Memory effects in the relaxation of ising models

Eu ophys. Le .,57 (2), pp. 171–177 (2002) EUROPHYSICS LETTERS 15 Janua y 2002 Memo y effec s in he elaxa ion o Ising models J. J. B ey and A. P ados F´ısica Te´o ica, Uni e sidad de Se illa Apa ado de Co eos 1065, E-41080, Se illa, Spain ( ecei ed 14 May 2001; accep ed in final o m 2 No embe 2001) PACS. 05.50.+q – La ice heo y and s a is ics (Ising, Po s, e c.). PACS. 05.70.Ln – Nonequilib ium and i e e sible he modynamics. PACS. 45.70.Cc – S a ic sandpiles; g anula compac ion. Abs ac . – I is analy ically shown ha he one-dimensional Ising model wi h Glaube dynamics exhibi s sho - ime memo y effec s when submi ed o an ab up change in empe a- u e. These effec s a e quali a i ely simila o hose expe imen ally obse ed in he compac ion o ib a ed g anula ma e ials. Mo eo e , a c i ical ime sepa a ing egimes o “no mal” and “anomalous” esponses o he pe u ba ion is ound. The dynamics o g anula sys ems exhibi s a ich phenomenology ha has been he subjec o in ensi e s udy in he las yea s [1]. In pa icula , when submi ed o a e ical ib a ion s a ing om a loose packing configu a ion, g anula media compac beha ing in a way ha is eminiscen o con en ional s uc u al glasses [2,3]. The glassy na u e o g anula compac ion clea ly shows up when he esponse o he sys em o a sudden change in he ib a ion in ensi y is measu ed [4–6]. The esul s show he p esence o ele an sho - e m memo y effec s ha a e in con as wi h he long- ime beha iou obse ed a cons an in ensi y. A dec ease (inc ease) in he ib a ion accele a ion leads o an inc ease (dec ease) o he compac ion a e a sho imes. Fo la ge imes, he p e ious his o y is o go en and he elaxa ion a e becomes he same as o a cons an -in ensi y p ocess. He e we s udy he exis ence o a simila “anomalous” esponse in he con ex o he one-dimensional Ising model wi h Glaube dynamics [7]. This is a model wi h sho - ange in e ac ions ha exhibi s many o he quali a i e ea u es o con en ional s uc u al glasses, like slow non-exponen ial elaxa ion, a kine ic phenomenon esembling a labo a o y glass ansi ion, aging, and hys e esis effec s [8–19]. On he o he hand, i is simple enough o allow exac analy ical calcula ions in si ua ions whe e mo e ealis ic models a e un ac able. In pa icula , we will be able o calcula e he sho - ime esponse o he sys em o a sudden small change in empe a u e, which plays in he model a ole analogous o he ib a ion in ensi y in g anula compac ion. As long as he change is no made a he ea ly s ages o he elaxa ion p ocess, he sys em shows an “anomalous” esponse. On he o he hand, i he pe u ba ion is in oduced e y soon, he esponse is “no mal”, in he sense ha he modifica ion o he elaxa ion a e has he same sign as he empe a u e change. Le us men ion ha hese wo quali a i ely diffe en ime egimes ha e been also iden ified wi hin a c EDP Sciences 172 EUROPHYSICS LETTERS gene al heo e ical amewo k de eloped o analyze memo y effec s in g anula compac ion [6], al hough i has no been expe imen ally e ified ye o he bes o ou knowledge. The ene gy o he one-dimensional Ising model in he configu a ion σ≡{σi},σ i=±1, is H(σ)=−J k (σkσk+1 −1) ,(1) whe e J>0 is he e omagne ic coupling cons an be ween nea es -neighbou spins. The Glaube dynamics o he model is o mula ed by means o a mas e equa ion, wi h he ansi ion a e om configu a ion σ o configu a ion Rkσ, diffe ing om σonly in he flip o spin σk, gi en by [7] ωk(σ)=α 21−γ 2σk(σk−1+σk+1).(2) He e γ= anh 2J kBT,(3) kBbeing Bol zmann’s cons an and T he empe a u e o he sys em. The quan i y αde e - mines he global ime scale o he sys em. Al hough i can be aken as empe a u e indepen- den , in he con ex o s uc u al glasses, whe e spin configu a ions a e associa ed wi h he minima o he po en ial ene gy [14], an A henius exp ession α(T)=α0exp −B kBT,(4) is o en conside ed [15, 20]. The cons an Bis a measu e o he ene gy ba ie sepa a ing ene gy minima in configu a ion space and α0can be employed o define a dimensionless basic ime scale. In he ollowing, we will use eq. (4) o he sake o gene ali y. The pa icula case o empe a u e-independen αcan be ob ained by aking he limi B→0. We will es ic ou sel es o homogeneous si ua ions, i.e. s a es whe e all he a e ages a e ansla ionally in a ian . I is use ul o in oduce he spin-spin co ela ions Cn( )≡σkσk+n = σ σkσk+np(σ, ),(5) whe e p(σ, ) is he p obabili y o configu a ion σa ime . In an infini e la ice, he ime e olu ion o hese co ela ions a cons an empe a u e is gi en by [7] d d Cn=−2αCn+αγ (Cn−1+Cn+1),(6) o n≥1, while C0= 1. The equilib ium co ela ions a e C(s) n(T)=ξ(T)n,ξ(T) = anh J kBT.(7) Deno ing by N he numbe o spins, a dimensionless ene gy pe spin can be defined as E( )≡H NJ =1−C1( ).(8) An equa ion o he ime e olu ion o E( ) ollows by pa icula izing eq. (6) o n=1. We will w i e i as an exp ession o he ins an aneous ene gy elaxa ion a e ( ), ( )≡− dE( ) d =α(T)[µ1( )−ε(T)µ2( )] ,(9) J. J. B ey e al.:Memo y e ec s in he elaxa ion o Ising models 173 whe e µ1( )=2E( )−1+C2( )≥0,µ 2( )=1+C2( )≥0,(10) and we ha e in oduced ε(T)=1−γ(T).(11) Bo h ε(T)andα(T) a e mono onic inc easing unc ions o he empe a u e. An equa ion ha ing he same s uc u e as eq. (9) was p oposed o desc ibe compac ion in apped g anula sys ems in e . [6]. The main diffe ence is ha he e we ha e conside ed he ime e olu ion o he ene gy, ins ead o he densi y. In he in e p e a ion o he Ising model as desc ibing s uc u al elaxa ion, an inc ease o he densi y co esponds o a dec ease o he ene gy. This is he eason why we ha e defined ( ) wi h a minus sign in eq. (9). Le us suppose we ca y ou he ollowing expe imen . S a ing om a gi en configu a ion, he sys em is allowed o elax a cons an empe a u e T. A a ce ain ime w, be o e he sys em has eached he s eady configu a ion co esponding o T, he empe a u e is suddenly changed o T+∆T. This p oduces a jump ∆ win he elaxa ion a e. Fo ∆TT, keeping only fi s -o de e ms in he pe u ba ion ∆T, i is easily ob ained ha ∆ w=λ( w)∆α, (12) wi h λ( )= ( ) α−αµ2( )dε dα= ( ) α−η(1 −γ2) 2µ2( ),(13) and η=4J/B. The unc ion λ( ) is defined along he elaxa ion cu e a cons an T. As long as ∆Tis small enough so ha he linea app oxima ion we a e conside ing is accu a e, he unc ion λ( ) o = wde e mines he ela i e beha iou o he elaxa ion a e jump ∆ w wi h espec o ∆α(o ∆T). To s udy he unc ion λ( ), we ha e o e alua e he elaxa ion a e ( ) and he co ela ion unc ion µ2( ) o a p ocess a cons an empe a u e. The gene al solu ion o he hie a chy (6) o an a bi a y ini ial condi ion can be analy ically ound [7,21]. We will conside p ocesses s a ing in he s a e o maximum diso de , i.e. he equilib ium s a e a infini e empe a u e. Simila esul s a e ob ained o o he ini ial condi ions, as long as hey co espond o highly diso de ed s a es. A simple calcula ion gi es [7,21] ∆n( )≡Cn( )−C(s) n(T)=−γ ππ 0 dqsin qsin nq 1−γcos qe−2α (1−γcos q).(14) Use o he abo e exp ession wi h n= 1 in o eq. (8) yields E( )−E(s)(T)=γ ππ 0 dqsin2q 1−γcos qe−2α (1−γcos q),(15) whe e E(s)(T)=1−C(s) 1(T) is he equilib ium alue o he dimensionless ene gy pe pa icle. The elaxa ion a e o he ene gy, defined in eq. (9), o he p esen p ocess is ( )=2αγ ππ 0 dqsin2qe −2α (1−γcos q)=e−2α I1(2αγ ) ,(16) whe e I1is he modified Bessel unc ion o he fi s kind [22]. To e alua e µ2( ) we use again eq. (14), now o n= 2. In his way we ge om eq. (10) µ2( )=1+ξ2−γ π ( ),(17) 174 EUROPHYSICS LETTERS whe e ( )=π 0 dqsin qsin 2q 1−γcos qe−2α (1−γcos q).(18) Subs i u ion o eqs. (16) and (17) in o eq. (13) leads o λ( )=e−2α I1(2αγ ) α −η(1 −γ2) 21+ξ2−γ π ( ).(19) In he ollowing we will conside ha he empe a u e Tis e y low, so ha ε1, γ→1, ξ→1, and α→0. I could be concluded ha he second e m in he igh -hand side o eq. (19) can be neglec ed as compa ed wi h he fi s one [23] in his limi . Ne e heless, his is no ue in gene al, and he ela i e ele ance o bo h e ms depends on he ime scale o in e es . In pa icula , o →∞ he fi s e m anishes, as well as ( ), and λ∞≡lim →∞ λ( )=−η(1 −γ2)(1 + ξ2) 2∼−2ηε < 0.(20) Since λ( ) is a con inuous unc ion o , he abo e esul implies ha , o la ge enough w, λ( w)<0 and, he e o e, ∆ wand ∆αha e opposi e signs. A sudden inc ease o he em- pe a u e p oduces a nega i e jump o he elaxa ion a e. This is he kind o “anomalous” esponse ound in compac ion p ocesses o g anula ma e ials [4–6]. I is con enien o define he unc ion Λ( )=λ( )−λ∞=e−2α I1(2αγ ) α +ηγ(1 −γ2) 2π ( ),(21) ha anishes o →∞. We a e going o analyze he beha iou o his quan i y bo h in he sho - and he la ge- ime egions. Fi s , conside “sho ” imes cha ac e ized by α ∼O(1),(22) so ha εα 1 o he low empe a u es we a e dealing wi h. On he ime scale defined abo e, we can app oxima e γby uni y in he exp ession o ( ), eq. (18), ob aining ( )=πe−2α [I0(2α )+2I1(2α )+I2(2α )] .(23) When his exp ession is subs i u ed in o eq. (21), i is easily ealized ha he e m in ol ing ( ) can be neglec ed because o he ηε ac o . Then, Λ( )∼e−2α I1(2α ) α ,(24) o α ∼O(1), ε1. In his ime window, λ( )∼Λ( )>0, whe e we ha e aken in o accoun ha λ∞is o o de ε(see eq. (20)). As a consequence, he esponse o he sys em o a small change ∆Tin he empe a u e a = wis “no mal” i αε w1. The modifica ion o he elaxa ion a e has he same sign as he empe a u e change. Abo e, i was shown ha o asymp o ically la ge imes he esponse was quali a i ely diffe en . Le us s udy in mo e de ail wha happens in he “la ge” ime egion. We define a slow ime scale by τ≡αε ∼O(1),(25) i.e., imes o he o de o he sys em elaxa ion ime. Fo hese imes, i is α =O(ε−1)1 and we can app oxima e I1(2αγ ) by i s asymp o ic beha iou o la ge a gumen [22], ge ing e−2α I1(2αγ ) α ∼ε3/2e−2τ 2π1/2τ3/2.(26) J. J. B ey e al.:Memo y e ec s in he elaxa ion o Ising models 175 This quan i y is much smalle han λ∞. Mo eo e , a simple asymp o ic analysis gi es ( )∼(2πε)1/2Γ(−1/2,2τ),(27) Γ(a, x) being he incomple e Gamma- unc ion [22]. Use o eqs. (26) and (27) in o eq. (21) leads o Λ( )∼ε3/2e−2τ 2π1/2τ3/2+η2 π1/2 Γ(−1/2,2τ),(28) alid o ε1, τ=O(1). In his long- ime egime, Λ( ) is o he o de o ε3/2,and |λ∞|Λ( ). Then, λ( )∼λ∞<0. The esponse o he sys em is anomalous, as we al eady knew om he discussion o he asymp o ic beha iou o →∞. Since we ha e ound ha λ( ) has diffe en signs in he sho - and long- ime egions, because o con inui y easons he e mus be a leas one c i ical ime ca which λ( c)=0. This ime sepa a es egions wi h no mal and anomalous beha iou . Fu he mo e, he abo e discussion indica es ha his ime mus belong o an in e media e ime scale be ween hose p e iously conside ed, namely hose gi en by eqs. (22) and (25), espec i ely. The ma ching ime ange whe e he wo solu ions, eqs. (24) and (28), a e alid is cha ac e ized by α 1,τ=εα 1.(29) On his ime scale, we can subs i u e he Bessel unc ion in eq. (24) by i s asymp o ic beha iou o la ge a gumen [22], wi h he esul Λ( )∼1 2π1/2(α )3/2.(30) The same exp ession is ob ained om eq. (28) in he app op ia e limi , showing ha he solu ions co esponding o he sho - and la ge- ime scales ma ch in he in e media e scale defined by eq. (29). A single uni o m app oxima ion Λuni ( ), alid o all imes, is ob ained by adding eqs. (24) and (28), and sub ac ing he exp ession in he ma ching egion, eq. (30), Λuni ( )=e−2α I1(2α ) α +e−2αε −1 2π1/2(α )3/2+ε3/2η2 π1/2 Γ(−1/2,2αε ).(31) In fig. 1 we compa e his esul wi h he exac exp ession, eq. (21) o ε=0.01 and J=B, i.e. η= 4. A e y good ag eemen is obse ed o all imes. The ele an physical quan i y we a e in e es ed in is he c i ical ime c, e i ying λ( c) = 0 o , equi alen ly, Λ( c)=−λ∞. This la e quan i y is also indica ed in he figu e. Since cis expec ed o belong o he in e media e scale, we ge om eqs. (20) and (30) α c∼4π1/2ηε−2/3.(32) This esul confi ms ha cis in he o e lapping egion in ag eemen wi h ou p e ious ansa z, since in he limi ε→0i isα c1andτc=εα c1. The e o e, eq. (32) is consis en wi h he uni o m exp ession (31). I is in e es ing o analyze he empe a u e dependence o c. By aking loga i hms in eq. (32), we find, o low T, ln(α0 c)=−2 3ln 8π1/2η+B kBT1+2η 3,(33) 176 EUROPHYSICS LETTERS Fig. 1 Fig. 2 Fig. 1 – Compa ison o he nume ical e alua ion o eq. (21) (ci cles) wi h he uni o m asymp o ic app oxima ion gi en by eq. (31) (solid line). Also plo ed (do ed line) is he asymp o ic long- ime alue |λ∞|. No e ha he quan i ies ep esen ed in bo h axes a e dimensionless. Fig. 2 – The dimensionless c i ical ime c, sepa a ing he egions o no mal and anomalous esponse, as a unc ion o he in e se o he educed empe a u e. The ci cles ha e been ob ained by nume ical e alua ion o eq. (21), while he solid line is he esul o he asymp o ic calcula ion, eq. (32). showing an A henius dependence. In fig. 2 he loga i hm o he c i ical ime c, gi en by eq. (32), is plo ed as a unc ion o he in e se o he educed empe a u e B/kBT, again o η= 4. Fo compa ison, also he nume ical alues ob ained om he exac exp ession, eq. (21), a e included. A linea p ofile, consis en wi h eq. (33), is obse ed. Le us now add ess he pa icula case o αbeing empe a u e independen ha can be ob ained by conside ing he limi B→0. Then ηgoes o infini y and c anishes. The e o e, he sys em always exhibi s an anomalous esponse o a empe a u e jump, independen ly o he ime ins an in which i is p oduced. The physical eason o his esul is clea , since eq. (12) educes in his si ua ion o ∆ w ∆T=−αµ2w dε dT<0.(34) Equa ion (34) eflec s ha , i αdoes no depend on he empe a u e, he a es o he p ocesses dec easing he ene gy do no change ins an aneously in a empe a u e jump. This can be di ec ly e ified, o ins ance, om eq. (9). In conclusion, he majo esul ob ained in his pape is he analy ical de i a ion o memo y effec s in he elaxa ion o he one-dimensional Ising model wi h Glaube dynamics. Mo e- o e , hese effec s a e quali a i ely simila o hose obse ed expe imen ally in g anula com- pac ion [4–6], o which he exis ence o a c i ical ime sepa a ing he egions o “no mal” and “anomalous” esponses has been heo e ically a gued [6]. The e o e, also in he p esen con ex he Ising model seems o be he simples ele an sys em beyond mean-field desc ip- ions. The esul s in his pape seem o indica e ha he memo y effec s and he p esence o a c i ical ime, as discussed he e, may be qui e gene al. In he Ising model, he c i ical ime lies in an in e media e ime window, being much smalle han he a e age elaxa ion ime o he sys em owa ds i s equilib ium s a e. J. J. B ey e al.:Memo y e ec s in he elaxa ion o Ising models 177 ∗∗∗ We acknowledge pa ial suppo om he Di ecci´on Gene al de In es igaci´on Cien ´ıfica y T´ecnica (Spain) h ough G an No. PB98-1124. REFERENCES [1] Jaege H., Nagel S. R. and Beh inge R.,Re . Mod. Phys.,68 (1996) 1259. [2] Knigh J. B., F andich C. G., Lau C. N., Jaege H. 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