kubo formula that is valid for systems with long-range or nonstationary correlations for which the standard approach is no longer valid. For the systemsunder consideration, the velocity autocorrelation function hv!t" !#v!t#i asymptotically exhibits a certainscaling behavior and the diffusion is anomalous, hx2!t#i≃ 2D"t". We show how both the anomalousdiffusion coefficient D" and the exponent " can be extracted from this scaling form. Our scaling Green-Kubo relation thus extends an important relation between transport properties and correlation functions togeneric systems with scale-invariant dynamics. This includes stationary systems with slowly decayingpower-law correlations, as well as aging systems, systems whose properties depend on the age of thesystem. Even for systems that are stationary in the long-time limit, we find that the long-time diffusivebehavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D" isnot unique, and we determine its values, respectively, for a stationary or nonstationary initial state. Wediscuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear frictioncorresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with externalnoise recently suggested to model active transport in cells, and the Lévy walk with numerous applications,in particular, blinking quantum dots. These examples underline the wide applicability of our approach,which is able to treat very different mechanisms of anomalous diffusion.

DOI: 10.1103/PhysRevX.4.011022 Subject Areas: Statistical Physics

I. INTRODUCTION

For many central results of statistical mechanics, thestationarity of the described quantities is crucial.Stationarity is conveniently defined in terms of the auto-correlation function hA!t" !#A!t#i of an observable A,where h::::i denotes an ensemble average. The quantity Ais called stationary if this correlation function is independentof the time t (called the age of the system) and only dependson the time lag !. Conversely, A is called aging if thecorrelation function depends explicitly on the age t.Generally, any system may show aging for short enoughtimes, but for a wide class of systems, relaxation into thestationary state is exponentially fast. They can thus be fullydescribed by their stationary behavior on time scales longerthan the characteristic relaxation time. However, this is nottrue for all physical systems. Some systems may possess astationary state, but the relaxation towards this state may beslow and thus important on experimentally relevant timescales [1–4]. Other systems may even exhibit aging on all

experimentally accessible time scales [5–7]. The extensionand generalization of known results for the stationary case tothe agingone is important to better understand and character-ize thebehaviorof aging systems [8]. In thiswork,wediscusssuch a generalization of the Green-Kubo formula.The Green-Kubo formula, a central result of nonequili-

brium statistical mechanics first discussed by Taylor [9],expresses the spatial diffusion coefficient D1, defined bythe equation hx2!t#i≃ 2D1t as a time integral of thestationary velocity correlation function [10,11],

D1 $Z

!

0d!hv!t" !#v!t#i: (1)

For a Brownian particle of mass m with Stokes’friction, the velocity correlation function is exponential,hv!t" !#v!t#i $ kBT=m exp!−#%!#, and we get theEinstein relation D1 $ kBT=!m#%# linking the diffusivityand the friction coefficient #% via the temperature T. TheGreen-Kubo formula offers a simple way of relating thediffusive properties of a system and its velocity dynamics.However, it is only valid as long as there exists a stationaryvelocity correlation function and its integral is finite. If, bycontrast, the velocity correlation function decays slowly oris nonstationary, Eq. (1) needs to be generalized.

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW X 4, 011022 (2014)

2160-3308=14=4(1)=011022(16) 011022-1 Published by the American Physical Society

We present a generalization of Eq. (1) to superdiffusivesystems, where hx2!t"i≃ 2Dνtν with ν > 1. This general-ized Green-Kubo relation will allow us to determine thediffusive behavior of the system by studying the scalingproperties of the velocity autocorrelation. The scalingexponent will be seen to be related to the diffusionexponent ν, while the anomalous diffusion coefficientDν, in the spirit of the original Green-Kubo formula, canbe expressed as the integral over a scaling function. Ourscaling Green-Kubo relation is applicable to a wide rangeof systems exhibiting anomalous scale-invariant dynamics.These include systems whose autocorrelations, while sta-tionary, exhibit a slow power-law decay, systems where thecorrelation function shows usual aging (i.e., the correlationtime increases linearly with the age of the system), andfinally, systems exhibiting what we refer to as superaging,where in addition to aging, the variance of the velocityincreases with time.One important finding is that even if a system’s velocity

correlation function is stationary in the long-time limit, thelong-time diffusive dynamics still may not be described bythis stationary correlation function. This implies that agingeffects can be important even in systems that are known topossess a stationary state; in particular, the diffusivity candepend strongly on the initial state of the system. This is incontrast to the established notion of transport coefficients asunique quantities that are independent of the initial prepa-ration of the system. Such a persistence of the initialcondition was previously observed by Zumofen and Klafterfor a certain class of dynamical systems [12]. Our scalingGreen-Kubo relation is able to treat both these persistenceeffects in stationary systems and in actual aging systems,offering new insights on the interrelation between diffusivebehavior and correlations.In the course of this work, we discuss three examples of

such systems with widespread applicability. The firstexample is diffusion under the influence of a nonlinearfriction force that is for large velocities inversely propor-tional to the velocity. This unusual kind of friction forceoccurs as an effective cooling force in the semiclassicaltreatment of cold atoms in dissipative optical lattices [13].The interaction of the atoms with the lattice dependsstrongly on their velocity, and the Sisyphus coolingmechanism described by the friction force is not effectivefor very fast atoms. This nonlinear behavior of the frictionforce on the velocity leads to anomalous dynamics,including velocity correlations that decay slowly in time[14] and exhibit superaging [15]. Our scaling Green-Kuborelation allows us to determine the spatial diffusionexponent and coefficient from the structure of the velocitycorrelation function. In a more general context, our resultsalso apply to diffusion in a logarithmic potential [15],which has applications in a wide array of physical systems,like vortex dynamics [16], long-range interacting systems[17], particles near a long charged polymer [18],

nanoparticles in a trap [19], self-gravitating particles[20], or the dynamics of denaturing DNA [21]. It has beennoted by Hirschberg et al. [22,23] that the distributionfunction evolves differently for different classes of initialconditions. By means of our scaling Green-Kubo relation,we study under which conditions this sensitivity on theinitial condition carries over to the diffusion coefficient.As our second example, we consider the fractional

Langevin equation with external noise. In this case, theanomalous dynamics are due to a power-law friction kernel,which describes a long-range hydrodynamic memory, andpower-law correlated noise [24,25]. The equilibrium frac-tional Langevin equation has been shown to provide a gooddescription for anomalous subdiffusion in a crowdedenvironment, like the cytoplasm of biological cells, inthermal equilibrium [26,27]. In Refs. [28,29], an externalnonequilibrium noise term was introduced to describe thesuperdiffusive behavior due to active forces in living cells.In this superdiffusive regime, we use our scaling Green-Kubo relation to determine the diffusive properties from thevelocity autocorrelation. In particular, we find that thediffusion coefficient is sensitive to whether the system isinitially in the stationary state or not, indicating that thenaive stationary description of the system may not providecorrect results for the measured transport coefficients.Our final example is a paradigm model for aging, the

Lévy walk [30,31], which is an extension of the continu-ous-time random walk [32,33] to superdiffusive dynamics.In its simplest form, this model describes a particle whosevelocity can have values #v0 with random switchingbetween them. The time between two consecutive changesin velocity is randomly selected according to a heavy-tailedpower-law waiting-time distribution, ψ!tw" ! t−μ−1w with0 < μ < 2. In particular, when 0 < μ < 1, the mean wait-ing time is infinite and the velocity correlation functionages [34,35]. The Lévy walk has been successfully appliedto various systems, including turbulence [36], searchpatterns of animals [37], and blinking quantum dots[38]. For the latter case, we show how our scalingGreen-Kubo relation offers a straightforward way to con-nect the anomalous photon statistics to the intensityautocorrelations.

II. GREEN-KUBO RELATION

We begin by generalizing the Green-Kubo formalismto treat nonstationary power-law correlations. Our startingpoint is the velocity correlation function Cv!t2; t1" $hv!t2"v!t1"i, which is defined as

Cv!t2; t1" $Z

"

−"dv2

Z"

−"dv1v2v1P!v2; t2jv1; t1"W!v1; t1":

(2)

Here, P!v2; t2jv1; t1" is the conditional probability density,i.e., the probability to find a particle with velocity v2 at time

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t2, provided it had velocity v1 at t1, and W!v1; t1" is theprobability density to find a particle with velocity v1 at t1.Equation (2) can be used to determine the mean-squaredisplacement hx2!t"i, an important transport characteristic.In the case where the process is stationary at long times,with stationary density Ws!v1" # limt1!!W!v1; t1", and ifP!v2; t2jv1; t1" # P!v2; t2 − t1jv1; 0", Eq. (2) tends to thestationary correlation function Cv;s!!" # hv!!"v!0"is,which only depends on the time lag ! # t2 − t1,

Cv;s!!"#Z

!

−!dv2

Z!

−!dv1v2v1P!v2;!jv1;0"Ws!v1": (3)

Since x!t" #Rt0 dt1v!t1", we can express the second

moment of the position as hx2!t"i #Rt0 dt2

Rt0 dt1 !

Cv!t2; t1" for a particle starting at the origin, x!t" # 0for t # 0. The (one-dimensional) diffusion coefficient D1

then follows as

D1 #1

2limt!!

ddthx2!t"i # lim

t!!

Zt

0d!Cv;s!!"; (4)

where the subscript 1 indicates that diffusion is normal,hx2!t"i≃ 2D1t1. The Green-Kubo formula (4) only holds ifthe velocity correlation function Cv!t2; t1" has a stationarylimit and if it decays fast enough (e.g., exponentially) sothat the time integral is finite. This condition defines acertain class of correlation functions. For this kind ofsystem, even if the initial state is not the stationary one, it isjustified to assume the validity of Eq. (4), as long as one isinterested in the dynamics on time scales much longer thanthe characteristic correlation time. In the following, we willintroduce a different class of correlation functions, forwhich the Green-Kubo formula (4) does not hold. Notethat, generally, there is also the possibility that the timeintegral in Eq. (4) vanishes. This leads to subdiffusion,which we will not consider here.We consider systems whose velocity correlation function

has the following asymptotic scaling form when both t(called the age of the system) and ! (the time lag) are largecompared to the system’s microscopic time scales,

Cv!t$ !; t"≃ Ct"−2#!!t

": (5)

This expression is valid for ! > 0; for ! < 0, one has toreplace ! by −! and t by t$ ! in order to respect thesymmetry of the correlation function with respect tointerchanging its arguments. Here, C > 0 is a constantand " > 1. The asymptotic behavior of the positive-valuedscaling function #!s" is limited by power laws,

#!s"<cls−$l with 2−"" $l < 1 for s! 0;

#!s"<cus−$u with $u> 1−" for s!!; (6)

where cl and cu are positive constants. These conditions donot require #!s" to behave as a power law at small and/orlarge arguments as long as the asymptotic behavior iswithin the above limits, so #!s" might, for example, decayexponentially for large s. Since the correlation functionEq. (5) explicitly depends on the age t of the system, it is anaging correlation function, in contrast to a stationarycorrelation function that only depends on the time lag !.The asymptotic mean-square displacement hx2!t"i for thistype of correlation function is then given by

hx2!t"i≃ 2CZ

t

0dt2

Zt2

0dt1t"−21 #

!t2 − t1t1

"; (7)

where we have used the symmetry of the correlationfunction (5). Introducing the variable s # !t2 − t1"=t1,we obtain

hx2!t"i≃ 2CZ

t

0dt2t"−12

Z!

0ds!s$ 1"−"#!s": (8)

We thus arrive at our first main result,

hx2!t"i≃ 2D"t"

with D" #C"

Z!

0ds!s$ 1"−"#!s": (9)

While the usual Green-Kubo relation (4) holds for normaldiffusion, this scaling Green-Kubo relation (9) is applicablefor " > 1, which corresponds to superdiffusion. Thus,hx2!t"i grows faster than linear in time, and the usualdiffusion coefficient in Eq. (4) is not defined. Instead, wehave the anomalous diffusion coefficient D", which,similarly to the original Green-Kubo formula (4), is givenin terms of an integral over a function of a single variable.Determining the diffusive behavior of a system from itscorrelation function thus amounts to determining theexponent " and the scaling function #!s".While we derived Eq. (9) in terms of position and

velocity, it holds for any two quantities where one is thetime integral of the other. An example for this is given inSec. III C.

A. Classification

The velocity correlation function Eq. (5) includes twoimportant special cases. The first case consists of long-range stationary correlation functions that exhibit apower-law decay for large !,

Cv;s!!"≃ Cs!"−2; (10)

with 1 < " < 2 and Cs > 0 a constant. This is obtainedfrom Eq. (5) if the scaling function is given by #!s" # s"−2.While this correlation function is stationary, its decay issuch that the time integral in Eq. (4) diverges in the infinite

SCALING GREEN-KUBO RELATION AND APPLICATION … PHYS. REV. X 4, 011022 (2014)

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time limit so that the usual Green-Kubo formula is notapplicable. From the scaling Green-Kubo relation Eq. (9),we obtain

hx2!t"i≃ 2D!;st! with D!;s #Cs

!!! − 1"; (11)

where the subscript s stands for stationary. In this case, theanomalous diffusion coefficient can also be expressed interms of a fractional derivative of the stationary velocityautocorrelation function as was shown by Kneller [39].However, as we will see below, the stationary correlationfunction may not be sufficient to describe the long-timediffusivity of the system. The second special case is ! # 2,where we have the usual type of aging correlation function,which is of the form [5,6,8,40]

Cv!t$ "; t"≃ C#!"t

": (12)

Since in this case, there exists no stationary correlationfunction, Eq. (4) is not applicable. For the aging type ofcorrelation function, Eq. (12), we find ballistic behavior,

hx2!t"i≃ 2D2t2

with D2 #C2

Z!

0ds!s$ 1"−2#!s": (13)

For ! > 2, we refer to the correlation function Eq. (5) assuperaging, since then, in addition to the dependence on theage of the system via the scaling function like in the usualaging case Eq. (12), the overall value of the correlationfunction also increases with the age of the system. A similartype of correlation function, although with a logarithmictime dependence of the prefactor, has been found for arandom walker in a random environment [41].Summarizing, we thus have four different classes ofcorrelation functions: Stationary, integrable correlationfunctions are covered by the usual Green-Kubo formulaEq. (4). By contrast, stationary, nonintegrable correlationfunctions and aging correlation functions, which we canfurther differentiate into aging and superaging ones, are allencompassed by the scaling Green-Kubo relation Eq. (9).Three important examples for physical models whosecorrelation functions belong to one of the latter threeclasses and thus follow the asymptotic behavior, Eq. (5),are discussed in Sec. III.The occurrence of the scaling function #!s" in Eq. (5)

captures the fact that the age t of the system also setsthe time scale over which the correlations decay; i.e., thecorrelation time increases linearly with the age of thesystem. For the usual exponential kind of correlationfunctions, the correlation time is constant and it providesa typical time scale. In aging systems, no constant corre-lation time exists, or it is much larger than the time scalesover which the system is observed. Then, the only relevant

time scale is the age of the system. If the correlation timeincreases slower or faster than linearly with the age of thesystem, the diffusion may be retarded (and even subdiffu-sive) or accelerated compared to Eq. (9), depending on theasymptotic behavior of the scaling function #!s". The casewhere the correlation time depends sublinearly on theage of the system is also referred to as subaging [42,43].This retarded and accelerated aging is discussed in theAppendix.

B. Application to experimental data

Before we start discussing the application of the scalingGreen-Kubo relation to specific physical systems, weoutline its application as a data analysis method for ageneric velocity correlation function that has been obtainedexperimentally or numerically. To this end, we use thevelocity correlation function generated by performingnumerical Langevin simulations of free diffusion underthe influence of a friction force that is asymptoticallyinversely proportional to the velocity (see Sec. III A fordetails). For now, we do not discuss the specifics andproperties of this system but instead just use it as a way togenerate the data for the correlation function. ApplyingEq. (9) to the measured correlation function requires us toidentify the scaling behavior of the latter. This is done in thefollowing way, as illustrated in Fig. 1: First, we obtain thecorrelation function as a function of the time lag " fordifferent initial times t and plot it as a function of "=t[Figs. 1a and 1b]. Then, we rescale the resulting curves byt!−2, the correct value of ! being obtained when the datacollapse onto a single curve [Fig. 1c]. This determines thediffusion exponent !. The single rescaled curve is then thefunction #!s" in Eq. (9). Multiplying #!s" by !1$ s"−! andnumerically integrating the resulting curve, we obtain thegeneralized diffusion coefficient D!. In Fig. 1d, wecompare the results obtained in this way to the mean-square displacement obtained directly from the numericalsimulations by integrating the velocity process and also tothe asymptotic analytical one [see Eqs. (31)and (32)], andwe find good agreement within the accuracy of thenumerical simulations. Compared to determining the dif-fusion exponent and coefficient directly from the data forthe mean-square displacement, this rescaling method hasthe advantage that it does not require any fitting of the data.

C. Persistence of initial conditions

Previously, we discussed the long-time limit of the mean-square displacement of a process starting at time t # 0.However, since the systems we are interested in exhibitlong-time correlations, the previous history of the system,i.e., the initial conditions, may have a significant impact onthe dynamics. In order to discuss the effect of the initialcondition, let us consider a slightly different situation. Westill assume that the long-time asymptotic form of thecorrelation function is given by Eq. (5). However, now we

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consider the relative mean-square displacementhΔx2!t"it0 # h!x!t$ t0" − x!t0""2i with respect to somelong aging time t0. We can think of t0 as an “equilibrationtime”: The process starts with a sharp initial condition [e.g.,v!0" # 0, x!0" # 0], and we let it evolve for a time t0before starting to measure the mean-square displacement.Note that, generally, we consider nonstationary systems,which possess no actual equilibrium state, hence thequotation marks on equilibration time. By definition, therelative mean-square displacement is given by

hΔx2!t"it0 # h!x!t0$ t"−x!t0""2i

#Z

t0$t

t0dt2

Zt0$t

t0dt1Cv!t2;t1"

≃2CZ

t

0dt2

Zt2

0dt1!t1$ t0"!−2"

!t2− t1t1$ t0

": (14)

Introducing the variables s # !t2 − t1"=!t1 $ t0" andz # t2=t, we can write this as

hΔx2!t"it0 ≃ 2Dt=t0! t!

with Dt=t0! # C

Z1

0dzz!−1

!1$ 1

z tt0

"!−1

!Z

z tt0

0ds!s$ 1"−!"!s": (15)

This resembles the scaling Green-Kubo relation (9), how-ever with an anomalous diffusion coefficient Dt=t0

! thatformally depends on the ratio t=t0. In the limit t ≫ t0, wecan neglect the second term in parentheses and take theupper bound of the second integral in Eq. (15) to infinity toobtain the expression given by the scaling Green-Kuborelation (9) independent of t0,

D∞! # D! #

C!

Z∞

0ds!s$ 1"−!"!s": (16)

For finite equilibration times, the scaling Green-Kuborelation, together with the aging correlation functionEq. (5), thus gives the “true” long-time limit for themean-square displacement.In the opposite limit t ≪ t0, on the other hand, the

behavior is governed by the small-s expansion of thescaling function "!s",

Dt=t0! ≃ C

!t0t

"!−1 Z 1

0dz

Zz tt0

0ds"!s": (17)

Generally, if the second moment of the velocity either isasymptotically constant or increases with time,hv2!t"i # C!t; t"≃ aCt#, with some positive constant aand 0 ≤ # < ! − 1, continuity demands "!s"≃ cls−$l with$l # 2 − !$ # for small s, and thus,

Dt=t0! ≃ clC

!! − # − 1"!! − #"

!t0t

"#: (18)

Consequently, we find for the relative mean-square dis-placement,

hΔx2!t"it0 ≃ 2clC

!! − # − 1"!! − #"t#0t

!−#: (19)

This is obviously different from Eq. (9); in particular, thediffusion exponent is different, and the anomalous diffusioncoefficient grows with the aging time t0. This shows thatthe history of the process and, by extension, the initial stateof the systems at time t0 influence the diffusive behavior.For # > 0, where the second moment of the velocityincreases with time and thus no stationary correlation

0 1000 2000 3000 400010-1

100

101

102

Cv(

t+!,

t)

Cv(

t+!,

t)

Cv(

t+!,

t)

t + !

t

0 5 10 1510-1

100

101

102

! /t

t

0 5 10 1510-2

10-1

100

/t2-

"

! /t100 1000

105

106

107

108

#x2 (t)

$

t

FIG. 1 (color online). Computation of the diffusion exponent and coefficient for cold atoms in optical lattices from the correlationfunction using the scaling Green-Kubo formula Eq. (9). (a) Correlation function as a function of the total time t$ % for different valuesof t. (b) Correlation function plotted as a function of %=t. (c) After rescaling by t!−2 with ! # 2.37. (d) Mean-square displacementobtained by integrating over the rescaled correlation function "!s" (empty circles) compared to the analytical result Eqs. (31) and (32)(red line) and the mean-square displacement obtained directly from the Langevin simulation (blue squares). From the rescaling andintegration procedure, we obtain !≃ 2.37 and D! ≃ 0.321, which agrees well with the asymptotic analytical results ! # 2.37 andD! # 0.323. The parameters for the simulation areD # 3.25, & # 5.10, and vc # 1.20 [see Eq. (22)], with 106 trajectories and 2.5 ! 105

integration steps per trajectory.

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function exists, this is not surprising. If we let theprocess evolve for a longer time, the overall velocity willincrease and thus diffusion becomes faster. Of particularinterest is the case ! ! " − 2 > 0 that occurs forthe superaging regime of the examples discussed inSecs. III A and III B. Here, we find from Eq. (19) quasi-ballistic diffusion with an anomalous diffusion coefficientthat increases with t0. In this case the scaling function isasymptotically constant, #"s#≃ cl for small s.If the stationary correlation function Eq. (10) exists, the

second moment of the velocity is asymptotically constantand thus ! ! 0. We then have, with Cs ! aC,

D0" !

Cs""" − 1#

! D";s: (20)

This is precisely the stationary result Eq. (11). In this case,we thus have two values for the anomalous diffusioncoefficient,

hΔx2"t#it0 ≃!

2D"t" for t ≫ t02D";st" for t ≪ t0;

(21)

both of which are independent of t0. Note that even thoughin this case the scaling function diverges at small s,#"s#≃ cl s"−2, the correlation function itself is of coursefinite in the limit $ ! 0 as Eq. (6) only gives the long timebehavior of the latter. A particular case of Eq. (21) is " ! 2,! ! 0, when the aging correlation function Eq. (12) tends toa constant that takes the place of the stationary correlationfunction for t ≫ $. An example for this is the agingcorrelation function for the Lévy walk (see Sec. III C).Which of the two values D" and D";s in Eq. (21) correctlydescribes the long-time diffusive behavior depends on theinitial preparation of the system: If the system is not initiallyin the stationary state (i.e., t0 is finite), then we need to usethe aging correlation function Eq. (5) and obtain D" for thediffusivity. On the other hand, if the system is initiallyprepared in the stationary state (i.e., infinite t0), then it iscorrectly described by the stationary correlation functionEq. (10), and we obtain D";s for the diffusivity. The initialcondition of the system thus persists even in the long-timelimit. This is in contrast to systems where the correlationsdecay exponentially and the long-time diffusivity is uniquelydetermined by the Green-Kubo formula (4) independent ofthe initial condition. Examples for this persistence of theinitial condition are discussed in Sec. III.

III. APPLICATION TO THREESCALE-INVARIANT SYSTEMS

A. Diffusion with a 1=v friction force

Our first example for a scale-invariant system to whichour scaling Green-Kubo formula is applicable is diffusionwith a friction force that is inversely proportional to thevelocity. The velocity dynamics are governed by theFokker-Planck equation,

∂tW"v; t# ! ∂v$−F"v#W"v; t## %D∂vW"v; t#&; (22)

for the velocity probability density W"v; t#, D being thevelocity diffusion coefficient. We take the friction force tobe asymptotically inversely proportional to the velocity,

F"v#≃− %v2cv

for v ≫ vc; (23)

with friction coefficient % and characteristic velocity vc. Aconcrete example would be

F"v# ! − %v1% v2=v2c

; (24)

which is linear for small velocities and has the desiredbehavior (23) for large velocities. This particular form ofthe friction force occurs in the semiclassical description ofcold atoms in dissipative optical lattices [13,14]. It turns outthat the long-time dynamics depend crucially on the ratio ofthe prefactor %v2c of the friction force and the diffusioncoefficient D. Specifically, we introduce the parameter

& ! %v2c2D

% 1

2; (25)

in terms of which the stationary solution of Eq. (22) for& > 1 is given by [44]

Ws"v#!1

Z

"1%v2

v2c

#12−&

with Z!$$$'

pΓ"&−1#vc

Γ"&−1=2#; (26)

where Γ"a# denotes the Gamma function. This stationaryvelocity distribution has power-law tails, meaning that,depending on the exponent, different moments of thestationary distribution will be infinite. In particular, thesecond moment diverges when & < 2, which impliesinfinite kinetic energy Ek ! mhv2i=2. For cold atoms inoptical lattices, both the power-law tailed probabilitydensity and the divergence of the kinetic energy have beenobserved in experiments [45,46]. Since an infinite kineticenergy is nonphysical, it is clear that a time-dependentsolution to the Fokker-Planck equation (22) is required tocapture the dynamics. Asymptotically for large velocitiesand long times, this time-dependent solution is given by theinfinite covariant density derived in Ref. [47],

WICD"v;t#≃8<

:

1ZΓ"&#

%jvjvc

&1−2&Γ

%&; v2

4Dt

&for &>1

1Γ"1−&#"4Dt#&−1jvj1−2&e− v2

4Dt for &<1;

(27)

with the upper incomplete Gamma function Γ"a; x#. Inconjunction with the stationary solution, the infinite covar-iant density determines the asymptotic behavior of allmoments, in particular,

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hv2!t"i≃

8>><

>>:

v2c2!!−2" for ! > 2

v2!−1cZ!!!"!2−!" !4Dt"2−! for 1 < ! < 2

!1 − !"4Dt for ! < 1.

(28)

For ! < 2, where the second moment of the stationarydistribution is infinite, the kinetic energy thus increaseswith time. The corresponding velocity correlation functionwas derived in Refs. [15,48] and has the asymptoticbehavior

Cv!t# "; t"≃8<

:v2!−1c

!!#

p

Z!!!"!!!#1"!4Dt"2−!f!!"t" for !> 1

#!!1−!"!!!#1"4Dtg!!"t" for !< 1;

(29)

with the functions f!!s" and g!!s" defined by

f!!s"$ s2−!Z

∞

0dyy2e−y2 1F1

"3

2;!#1;y2

#!!!;y2s";

g!!s"$ sZ

∞

0dyy2e−y2 1F1

"3

2;!#1;y2

#e−y2s; (30)

where 1F1!a; b; x" is a hypergeometric function.For ! < 3, the correlation function Eq. (29) is precisely

of the type (5), and we can apply our scaling Green-Kuborelation (9) and immediately obtain, for the diffusionexponent,

$ $$4 − ! for 1 < ! < 3

3 for ! < 1.(31)

Similarly, the anomalous diffusion coefficient isgiven by

D$%$

&2$

8<

:

42−!!!!−12"

!!!#1"!!!"!!!−1"!4−!"

Z∞

0ds!s# 1"!−4f!!s" for 1 < ! < 3

4!!#

p

3!!!"!!1−!"

Z∞

0ds!s# 1"−3g!!s" for ! < 1;

(32)

with the time and length scales % $ v2c=D and & $ v3c=D.For ! > 3, the usual Green-Kubo formula (4) holds. We canthus identify three different diffusion phases [49]: normaldiffusion with hx2!t"i ∼ t when ! > 3, anomalous super-diffusion with hx2!t"i ∼ t4−! when 1 < ! < 3, andRichardson-like diffusion with hx2!t"i ∼ t3 when ! < 1,which is known from turbulence [50] and has recently beenobserved as a transient behavior in ordinary Brownianmotion [51].As the age of the systems tends to infinity, t ≫ ", the

velocity correlation function Eq. (29) attains a stationarylimit for ! > 2 [14],

Cv;s!""≃ v2!−1c#!!! − 2"4Z!2!! − 1

2"!4D""2−!: (33)

Applying the scaling Green-Kubo relation Eq. (9) to thestationary correlation function Eq. (33), we find for2 < ! < 3, by virtue of Eq. (11),

D$;s

&2%−$$ 41−!

!!!#

p

!!! − 12"!! − 2"!4 − !"!3 − !"

: (34)

Both the stationary and nonstationary expressions for theanomalous diffusion coefficient are plotted in Fig. 2.Clearly, the two results differ substantially as the systemapproaches the aging phase ! < 2. Both results describe the

1 2 3 4 5 60.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

D1 = D1,sD!

D!

/( "2

#-!)

$

D! ,s

FIG. 2 (color online). Diffusion coefficient D$=!&2%−$" (32) forcold atoms in optical lattices as a function of the parameter ! (redline); the result for the normal diffusive case ! > 3 was takenfrom Ref. [53]. The result D$;s from using the stationarycorrelation function Eq. (34) is shown in blue. The coefficientD$ diverges at the transition from normal to anomalous super-diffusion (! $ 3) and vanishes at the transition from super-diffusion to t3 diffusion (! $ 1). While the values obtained fromthe aging and stationary correlation functions agree if thesystem is in the normal phase ! > 3, they differ significantlyas the system becomes more superdiffusive, clearly signifying thedependence on the initial condition. As ! approaches thetransition to aging behavior at ! $ 2, the stationary correlationfunction ceases to provide a good description of the finite timesystem and the corresponding diffusion coefficient diverges.

SCALING GREEN-KUBO RELATION AND APPLICATION … PHYS. REV. X 4, 011022 (2014)

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asymptotic long-time behavior but correspond to differentinitial conditions. For the stationary expression D!;s,Eq. (34), the initial velocity probability distribution isthe stationary power-law distribution, Eq. (26). By con-trast, if the system starts out with a narrow (e.g., Gaussian)velocity initial condition, the asymptotic mean-squaredisplacement is described by D!, Eq. (32). InRefs. [22,23], it was observed that these two classes ofinitial conditions lead to a qualitatively different timeevolution of the velocity probability density. Our scalingGreen-Kubo relation shows that for 2 < " < 3, the initialcondition also persists in the diffusive behavior of thesystem in the form of different expressions for theanomalous diffusion coefficient. For " < 2, there is nostationary velocity correlation function and the systemages. Here, Eq. (31) and (32) describe the diffusivebehavior on time scales that are much longer than theinitial relaxation time of the system [see also Eq. (15)]. Forlong relaxation times, the diffusion becomes quasiballistic,as was discussed in Ref. [52].The diffusion coefficient D! (32) can be seen (Fig. 2) to

diverge at the transition from normal to superdiffusion(" ! 3) and, in contrast, to vanish at the transition fromsuperdiffusion to t3 diffusion (" ! 1). This irregularbehavior of the diffusion coefficient at the transitionsjustifies referring to the three different regimes as diffusionphases, as it indicates a change in the qualitative dynamicsof the system.

B. Fractional Langevin equation

Next, we will consider the motion of a tracer particle in acrowded environment under the influence of external noise.The motion is governed by the fractional Langevin equa-tion [24,25,28,54–59]

v: "t# ! !

Zt

0dt0#"t ! t0#v"t0# $ 1

m$"t#; (35)

where the memory kernel #"t# describes the retardingfriction experienced by the tracer in a crowded environment[27], like the cytoplasm of biological cells. This frictionkernel is given by

#"t# !#%

!"%#t%!1; (36)

with 0 < % < 1 and a generalized friction coefficient #%.If the noise $"t# is thermal equilibrium noise, it is relatedto the friction kernel via the fluctuation-dissipationtheorem

h$"t2#$"t1#i ! mkBT#"t2 ! t1#; (37)

which expresses the fact that both friction and internalnoise are due to interaction with the thermal bath. In thiscase, Eqs. (35) and (36) describe the motion of the tracerin a crowded equilibrium environment, which is known toexhibit subdiffusion with exponent ! ! 1 ! % [24,25]. If$"t# is considered to be an external noise driving thesystem out of equilibrium, the fluctuation-dissipationtheorem does not hold, and the system may becomesuperdiffusive [28,29]. In living cells, this external noiseis due to the action of molecular motors and it drivesactive transport [28]. It requires an external source ofenergy (e.g., ATP) to continuously keep the system out ofequilibrium [60].Like the equilibrium noise, we take the external noise to

be a Gaussian, power-law correlated process. However, weallow the external noise to be nonstationary, which, asshown in Ref. [29], is necessary to explain the super-diffusion exponent observed in several experiments[60,61]. More precisely, we define the autocorrelation ofthe noise in terms of its Laplace transform ~$"s# !R∞0 dte!st$"t# [29],

h~$"s2#~$"s1#i ! F &"s2s1#!

&2

s2 $ s1; (38)

with & > % and the constantF & describing the magnitude ofthe external noise. The condition & > % ensures that theexternal noise correlations decay more slowly than those ofthe internal noise. In the time domain, the noise correlationsread

h$"t$ '#$"t#i ! "!1#!&2F &

!2"&2#'&!1B

!! t';&2;&2

"; (39)

where B"x; a; b# is the incomplete Beta function. The noiseprocess has a stationary autocorrelation for & < 1 in thelimit t ! ∞, specifically, h$"t$ '#$"t#i ∝ '&!1. For & > 1,on the other hand, the noise correlations are nonstationary;here, the variance of the external noise increases with timeas h$2"t#i ∝ t&!1 [62].The velocity autocorrelation can be obtained by solving

Eq. (35) in Laplace space and then inverting theLaplace transform. The exact solution can be expressedin terms of the generalized Mittag-Leffler functionEa;b"x# [29],

Cv"t$ '; t# ! F &

m2

Zt

0dt0G%;&"' $ t0#G%;&"t0#

with G%;&"t# ! t&2E%$1;&2$1"!#%t%$1#: (40)

The scaling properties of this expression becomeapparent when considering the limit of long times', t ≫ #!1="1$%#

% ,

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Cv!t" !; t#≃ F "

m2#2$!!"2 − $#!!"2 − $" 1#

!!t" !#"2−$−1t"2−$ " !−1#$−"

2

""2− $ − 1

#!"−2$−1

! B"− t!;"2− $" 1;

"2− $ − 1

#$: (41)

Casting this in the form of Eq. (5), we have

Cv!t" !; t#≃ Ct%−2&"!t

#with % $ " − 2$" 1; C $ F "

m2#2$!!%−12 #!!%"12 #

and &!s# $ !1" s#%−32 " !−1#1−%2 % − 3

2s%−2B

"− 1

s;%" 1

2;% − 3

2

#: (42)

We can immediately read off the diffusion exponent% $ " − 2$" 1 and find that the system is superdiffusivefor " > 2$. Using our scaling Green-Kubo relation (9), wecan then calculate the anomalous diffusion coefficient byintegrating the scaling function &!s#,

D% $F "

m2#2$

1

2%!2!%"12 #

: (43)

In the regime 1 < % < 2, that is, for sub-ballistic super-diffusion, the velocity autocorrelation Eq. (41) attains astationary limit,

Cv;s!!#≃ F "

'm2#2$!!2 − %# sin

"'% − 1

2

#!%−2: (44)

Similarly to Sec. III A, we can apply the scaling Green-Kubo relation Eq. (9) to both Eqs. (42) and (44). From thelatter, we obtain

D%;s $F "

m2#2$

!!2 − %# sin !' %−12 #

'%!% − 1#: (45)

The results for the anomalous diffusion coefficient areshown in Fig. 3. Once again, we observe significantdifferences between using the stationary and aging corre-lation functions as the diffusion exponent increases. Incontrast to the system discussed in Sec. III A, the stationarystate is not fully described by the corresponding velocitydistribution. Since the fractional Langevin equation (35) isnon-Markovian, we need to specify the entire history of theprocess to determine the correct stationary state. Evenif the velocity is initially distributed according to thestationary distribution, which is Gaussian with widthhv2is $ !F "=m2#

R!0 dtG2

$;"!t#, we still obtain the nonsta-tionary result Eq. (43) for the diffusion coefficient. Thestationary result Eq. (45) is only obtained if we explicitly letthe process evolve for some long time t0 before starting themeasurement of the mean-square displacement.In the regime % > 2, the velocity autocorrelation Eq. (41)

is superaging and the variance of the velocity increases with

time. The results for the diffusion exponent % $ " − 2$" 1and the diffusion coefficient D%, Eq. (43), remainunchanged. However, similarly to the superaging case inSec. III A, we then find faster-than-ballistic diffusion in thisregime.

C. Blinking quantum dots and Lévy walk

As our final example for the application of the scalingGreen-Kubo relation, we study the time-dependent inten-sity I!t# of the light emitted by nanocrystals embedded in adisordered medium. This deviates from the previous dis-cussion insofar as the observables of interest are no longervelocity and position. Instead, we identify the velocity withthe intensity and the total number of emitted photons,which is proportional to the time integral over the intensity,as the position. The intensity exhibits a phenomenon calledblinking: The quantum dot will emit light during some

1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

D!

D! ,s

D!

/(F "

# $-2)

!

FIG. 3 (color online). The anomalous diffusion coefficientD%=!F "#−2$ # for the fractional Langevin equation as a functionof the diffusion exponent %. The red line is the result using theaging correlation function Eq. (42), and the blue line is the onefrom the stationary correlation function Eq. (44). Close to thethreshold to normal diffusion % $ 1, both values agree; however,as the diffusion exponent increases, the two results deviate fromeach other. At the transition to the nonstationary phase (% $ 2),the stationary result diverges.

SCALING GREEN-KUBO RELATION AND APPLICATION … PHYS. REV. X 4, 011022 (2014)

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periods and remain dark during others [63–65]. Thedurations of these on periods and off periods are foundto be distributed according to a power law whose exponentis such that the average on and off time is infinite (i.e., ofthe order of the measurement time), leading to aging in theintensity autocorrelation function [7,66]. This model is thefamiliar Lévy walk [12,31,36,67], which is a popularstochastic framework with many applications. As men-tioned in the Introduction, Zumofen and Klafter [12]showed that the diffusivity in this model is sensitive tothe initial conditions. Here, we demonstrate the applicabil-ity of our scaling Green-Kubo relation, while previousworks used renewal theory approaches [7,34].We consider the special case where the on (!) and

off (−) times of a blinking quantum dot follow thesame power-law distribution ψ!"t# $ ψ−"t# $ ψ"t#∼"t=tc#−μ−1 for t ≫ tc with 0 < μ < 2. Thus, the dot is inthe on state with intensity I $ 1 for a waiting time drawnfrom ψ"t#, after which it switches to the off state I $ 0,again with a dwell time drawn from ψ"t#, and so on. Theaverage on/off time t̄ only exists for μ > 1 and is infinite for0 < μ < 1. For this form of the distribution, the agingintensity autocorrelation for long times and 0 < μ < 1 hasbeen explicitly obtained in Refs. [7,34] by mapping theproblem onto a Lévy walk,

hI"t! τ#I"t#i≃ 1

2

!1 − sin"πμ#

2πB!

1

1! tτ

; 1 − μ; μ""

:

(46)

Comparing this to the general scaling form of the corre-lation function Eq. (5), we find ν $ 2, C $ 1=2, and

ϕ"s# $ 1 − sin"πμ#2π

B!

s1! s

; 1 − μ; μ": (47)

Figure 4 shows the correlation function Eq. (46) for μ $ 0.6obtained from numerical simulations. Plotting the intensityautocorrelation as a function of τ=t results in a data collapsewithout any rescaling, indicating the pure aging behaviorfor 0 < μ < 1. The total number of photons n"t# emitted bythe quantum dot during time t is proportional to the timeintegral of the intensity, n"t# $ I0

Rt0 dt

0I"t0#, with I0 thephysical intensity of the dot in the on state. Consequently,ν $ 2 in the scaling Green-Kubo relation Eq. (9) immedi-ately implies that the second moment of the photon numberexhibits ballistic scaling [see Eq. (13)] [31]:

hn2"t#i≃ I20

2

!1 − μ

2

"t2: (48)

Since the average intensity is asymptotically constant,hI"t#i≃ 1=2, we find, for the variance of the photonnumber,

hδn2"t#i $ hn2"t#i − hn"t#i2 ≃ 2D2t2

with D2 $I20

8"1 − μ#; (49)

which also scales ballistically. This agrees with the resultobtained in Ref. [38] using the probability distributionP"n; t# of the photon number and with the experimentalobservation of a linear increase of Mandel’s Q parameterQ $ hδn2"t#i=hn"t#i [68].For 1 < μ < 2, we find, from the general results of

Ref. [7],

hI"t! τ#I"t#i≃1

4

#1! tμc

"μ−1#t̄%τ1−μ− "t! τ#1−μ&

$; (50)

which for long times t ≫ τ approaches the stationarylimiting form

hI"t! τ#I"t#is ≃1

4

#1 − tμc

"μ − 1#t̄τ1−μ

$: (51)

Because of the constant term, these expressions are notprecisely of the scaling form (5), respectively, (10). Thisconstant term leads to a ballistic contribution in the secondmoment of the photon number. However, by consideringthe fluctuations of the intensity δI"t# $ I"t# − 1=2, whichare related to the fluctuations of the photon number byδn"t# $ I0

Rt0 dt

0δI"t#, we can once again apply the scalingGreen-Kubo relation (9) to find sub-ballistic scaling for thevariance of the photon number [31]

hδn2"t#i≃ 2Dνtν; (52)

0 5 10 150.25

0.30

0.35

0.40

0.45

0.50t = 5000t = 10000t = 25000t = 50000

!I(t+

")I(t

)#

"/t

FIG. 4 (color online). Aging correlation function for theblinking quantum dots. The plots were obtained from numericalLévy walk simulations with tc $ 1 and μ $ 0.6. Since thecorrelation function is of the usual aging type (ν $ 2), norescaling is necessary for the curves to collapse onto a singleone. Integrating this curve, we obtain the coefficient D2 ≃ 0.043in good agreement with the result D2 $ 0.05 from Eq. (48).

A. DECHANT et al. PHYS. REV. X 4, 011022 (2014)

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with ν ! 3 ! μ, and

Dν !I20t

μc

4"3 ! μ#"2 ! μ#t̄; (53)

respectively,

Dν;s !I20t

μc

4"3 ! μ#"2 ! μ#"μ ! 1#t̄: (54)

Similarly to our previous two examples, we have twodifferent expressions for the anomalous diffusion coeffi-cient depending on the initial preparation of the system. Asnoted, this is in agreement with the result obtained inRef. [12] using a different approach. Both expressions forthe diffusion coefficient are shown in Fig. 5. In contrast tothe cases discussed in Secs. III A and III B, the agingcorrelation function Eq. (46) does not increase as a functionof t for t ≫ τ but tends to a constant value. As aconsequence, the stationary diffusion coefficient does notdiverge as we approach the aging regime μ < 1. For μ < 1,there is no actual stationary correlation function; however,depending on whether the measurement time is large orsmall compared to the aging time t0, we still obtain twodifferent values for the anomalous diffusion coefficient (seeSec. II C). Finally, for μ > 2, the usual Green-Kubo relationis applicable and the variance of the photon numberincreases linearly with time.While the explicit calculation of the intensity correla-

tions and photon statistics is possible for the two-statemodel used in Refs. [7,38], this is no longer the case formore realistic models that take into account that there may

be more than single on-state or exponential cutoff on thepower-law statistics of on and/or off times [69]. Since ourscaling Green-Kubo relation Eq. (9) only relies on theasymptotic scaling of the intensity correlation function, itcan be used to relate intensity correlations and photonstatistics even for models that are only tractablenumerically.

IV. DISCUSSION

The scaling Green-Kubo relation (9) extends the usualGreen-Kubo formula (4) to the superdiffusive, nonsta-tionary regime. It enables the direct evaluation of themean-square displacement hx2"t#i from the knowledgeof the scaling properties of the velocity correlation functionCv"t$ τ; t#. In particular, a scaling function replaces thestationary autocorrelation in the usual Green-Kubo formulafor the determination of the diffusion coefficient. Thescaling Green-Kubo relation is applicable to slowlydecaying stationary power-law correlation functions, aswell as nonstationary ones exhibiting aging or even super-aging. For the latter two classes, it relates, for the first time,the anomalous diffusion coefficient to the scaling of thecorrelation function in a simple way.This relation between an aging correlation function and

the diffusion coefficient turns out to be important even incases where a stationary velocity correlation function doesexist but decays slowly. Different classes of initial con-ditions lead to different expressions for the diffusivity: Dν;s[Eqs. (34), (45) and (54)] for a stationary initial conditionand Dν [Eqs. (32), (43) and (53)] if the system is notinitially in the stationary state (see also Figs. 2, 3 and 5).Generally, which one of these two expressions for thediffusivity is appropriate depends on the time scales one isinterested in: As long as the measurement time is smallcompared to the aging time t0 [see Eq. (15)], the anomalousdiffusion coefficient is given by the stationary value Dν;s.Once the measurement time becomes comparable to t0, thediffusion coefficient will start to deviate from its stationaryvalue, eventually reaching the nonstationary value Dν fortimes much longer than t0. Because of this sensitivity on theinitial conditions, the measurement and interpretation ofanomalous transport coefficients is not as straightforwardas for normal transport, where the coefficients are generallyunique. For cold atoms in optical lattices (Sec. III A) andthe fractional Langevin equation (Sec. III B), the stationarydiffusion coefficientsDν;s diverge as we approach the agingregime. The persistence of the initial condition is directlyrelated to the scale invariance of the velocity correlationfunction: There exists no typical time scale on which thecorrelations decay, and thus, the only relevant time scale isset by the aging t0, which is not an intrinsic property of thesystem but an external parameter determined by its initialpreparation.Here, we discussed the persistence of the initial con-

dition in the context of superdiffusive dynamics. Recently,

0.0 0.5 1.0 1.5 2.00.0

0.2

0.4

0.6

0.8

1.0

D!/

(A2

t cµ-

1 )

µ

D! ,s

D!

FIG. 5 (color online). The anomalous diffusion coefficientDν="I2

0tμ!1c # for the variance of the photon number. Here, we

assumed a waiting-time distribution of the form ψ"t# ! 0 for t <tc and ψ"t# ! μ=tc"t=tc#!μ!1 for t > tc. The stationary result Dν;s[Eq. (54), blue line] differs from the nonstationary one Dν[Eq. (53), red line] but is finite at the transition to the agingphase μ < 1. The blue dashed line is the result obtained for longaging times t0 in the aging regime.

SCALING GREEN-KUBO RELATION AND APPLICATION … PHYS. REV. X 4, 011022 (2014)

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the dependence of the diffusion coefficient on the initialcondition has been quantified for single-file diffusion,which is subdiffusive and can be related to fractionalBrownian motion [70], indicating that this effect may beof general importance for long-range correlated systems.Another interesting question is how the sensitivity to initialconditions translates to the time-averaged diffusivity,where, instead of averaging over an ensemble of trajecto-ries at a given time, the mean-square displacement iscomputed from a time average over a single trajectory.The dependence of the time-averaged diffusivity on theinitial conditions has so far only been discussed for specialcases [71–74], and whether it will correspond to thestationary or nonstationary expression obtained here orto neither is an open question.We have shown that the three different types of scaling

correlation functions (stationary, aging, and superaging)

appear in a variety of physical models. The three examplesconsidered in this work are of rather different types ofstochastic models, yet they can all be treated using thescaling Green-Kubo relation: The diffusion with a 1=vfriction force obeys a Markovian, nonlinear, non-GaussianLangevin dynamics, while in the case of the fractionalLangevin equation, the dynamics is linear and Gaussian butnon-Markovian. The power-law blinking of quantum dotscan be mapped onto a Lévy walk, which represents yetanother class of stochastic models and is neither Markoviannor Gaussian. Sample trajectories for the three models areshown in Fig. 6, highlighting the differences between them.Yet because of their underlying scale invariance, the scalingGreen-Kubo relation can be used in all three cases todetermine the diffusive ensemble properties, which aresummarized in Table I. Our scaling Green-Kubo relationthus appears to be widely applicable and should therefore

0 20 40 60 80 100-4

0

4

time

-4

0

4

velo

city

-4

0

4

0 20 40 60 80 100

-20

0

20

time

-20

0

20

posi

tion

-20

0

20

FIG. 6. Sample trajectories for the three systems discussed in Sec. III. Top: Diffusion with a nonlinear 1=v friction force. Middle:Fractional Langevin equation with external noise. Bottom: Lévy walk alternating between !v0. The left figure depicts the velocity of asingle trajectory as a function of time; the right figure shows the corresponding position. The parameters were chosen such that thediffusion exponent for all three systems is ν≃ 1.5 and the (dimensionless) anomalous diffusion coefficient isDν ≃ 0.25. Despite similardiffusive properties, the actual trajectories are apparently very different.

TABLE I. Summary of the aging and diffusive properties for the three systems discussed in Secs. III A, III B and III C. The firstcolumn lists the system and the relevant parameters. The second column states the parameter range in which the scaling Green-Kuborelation is applicable; the resulting diffusion exponent is given in the third column. The aging properties of the correlation function aregiven in the fourth column. The fifth column states whether the diffusion coefficient depends on the initial condition and, if so, whether itexplicitly depends on the aging time t0.

ScalingGreen-Kubo Diffusion exponent ν Correlation function Diffusion coefficient

Nonlinear 1=v friction Frictionparameter 1

2 < α < !α < 3 1 for α > 3

4 − α for 1 < α < 33 for α < 1

Stationary for α > 2Superaging for α < 2

Unique for α > 3Dν " Dν;s for 2 < α < 3Dν " Dt=t0

ν for α < 2Fractional Langevin equationFriction exponent 0 < ρ < 1Noise exponent λ > ρ

λ > 2ρ λ − 2ρ" 1 Stationary for λ < 2ρ" 1Superaging for λ > 2ρ" 1

Dν " Dν;s for λ < 2ρ" 1Dν " Dt=t0

ν for λ > 2ρ" 1

Blinking quantum dotsOn/off time exponent μ > 0

μ < 2 1 for μ > 23 − μ for 1 < μ < 2

2 for μ < 1

Stationary for μ > 1Aging for μ < 1

Unique for μ > 2Dν " Dν;s for α < 2

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constitute an important tool for investigating the diffusiveproperties of many systems.

ACKNOWLEDGMENTS

This work was supported by the Israel ScienceFoundation, the Focus Area Nanoscale of the FU Berlinand the DFG (Contract No. 1382/4-1). A. D. also thanks theElsa-Neumann Graduate Funding for support.

APPENDIX: RETARDED ANDACCELERATED AGING

Our scaling correlation function Eq. (5) is of the form

C!t; t" !#≃ Ct"−2#!!t

": (A1)

In this case, the correlation time tc is given by the age t ofthe system. A more general case is a correlation time thathas a different functional dependence on the age; i.e., tc!t#is some monotonously increasing function with tc!0# $ 0.The corresponding correlation function is then of the form

C!t; t" !#≃ Ct"−2#!

!tc!t#

": (A2)

The mean-square displacement is given by

hx2!t#i≃ 2CZ

t

0dt2

Zt2

0dt1t"−21 #

!t2 − t1tc!t1#

": (A3)

We define s $ !t2 − t1#=tc!t1# and obtain

hx2!t#i≃ 2CZ

t

0dt2

Z∞

0ds!−∂t1!s#

∂s"t"−21 !s##!s#; (A4)

where t1!s# is the solution of the equation,

t2 − t1tc!t1#

$ s: (A5)

For general tc!t#, this equation is not solvable analytically.However, the asymptotic behavior of t1!s# is given by

t1!s#≃8<

:t2 − stc!t2# for s ≪ t2

tc!t2#

~tc!t2s# for s ≫ t2tc!t2#

;(A6)

where ~tc!x# denotes the inverse function of tc. For theparticular choice

tc!t# $ ta

!tta

"$; (A7)

with some time scale ta and 0 < $ < 2, this gives us

t1!s#≃8<

:

t2 − sta#t2ta

$$

for s ≪#t2ta

$1−$

ta#s tat2

$−1$ for s ≫

#t2ta

$1−$

:(A8)

For the expression appearing in Eq. (A4), we have

!−∂t1!s#

∂s"t"−21 !s#≃

8<

:

t1−$a t""$−22 for s≪

#t2ta

$1−$

1$ t

!"−1#!$−1#$

a t"−1$2 s

1−"−$$ for s≫

#t2ta

$1−$

:

(A9)

Plugging this into Eq. (A4), we obtain

hx2!t#i≃ 2CZ

t

0dt2

%t1−$a t""$−2

2

Z!t2ta#

1−$

0ds#!s#

" 1

$t!"−1#!$−1#

$a t

"−1$2

Z∞

!t2ta#1−$

dss1−"−$

$ #!s#&: (A10)

The asymptotic behavior of this expression depends on theasymptotic behavior of #!s#, which we assume to be of thefollowing form,

#!s#≃'

cls−$l with $l < 1 for s ≪ 1

cus−$u with $u > 1−"$ for s ≫ 1:

(A11)

The case $ < 1.—For $ < 1, the upper bound of the firstand the lower bound of the second s integral are large fort2 ≫ ta. If $u > 1,wemay thenextend theupperboundof thefirst integral to infinity; otherwise, the integral is dominatedby the value at the upper bound and thus the large-s behaviorof the scaling function. The s integral in the second termonlydepends on the large-s behavior. We then have

hx2!t#i≃ 2C !

8<

:

1""$−1 t

1−$a t""$−1 R∞

0 ds#!s# for $u > 1

cu!1"1$#

""!$−1#$u t!1−$#$ua t""!$−1#$u for $u < 1.

(A12)

We thus have two different results for the diffusioncoefficient and exponent depending on the large-s behaviorof the scaling function: For $u < 1, i.e., when the scalingfunction #!s# decays slowly or even increases at large s, thediffusion coefficient is determined by this large-s behavior.This agrees with the physical intuition, since for $ < 1, thesystem ages more slowly and thus the low-age behavior(corresponding to t ≪ !, respectively, large s in thecorrelation function) is important. For 0 < $u < 1, thediffusion is retarded (i.e., the diffusion exponent is smallerthan "), while it is accelerated for $u < 0. On the otherhand, if the scaling function decays very fast ($u > 1), thecontribution from the low-age regime is numerically smalland of the same order as the high-age one. Thus, the

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diffusion coefficient depends on the detailed shape of !!s"and not only on the large-s expansion. In this case, thediffusion is always retarded and the system may evenbecome subdiffusive for " < 2 − #.The case " > 1—For " > 1, the upper bound of the first

and the lower bound of the second s integral are small for

t2 ≫ ta. If "l < !1 − #"=", we may then extend the lowerbound of the second integral to zero; otherwise, the integralis dominated by the value at the lower bound and thus thesmall-s behavior of the scaling function. The s integral inthe first term only depends on the small-s behavior. Wethen have

hx2!t"i≃ 2C !

8<

:

"##"−1 t

!#−1"!"−1""

a t##"−1

"R∞0 dss

1−#−"" !!s" for "l < 1−#

"

cu!1#1""

##!"−1""l t!1−"""la t##!"−1""l for "l > 1−#

" :(A13)

Similarly to the case " < 1, we find two qualitativelydifferent behaviors, though now depending on the small-sexpansion of the scaling function, as the system now agesfaster and thus the high-age behavior is important. For"l > !1 − #"=", the scaling function diverges at small s orslowly tends to zero, and the high-age behavior (t ≫ $ orsmall s) dominates the diffusion coefficient. Depending onthe sign of %l, the diffusion may be retarded or accelerated.For "l < !1 − #"=", the scaling function rapidly tends tozero for small s, and thus the high-age contribution and thelow-age one are of the same order. As before, the diffusioncoefficient depends on the entire domain of !!s".For " $ 1, both the low-age and high-age contributions

are always of the same order, and the behavior of !!s" overthe whole range of its argument determines the diffusioncoefficient. This underlines the special importance of the“true” scaling type of correlation functions with " $ 1discussed in the main body of the paper and in theexamples.

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