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Multiple scaling regimes in simple aging models Bernd Rinn,1 Philipp Maass,1,2 and Jean-Philippe Bouchaud2 1Fakultät fu¨r Physik, Universität Konstanz, D-78457 Konstanz, Germany 2Service de Physique de l’Etat Condensé, CEA Saclay, 91191 Gif sur Yvette Cedex, France 0 (January 12, 2000) 0 We investigate aging in glassy systems based on a simple model, where a point in configuration 0 space performs thermally activated jumpsbetween theminima ofa random energylandscape. The 2 model allows us to show explicitly a subaging behavior and multiple scaling regimes for the corre- n lation function. Boththeexponentscharacterizingthescalingofthedifferentrelaxationtimeswith a thewaitingtimeandthosecharacterizingtheasymptoticdecayofthescalingfunctionsareobtained J analytically by invokinga ‘partial equilibrium’ concept. 2 1 PACS numbers: 02.50.-r, 75.10.Nr, 05.20.-y ] h c The dynamics of glassy materials can be strongly de- ble occurrenceofsubagingbehaviorandmultiple scaling e pendent on the history of glass formation [1,2]. Gen- regimes. This model, which has a strong resemblance to m erally speaking, one finds that the relaxation dynamics the previously studied “trap model” [8,9], is motivated - becomes increasingly slower with the “age” of the sys- by the simple and widespread view that glassy dynam- t a tem, that means with the time tw expired since the ma- ics may be described by a thermally activated motion of t terialwas broughtinto the glassystate. Such aging phe- a point (“particle”) that jumps among the deep (free) s t. nomena have been identified in many systems and vari- energy minima Ei ofa complex configurationspace. Ac- a ous dynamical probes (for a recent review, see e.g. [3]). cording to extreme value statistics one may expect the m Prominentexamplesareshearstressrelaxationsinstruc- distribution ρ(E) of these deep minima to be exponen- - turalglasses[4], thermoremanentmagnetizationsin spin tial, and indeed, mean-field theories of spin glasses [10] d glasses[5],andelectricfieldrelaxationsindipolarglasses and recent results from molecular dynamics simulations n [6]. A convenient way to quantify aging in such exper- [11] suggest this to be the case. o iments is to disturb the probe at time t by a sudden To be specific, let us consider a d-dimensional cu- c w [ change of external field, and to measure the response bic lattice and assign to each lattice site i an en- R(tw +t,tw) at a later time tw +t. Often, the charac- ergy Ei, −∞ < Ei ≤ 0, drawn from the distribution 1 teristic relaxation time grows proportionally to the age ρ(E)=T−1exp(E/T ). The particle jumps among near- v g g t . When t is larger than all microscopic times asso- est neighbor sites only, and the jump rate from site i to 1 w 6 ciated with fast time-translational invariant relaxations, a neighboring site j is 1 one then expects R(tw + t,tw) to depend on the ratio w =νexp(−β[αE −(1−α)E ]) , (1) 1 t/t only, i.e. R(t +t,t )=F(t/t ). i,j j i w w w w 0 In principle, however, one cannot rule out other scal- where the “attempt frequency” ν≡1 sets our time unit, 00 ing forms, as e.g. R(tw+t,tw) = F(t/tµw) with µ > 0 β−1 ≡ T is the temperature (or thermal energy), and being different from one. In particular the case µ < 1, the parameter α specifies how the energies of the ini- / t which has been called ‘subaging’ because the effective tial and target site are weighted. In order for the w a i,j m relaxation time grows more slowly than the age of the to obey detailed balance, α can assume any real value, system, seems to be of experimental relevance [3]. It is but on physical grounds it is reasonable to restrict α to - d moreoverpossiblethatthereexist,forgivenwaitingtime the range 0≤α≤1. Independent of α, the system un- n t , various scaling regimes in time t, which are governed dergoes a “dynamical phase transition” at T = T : In w g o by different relaxation times ∝ tµs, s = 1,2,.... More the high-temperature phase, where T >T , the proba- w g c precisely, depending on how t is scaled with t , one can bility ϕ(E)∝ρ(E)exp(−βE) for finding the system in a : w v obtain different asymptotic scaling functions in the limit statewith energyE is normalizableandthermalequilib- i t →∞. For example, for µ >µ > 0, one may find rium will be approached with a characteristic equilibra- X w 1 2 r R(tw+Λ1tµw1,tw)∼F1(Λ1)andR(tw+Λ2tµw2,tw)∼F2(Λ2) tion time that diverges for T ցTg. By contrast, in the a whentw →∞. Infact,theoccurrenceofdifferentscaling glassyphase,whereT<Tg andϕ(E)isnotnormalizable, functions being associatedwith varioustime regimeshas the dynamics never becomes stationary but ‘ages’. recently been conjectured on the basis of analytical re- It is important at this point to stress the differences sults for the Langevin dynamics of mean-field spin glass between the above model and the earlier studied trap models [7,3]. So far, however, it was not possible to val- model [8]. In the latter, the jump rates depend only on idate these conjectures, or to exemplify them in some the energy of the initial site corresponding to α=0 in reasonable phenomenologicalmodels. eq.(1),andthisallowsastraightforwardmappingontoa In this Letter we will discuss a model that allows us continuoustime-randomwalkwitha waitingtime distri- to demonstrate for the first time explicitly the possi- bution decaying as a power law. Much more important, 1 Π(t +t,t ) 1−Π(t +t,t ) change state within time t, where the sum over nj runs 100 w w w w over all nearest neighbor sites of j. Exactly two of these neighboring sites are considered to belong to the Brow- nian path in view of its one-dimensional topology. The )w remaining 2(d−1) neighboring sites are assumed to have ,t10−1 not been visited before. Hence, we may deduce the scal- t+tw ~tε ~t−δ ing properties of Π(tw+t,tw) from ( Π 10−2 d=1 Π˜(t +t,t )≡ Sj=(t1w)τjexp −tτjα−1 njτnαj , (2) w w *P (cid:16)Sk=(t1w)τk P (cid:17)+ whereh...idenotesanaveraPgeover(2d−1)S(t )uncor- w 10−1 ε ~t−δ troelathteedproawnderomlawnuφm(τb)er=sτθjτt−h1a−tθarweitdhistθr≡ibTut/eTdg.according ~t Clearly, the partial equilibrium concept is an approxi- mation that needs to be tested. To this end we have de- 10−2 d=3 terminedΠ(t +t,t )andS(t )ind=1,3forvariousval- w w w uesofθandαbyMonteCarlosimulations. Thenwetook 10−5 10−3 10−1 101 103 105 S(t ) from these simulations to calculate Π˜(t +t,t ) w w w µ t/t from eq.(2). The disorder averagein this simple numer- 1 w ical evaluation was performed over a set of realizations FIG. 1. Double-logarithmic plot of Π(tw +t,tw), 1− independent of the ones taken in the simulations. Π(tw + t,tw) (symbols ◦, (cid:3), △, ▽, ⋄) and Π˜(tw + t,tw), Figure 1 shows the results for one representative pa- 1−Π˜(tw+t,tw) (symbols ⊲, +, ×) as functions of t/tµw1 in rameter pair (θ,α)=(1/4,3/8). The data have been col- d=1 and d=3 for (θ,α)=(1/4,3/8). Different symbols cor- lectedasfunctions oft for abroadrangeoffixedwaiting (irn△es)pd,o=1n0d81,t(oa(cid:3)sd),iwff1ee0lrl9ena(ts◦w)twaai=ntidn1g0t6wti(m=⋄)e1,s0,18t0w7(⊲=()▽,1)01,5011(10⋄8)(,+(1△)0),6,1(10▽0193),((1(cid:3)×0)7), ttsiipomencesisfiotefwdΛba1neld=owatr./eAtpµws1locwtatnietdhbeaelsxreepeaondnyfernoitnmssµctha1le=edfiµgfu1o(rrθem,,αtah)sebfdueainntcga- 1010 (◦) and tw=1011 (⊲), 1013 (+), 1015 (×) in d=3. The forbothΠ(t +t,t )andΠ˜(t +t,t )collapseontosingle solidlineshaveslopeǫ(smalltbehavior)andslope−δ (large curves F (Λw) andw F˜ (Λ ), wrespecwtively. Although the t behavior). For thevalueof theexponentsǫand δ, see text. 1 1 1 1 two scaling functions are different, their asymptotic behavior for large Λ is the same, F (Λ )∼F˜ (Λ )∼Λ−δ thetrapmodelwassofarconsideredonlyonamean-field 1 1 1 1 1 1 (seethesolidlinesinFig.1). Thevaluesoftheexponent levelcorrespondingtoan“annealedsituation”,wherethe δ=δ(θ,α) are given below. site energies are drawn anew after each jump. It is important to inspect also more closely the ‘short’ In order to study aging effects in the glassy phase, we time regime, where the complementary probability 1− focus, as in the trap model, on the (disorder averaged) Π(t +t,t ) for the system to change state between t w w w probabilityΠ(tw+t,tw) thatthe systemdoes not change and tw+t is small. This complementary probability can its state betweentw andtw+t. Forparticleshopping on be as relevant as Π(tw+t,tw) dependent on the physical a lattice, this can be interpreted as a dynamical struc- quantitybeingmeasured. Scalingplotsof1−Π(t +t,t ) w w ture factor [8]. Initially, the particle is located on any of and 1−Π˜(tw+t,tw) as functions of Λ1 =t/twµ1 in d=1,3 the sites and then it starts to explore the configuration are also shown in Fig. 1. Again there is a good data col- spaceatsome T<Tg. Physically,this means that weare lapseandforΛ →0wefind1−F (Λ )∼1−F˜ (Λ )∼Λǫ 1 1 1 1 1 1 considering an instantaneous quench from T =∞. with exponents ǫ = ǫ(θ,α) given below. An analogous Our idea to explore the scaling properties of Π(tw+ overallbehavioras displayedinFig. 1 wasfound also for t,tw) for both t and tw becoming large is based on other pairs (θ,α) (with 0< θ<1, 0≤α≤1). We thus the following “partial equilibrium” concept: From the conclude that the partial equilibrium concept not only time after the quench up to tw the particle has fol- yields the correct scaling behavior (same µ1 exponents) lowed a Brownian path in configuration space that on but also the correct asymptotics of the scaling functions average consists of S(tw) distinct and mutually con- [12]. However, a study of the ‘participation ratios’ (see nected sites. On a typical path with S ≃ S(tw) sites [13] for their definition) shows that the partial equilib- then, it is reasonable to think that the particles should rium concept is not exact, even for large times [14]. effectively equilibrate, i.e. the probability to be on a Now we turn to the analytical study of eq. (2). Since particular site j of the path may be approximated by Π˜(t +t,t )depends ont onlyviaS(t )letusfirstdis- w w w w τj/ Sk=1τk, where τj ≡ exp(−βEj) (1≤τj< ∞). Con- cuss the scaling of S(tw) with time tw. For α=0 this ditioned on being at the site j, the system has a proba- problemhasbeenaddressedsometimeago(seee.g.[15]) bilitPy exp(−t njwj,nj) = exp(−tτjα−1 njτnαj) not to and for tw→∞ one finds P P 2 dθ , 1≤d<2 tγ [1−Π(t +t,t )] S(tw)∼tγw, γ = d+(2−d)θ (3) w w w θ, d>2  102 ) θ/α (In d = 2 there are logarithmic corrections, S(tw) ∼ tw ~t [tw/logtw]θ.) For0<α≤1oneexpects(3)nottochange, +t, 100 since α only affects the nearest-neighbor hopping rates w but not the transport properties on large length scales. (t ~t d=1 Π 10−2 Infact,inoursimulationswealwaysfoundeq.(3)tohold true. Let us note also that in d=1 one can give a simple finite-size scaling argumentto show that (3) remains 102 θ/α ~t valid for 0<α≤1. NextwederivethescalingpropertiesofΠ˜ asafunction 100 of t and S = S(t ), and then use eq. (3) to obtain the w corresponding scaling properties of Π˜ as a function of t 10−2 ~t d=3 and t . When replacing the denominator in eq. (2) by w 0∞dλ exp(−λ Sk=1τk),theaverageovertheτj cantoa 10−6 10−4 10−2 100 102 104 large extent be factorized, and one obtains the following µ t/t Rasymptotic formPula valid in the limit of large S w2 θ3S˜ ∞ ∞ dτ ∞ dτ FIG. 2. Double-logarithmic plot of 1−Π(tw+t,tw) and Π˜ ∼= κ dλe−λθS˜ τ1+1θ e−λτ1 τ1+2θ e−λτ2 1−Π˜(tw+t,tw)asfunctionsoft/tµw2 in d=1andd=3forthe Z0 Z1 1 Z1 2 same parameters as in Fig. 1 (the same symbols have been × ∞dτ e−λτ f t 2(d−1)exp −tτ1α+τ2α , used for the various waiting times tw). The solid lines have τθ τ1−α τ1−α slopeoneandθ/α,andindicatethelimitingbehaviorofF2(.) Z1 ∞ hdτ(cid:0) (cid:1)i (cid:16) (cid:17) according toeq. (11). f(x)≡θ exp(−xτα). (4) τ1+θ Z1 and has the limiting behavior Hereκ≡Γ(1−θ),whereΓ(.)istheGammafunction,and S˜ ≡ κS. Note that for t=0 and S˜→∞, eq. (4) yields (1−θ)/(1−α) 1−c Λ , θ >α theAcftoerrretcwtontorramnasfloizramtiaotnioΠn˜s→λ1→. S˜λθ and τ→S˜1/θτ we F˜1(Λ1)∼ 1−c<>Λθ11/α, θ <α Λ1 →0 (7) can identify Λ1=t/S˜(1−α)/θ as a scaling variable corre-  c Λ−θ/(1−α) Λ →∞ spondingtoafirstcharacteristictimet ∼S˜(t )(1−α)/θ ∼ ∞ 1 1 1 w tγw(1−α)/θ ≡tµw1. We thus obtain  The constants c , c , and c can be expressed in terms > < ∞ γ(1−α) of α, θ and d but are not of interest here. Equation (7) µ = (5) 1 θ yields the exponents δ=θ/(1−α) and ǫ=(1−θ)/(1−α) takenin Fig. 1 to characterizethe decay of Π and rise of with γ from eq. (3). This exponent µ has been used to 1 1−Π, respectively. For α=0, one recoversthe results of collapsethedatainFig.1,i.e. wetookµ =(1−α)/(1+θ) 1 the annealed model [8], namely δ=θ and ǫ=1−θ. in d=1 and µ = (1−α) in d=3. Based on the equi- 1 librium concept it is is easy to show that t1 has a sim- For θ > α the function F˜1(Λ1) describes the scaling ple physical interpretation: It scales as the typical max- properties of Π˜ completely. However, based on eq. (4) imum trapping time t (t ) encountered after t . In one finds that for θ < α < 1/2, there exists a second max w w d=1 one thus finds a subaging behavior even for α=0 scaling variable Λ2=t/S˜(1−2α)/θ ∼ t/twγ(1−2α)/θ ≡ t/tµw2 since the deepest trap is visited a large number of times yielding N(t )∼tθ/(1+θ), so that t ∼t (t )∼t /N(t )≪t . w w 1 max w w w w Similarly, for α6=0, the deepest trap is revisited a large γ(1−2α) numberoftimesinalldimensionsdduetoastrongback- µ2 = . (8) θ wardjumpcorrelationwhentheparticleleavesasitewith low energy. Therefore, a second characteristic time scale t ∼ The scalingfunction in the firsttime domaint=Λ1tµw1 S(1−2α)/θ, diverging when t → ∞, governs the be2hav- reads w ior when Π˜ is close to one. (Note that for fixed Λ , F˜ (Λ )= θ3 ∞dλe−λθ ∞ dτ1 ∞ dτ2 (6) Λ1=S˜−α/θΛ2→0 for S˜→∞). In the new time domai2n 1 1 κ τ1+θ τ1+θ one finds the following generalized scaling form, Z0 Z1 1 Z1 2 ∞ dτ Λ 2(d−1) τα+τα ×Z0 τθe−λτhf(cid:0)τ1−1α(cid:1)i exp(cid:16)−Λ1 1τ1−α2 (cid:17), S˜(1−Π˜)=F˜2(Λ2), θ<α<1/2, (9) 3 where F (.) is given by of several distinct scaling regimes in the two-time plane. 2 Such a possibility is of crucial importance for the inter- F˜ (Λ )= ψ∞θ2Λ2 ∞dλe−λθ ∞dτe−λτ g(Λ λ1−ατα), pretationofexperiments,sincethewaitingtime cantyp- 2 2 (1−α)κ λα−θ τ1−α+x 2 ically be varied between one minute and a few days only ∞ Z0 Z0 (with some notable exceptions [4]). The occurrence of g(x)≡xα1−−αθ du (1−e−u) exp[−(x/u)(1−1α)]. (10) several time regimes then may get masked by an appar- Z0 u1+11−−αθ ent rescaling of the relaxation curves by a single effective value of µ. From a theoretical perspective, it would Here ψ∞≡limτ¯→∞ψ(τ¯)/(θτ¯−1−θ), and ψ(τ¯) denotes the be interesting to study the existence of a generalized probability distribution for the variable τ¯≡(τα+τα)1/α, 1 2 Fluctuation-DissipationTheoreminthe agingregime,as i.e. ψ(τ¯)≡hδ(τ¯−[τα+τα]1/α)i. From (10) follows 1 2 it is predicted by mean-field spin-glass models [16]. This problemisintimatelyrelatedtothevalidityofthepartial c Λ , Λ →0 F˜ (Λ )∼ 0 2 2 (11) equilibrium concept introduced above. 2 2 ( c<Λθ2/α, Λ2 →∞ We should like to thank W. Dieterich and E. Pitard forstimulatingdiscussions. P.M.gratefullyacknowledges where c0 is a constant dependent on θ and α. We note financialsupportfromthe DeutscheForschunggsgemein- that 1−Π˜ matches continuously as one leaves the short schaft (Ma 1636/2-1). time scaling regime (t ∼ tµw2) described by F2 to enter the regime described by the scaling function F (where 1 t∼tµ1). Figure 2 shows the short time behavior of Π˜ w and Π, rescaled as in (9) for the same parameters as in [1] GlassesandAmorphousMaterials,MaterialsScienceand Fig.1. Ascanbeseenfromthefigure,thedataapproach Technology, Vol. 9, edited by J. Zarzycki (VCH, Wein- the two power laws predicted by eq. (11) for large t . w heim, 1991). Forclarity,weillustratethe overallbehaviorofΠ(t + w [2] G. Parisi, e-print cond/mat 9910375. t,t ) as a function of t in Fig. 3. For (θ,α) values in w [3] J. P. Bouchaud, L. Cugliandolo, J. Kurchan, and the two-time-scaling region 0<θ<α<1/2 (shaded area M. Mézard, Out of Equilibrium dynamics in spin-glasses of the α-θ-diagramshown in the inset), there exist three and other glassy systems, in ‘Spin-glasses and Random different t regimes: (I) Π(tw+t,tw)∼1−const.t−wγ[t/tµw2] Fields’,A.P.YoungEditor,(WorldScientific,Singapore, for t≪tµw2, (II) Π(tw+t,tw)∼1−const.′t−wγ[t/tµw2]θ/α = 1998), and refs. therein. 1−const.′[t/tµw1]θ/α for tµw2 ≪t≪tµw1, and (III) Π(tw+ [4] L. C. E. Struick, “Physical Aging in Amorphous Poly- t,tw)∼[t/tµw1]−δ for tµw1≪t. When (θ,α) lies in the un- mers and OtherMaterials” (Elsevier, Houston, 1978). shaded area of the α-θ-diagram, the first regime t≪tµw2 [5] E. Vincent,J. Hammann, M. Ocio, J. P. Bouchaud,and doesnotexist(or,moreprecisely,itthenbecomes irrele- L.Cugliandolo,inComplexBehaviourofGlassySystems, vant in the limit of large tw). Note that for tw→∞ and M.RubiEditor,LectureNotesinPhysics(SpringerVer- Λ2=t/tµw2 fixed,thelong-timeregimeinFig.3“movesto- lag, Berlin,1997), Vol.492, pp.184-219, andrefs.therein. ward infinity” and the behavior is fully described by the [6] F. Alberici, P. Doussineau, and A. Levelut, J. Phys I second scaling function F (Λ ), while for t →∞ and (France) 7, 329 (1997); F. Alberici, P. Doussineau, and 2 2 w Λ1=t/tµw1 fixed the short-time regime in Fig. 3 “moves A. Levelut,Europhys. Lett. 39, 329 (1997). toward zero”, and the behavior is fully described by the [7] L. Cugliandolo and J. Kurchan, J. Phys. A 27, 5749 first scaling function F (Λ ). (1994). 1 1 In summary we have shown (i) that generalized trap [8] J.P.BouchaudandD.S.Dean,J.PhysiqueI(France)5, modelscanexhibitsubagingbehavior,inducedbymulti- (1995) 265, C. Monthus and J. P. Bouchaud, J. Phys. A 29, 3847 (1996). plevisitstothesametrap,and(ii)thepossibleexistence [9] M. Feigel’man and V. Vinokur, J. Physique I (France) 49, 1731 (1988). 1 [10] J. P. Bouchaud and M. Mézard, J. Phys. A 30, 7997 θ (1997). [11] J. C. Schön and P.Sibani, e-print cond/mat 9911197. ) w 0 +t,tw 0 α 1 [12] Ocenaesemstigohbteovbajleidctatbhoavtetshoempeacrrtiitaiclaelqduiimlibenrastioionndcco.nHcoewpt- Π(t ever,for(θ,α)=(1/3,1/4)wetestedtheconceptind=10 g and d=100also, and found it to hold true. lo I II III [13] A. Compte and J. P. Bouchaud, J. Phys. A 31, 6113 (1998). log[twµ 2 ] log[twµ 1 ] log t [[1145]] BH..RHianrnd,ePr,.SM.aHaassv,liann,danJd. PA..BBouucnhdaeu,dP,hinysp.rRepeva.raBtio3n6., FIG.3. SketchofthebehaviorofΠ(tw+t,tw)asafunction 3874 (1987). of timetin thethreeregimes (I-III).Theshadedarea in the [16] L. Cugliandolo, J. Kurchan, and L. Peliti, Phys. Rev. E α-θ-diagram marks the two-time scaling region. 55, 3898 (1997), and refs. therein. 4