Glassy behavior in a simple model with entropy barriers

Glassy beha io in a simple model wi h en opy ba ie s A. P ados and J. J. B ey Fı ´sica Teo ´ ica, Facul ad de Fı ´sica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080 Se illa, Spain B. Sa ´nchez-Rey Escuela Poli e ´cnica Supe io , Uni e sidad de Huel a, E-21819 La Ra ´bida, Huel a, Spain ~Recei ed 13 Sep embe 1996! We s udy he dynamical beha io o a sys em wi h a a iable numbe o pa icles n. The emp y s a e n50 is he g ound s a e, while all he o he s a es n.0 a e degene a e in ene gy. In equilib ium, he mean numbe o pa icles is equal o uni y, independen ly o he empe a u e. The s a ic p ope ies a e he same as o he Backgammon model ecen ly p oposed by Ri o @Phys. Re . Le . 75, 1190 ~1995!#, while a a ia ion o he kine ics is conside ed. The elemen a y dynamical p ocesses a e he a i al and depa u e o a pa icle. The a e o he depa u e p ocess is cons an , while he a i al a e is ob ained om he de ailed balance condi ion. Thus, he e is no ene gy ba ie sepa a ing he g ound s a e n50. Ne e heless, glassy beha io appea s due o he p esence o e ec i e en opy ba ie s. A low empe a u es, he esponse unc ions a e shown o obey ( ).exp@2( / ) g #. In he mal cycles o cooling and ehea ing om low empe a u es, he sys em shows hys e esis, which ollows om he end o he sys em o app oach he no mal cu e cha ac e izing he hea ing p og am. @S0163-1829~97!03710-7# I. INTRODUCTION The s udy o glassy beha io has been qui e an ac i e ield in ecen yea s. A e iew o he main ea u es obse ed in eal glasses, and se e al mic oscopic models showing simi- la i y wi h hem, can be ound in Re s. 1 and 2. In elaxa ion expe imen s, he linea esponse unc ions show nonexpo- nen ial beha io . In pa icula , a Kohl ausch-Williams-Wa s ~KWW!decay is usually ound. In cooling expe imen s a labo a o y glass ansi ion, in which he p ope ies de ining he s a e o he sys em become ozen, is obse ed. The an- si ion is associa ed o a as inc ease o he elaxa ion ime as he empe a u e is lowe ed. Du ing ehea ing, hys e esis e - ec s show up, wi h he sys em e u ning o equilib ium ol- lowing a pa h which is di e en om he cooling one. A mo e de ailed discussion o he ich phenomenology o glasses is a ailable in Re s. 3 and 4. The e is a g ea a ie y o models ying o explain glassy beha io . The simples one is a wo-le el sys em ~TLS!, whe e an ene gy ba ie mus be su passed in o de o go om he exci ed o he g ound s a e.5,6 In some models, he inc ease o he elaxa ion ime is associa ed o he in oduc- ion o coope a i i y in he dynamics o he sys em,7bu he e is also an ene gy ba ie sepa a ing he g ound s a e om he exci ed ones. This ba ie plays an essen ial ole in he di e gence o he elaxa ion ime a low empe a u es. On he o he hand, en opy is known o play an impo an ole in he desc ip ion o glassy beha io since he pionee ing wo k by Adam and Gibbs.8 Recen ly, Ri o 9,10 has p oposed a model wi hou ene gy ba ie s, in he sense ha he sys em can always each he g ound s a e wi hou any ene gy-ac i a ed p ocess. The dy- namical s udy o he model has ocused on he mal cycles o cooling and ehea ing, and ze o- empe a u e p ope ies as ag- ing. The sys em displays glassylike beha io , despi e he ab- sence o ene gy ba ie s, and he di e gence o he elaxa ion ime is due o he en opic con ibu ion o ee ene gy ba i- e s. These appea because o he small numbe o di ec ions in phase space along which he ene gy dec eases. Slow e- laxa ion shows up because he sys em has o explo e a wide phase space egion be o e eaching he g ound s a e. Because o he ules go e ning i s dynamics, he model has been e e ed o as he Backgammon ~BG!model. I can be isualized in se e al di e en , al hough equi alen , ways. He e we p esen one o hem, while ano he one is discussed in he inal sec ion. Suppose we ha e a wo-dimensional la - ice wi h a pa icle a each si e. Then, an ex e nal mechanism is in oduced such ha pa icles end o agg ega e in he di- ec ion pe pendicula o he la ice. Pa icles emaining on he la ice ha e a la ge ene gy han hose which a e agg e- ga e o hem, so ha he minimum ene gy is eached when all pa icles o m a unique agg ega e a a gi en si e. All si es and pa icles being equi alen , his s a e has a degene a ion gi en by he numbe o si es ~o pa icles!. The dynamics o he sys em is de ined by means o a Ma ko p ocess in which each pa icle can mo e o any o he si e, wi h ansi ions a es gi en by Me opolis dynamics. Since he spa ial a - angemen o he si es in he plane plays no ole a all, he model is o a mean- ield ype. Mean- ield app oxima ions a e no accu a e o desc ibe elaxa ion h ough ene gy ba ie s in eal s uc u al glasses, because o he nuclea ion p ocesses aking place in hem. Ne e heless, as poin ed ou by Ri o ,9 he e ec o en opy ba ie s should no depend e y s ongly on he ange o he in e ac ions and he in o ma ion ob ained om his kind o model is expec ed o be ele an also in he case o sho - anged in e ac ions. In his wo k we in oduce a model ha keeps he main cha ac e is ic o Ri o ’s model, namely he absence o en- e gy ba ie s o ansi ions o he g ound s a e, and allows an analy ical ea men o he dynamics. We conside a sys em wi h a a iable numbe o pa icles n, in which he g ound s a e has no pa icles, n50, while all he s a es wi h n.0 PHYSICAL REVIEW B 1 MARCH 1997-IIVOLUME 55, NUMBER 10 55 0163-1829/97/55~10!/6343~13!/$10.00 6343 © 1997 The Ame ican Physical Socie y a e degene a e. The s a ics o his sys em is equi alen o he BG model wi h indis inguishable pa icles.11 The dynamics is o mula ed by means o a mas e equa ion wi h ansi ion a es e i ying he de ailed balance condi ion. The equa ion can be exac ly sol ed o cons an empe a u e p ocesses, allowing he iden i ica ion o he mechanisms leading o non- exponen ial elaxa ion and o he di e gence o he elaxa ion ime. Fo cooling p ocesses we show he ele ance o he elaxa ion modes o he mas e equa ion and o he ene gy elaxa ion ime o cha ac e ize he labo a o y glass ansi ion and he eezing empe a u e, espec i ely. Along hea ing, he dynamical beha io o he model is unde s ood om he end o he sys em owa ds a ‘‘no mal’’ cu e.12 In pa icu- la , he hys e esis e ec , which is so cha ac e is ic o glasses, is di ec ly ela ed o he app oach o his no mal cu e. The exis ence o such a cu e is a qui e s ong p edic ion o mod- els bases on a mas e equa ion o mula ion o he dynamics. Whe he he e is a no mal cu e also o eal s uc u al glasses emains an open ques ion. The esul s ob ained will be compa ed o o he p e iously conside ed models and, in pa icula , he one-dimensional Ising model wi h Glaube dynamics.13 Le us men ion ha he Ising model may be ele an in he con ex o s uc u al glasses, since i has been p o ed o accu a ely desc ibe he e olu ion o he con igu a ion o a one-dimensional sys em o pa icles wi h anha monic and compe ing in e ac ions.14 Al hough ene gy ba ie s exis in he model s udied in Re . 13, glassy beha io appea s in bo h cases o simila easons. P obably, his is also he case o any model showing glassy beha io , as long as i s dynamics is desc ibed by a mas e equa ion. The plan o he pape is he ollowing. In Sec. II he model is o mula ed, and he mas e equa ion desc ibing i s dynamics is sol ed o he cons an empe a u e case. Relax- a ion p ope ies a e conside ed in Sec. III, ocusing on he s e ched exponen ial decay ound a low empe a u es. Sec- ion IIIA is de o ed o he s udy o he equilib ium ime au oco ela ion unc ion o he ene gy, while linea elax- a ion a e a empe a u e pe u ba ion is he subjec o Sec. IIIB. The mal cycles a e s udied in Sec. IV, and cooling p ocesses a e conside ed in Sec. IVA, whe e he labo a o y glass ansi ion is analyzed in de ail. Sec ion IVB deals wi h hea ing p ocesses. The no mal cu e associa ed wi h a gi en hea ing p og am is de ined, and i s ela ion o he obse ed hys e esis e ec is discussed. Finally, he main conclusions o he pape a e summa ized in Sec. V. II. THE MODEL The model we conside has a a iable numbe o pa icles n. This numbe comple ely speci ies he s a e o he sys em. The emp y s a e, n50, has ze o ene gy, e 050, and all he s a es wi h n.0 a e degene a e, wi h ene gy e n5 e . The sys- em is in con ac wi h a hea and pa icle ba h cha ac e ized by a empe a u e Tand ugaci y z [exp(2 a ). The e o e, he equilib ium p obabili y o inding he sys em in s a e nis pn ~0!5Ce2 be ne2 a n,~2.1! whe e b 5(kBT)21,kBbeing Bol zmann’s cons an . The cons an Cis de e mined om he no maliza ion condi ion, and i is gi en by C512e2 a 12e2 a ~12e2 be !.~2.2! Now, we assume ha he ba h is such ha he equilib ium a e age numbe o pa icles is uni y, independen ly o he empe a u e, i.e., ^ n & 05( n50 ` npn ~0!51. ~2.3! This p o ides a ela ionship be ween he ugaci y and he empe a u e, namely a 5ln~11e2 be /2!.~2.4! We no ice ha exp essions o his kind a e ypical when passing om a canonical desc ip ion o a g and-canonical one, and he la e is equi ed o co ec ly ep oduce he num- be o pa icles in he sys em. Using his ela ion, Eq. ~2.2! educes o C5exp(2 a ), and he equilib ium dis ibu ion can be w i en p0 ~0!5e2 a ,~2.5a! pn ~0!5e2 be 2 a ~n11!,n>1. ~2.5b! The in oduc ion o a ba h e i ying Eq. ~2.4!has been s imula ed by he wo k ca ied ou in Re s. 9–11, whe e wo a ia ions o he BG model a e s udied. In hese models, N pa icles can occupy Ndi e en ‘‘abacuses’’ 51,...,N. While in one o he models9,10 he pa icles a e conside ed as dis inguishable, in he o he one11 hey a e ea ed as indis- inguishable. This is he only di e ence be ween bo h mod- els. Excep o an addi i e cons an , he ene gy o a gi en con igu a ion is p opo ional o he numbe o occupied aba- cuses. The e is no limi a ion in he numbe o pa icles n being in a pa icula abacus , excep he one ollowing om he o al numbe o pa icles, ( n 5N. Since all he aba- cuses a e equi alen , he a e age numbe o pa icles in each o hem mus be uni y a equilib ium. The model desc ibed abo e mimics he equilib ium p op- e ies o he BG model wi h indis inguishable pa icles. A b ie discussion o his is gi en in Appendix A. The idea is o ocus on one o he abacuses, conside ing he emainde o hem as a ba h in he limi N→`. The condi ion gi en by Eq. ~2.4!gua an ees ha his limi is aken keeping he same bo h he numbe o abacuses and he numbe o pa icles. F om Eq. ~2.5!i is s aigh o wa d o ob ain he equilib- ium p ope ies o he sys em, as unc ions o he empe a- u e. The a e age ene gy is ^ E & 05( n50 ` e npn ~0!5 e ~12p0 ~0!!5 e e2 be /2 11e2 be /2 ,~2.6! and i s luc ua ions a e gi en by s E 25 ^ E2 & 02 ^ E & 0 25 e 2e2 be /2 ~11e2 be /2!2.~2.7! Fluc ua ions in he numbe o pa icles a e s N 25 ^ n2 & 02 ^ n & 0 252e be /2.~2.8! 6344 55A. PRADOS, J. J. BREY, AND B. SA ´NCHEZ-REY This quan i y di e ges in he low- empe a u e limi , whe e mos o he p obabili y co esponds o he g ound s a e. Due o he condi ion o he mean numbe o pa icles being equal o uni y, he p obabili y dis ibu ion has a long ail as a unc- ion o n. The e o e, he e is an e ec i e co ela ion leng h associa ed o he di e gence o he luc ua ions o he numbe o pa icles. Finally, he equilib ium en opy eads S kB 52 ( n50 ` pn ~0!lnpn ~0!52ln~11e2 be /2!1 be e2 be /2 11e2 be /2 . ~2.9! This exp ession coincides wi h he en opy pe abacus in he BG model wi h indis inguishable pa icles. I is ee o he pa hological beha io shown by he en opy in he case o conside ing he pa icles as dis inguishable, whe e i be- comes nega i e a low empe a u es.9,11 Nex , we p oceed o o mula e he kine ics o he model. The elemen a y dynamical p ocesses we will conside a e he a i al o he depa u e o one pa icle, and he e o e he dynamical e olu ion o he sys em will be gi en by a one- s ep p ocess15 mas e equa ion, dpn d 5 n11pn111gn21pn212~ n1gn!pn,~2.10! whe e pn( ) is he p obabili y ha he sys em has npa icles a ime , nis he ansi ion a e om s a e n o s a e n21 ~loss o one pa icle!, and gnis he ansi ion a e om s a e n o s a e n11~gain o a pa icle!. O cou se, he s a e n50 is a e lec ing bounda y, 050. ~2.11! As we do no wan o in oduce any ene gy ba ie ob- s uc ing he elaxa ion o he sys em owa ds he g ound s a e, we will ake n5 n ,n.0, ~2.12! whe e n is a cons an pa ame e wi h dimensions o e- quency. The ansi ion a es gna e chosen in o de o e i y he de ailed balance condi ion, i.e., g05 n e2 be 2 a ,~2.13a! gn5 n e2 a ,n.0. ~2.13b! Since he g ound s a e can be eached a any empe a u e om any o he s a e wi hou su moun ing any ene gy ba ie , possible di e gence o he cha ac e is ic elaxa ion ime and glassy beha io can only appea in he model due o he p esence o en opy ba ie s. In ac , glassy beha io is o be expec ed, because a low empe a u es a →0 acco ding o Eq. ~2.4!, and he leading beha io o he ansi ion a es is gi en by gn. n5 n ,n.0, ~2.14a! g0. n e2 be .~2.14b! One can a gue whe e he model, as o mula ed he e, inco - po a es he en opy ba ie s which a e so e iden in he o igi- nal BG model. A low empe a u es, he p obabili y o ind- ing he sys em in an exci ed s a e a om n51, which is he only one om which he ene gy can dec ease, is o he same o de as p1up o n5O( a 21). This e lec s he equi alence o all he abacuses in he o iginal BG model. The elaxa ion slows down because he andom walk pe o med by namong he exci ed s a es con ibu ing o he ene gy is symme ic, and i akes a e y la ge ime o he sys em o elax om s a es n5O( a 21). Ve y ecen ly, some andom walk models ha e been p o- posed o mimic he ze o- empe a u e dynamics o he BG model.16,17 To pu ou wo k in a p ope con ex , i is impo - an o no e ha , i s , we will s udy he e he ini e empe a- u e kine ics o ou model and, secondly, ha we a e using a g and-canonical ensemble desc ip ion. Fo his eason ou andom walk is no symme ic, excep in he limi T→0. Ou aim is no o p opose a model ha exac ly ep oduces he dynamics o he BG model. Ins ead, we wan o e ain i s main ea u es in a sol able model, in o de o iden i y he ele an mechanisms leading om en opy ba ie s o glassy beha io . The solu ion o he mas e equa ion in he case o ime independen empe a u e can be ob ained by using s anda d p ocedu es.15 The cons an n in he ansi ion a es will be used o se up he ime scale, and hus i will be aken equal o uni y in he ollowing. We look o he eigen alues and eigen ec o s o he p oblem. The o me a e gi en by l~q!511e2 a 22e2 a /2cosq,~2.15! whe e q uns in he in e al @0, p #. Besides, he e is he eigen alue l50, whose eigen ec o is he equilib ium dis- ibu ion gi en by Eq. ~2.5!. All he eigen alues lo he han l50 a e s ic ly posi i e, as i mus be he case o a mas e equa ion wi h ansi ion a es e i ying de ailed balance. The eigen ec o associa ed o l(q)is j 0 ~ q ! 5 S 2 p D 1/2 e~ be 2 a !/2cos h ~q!,~2.16a! j n~q!5 S 2 p D 1/2 e2[ be 1 a ~11n!]/2cos@nq1 h ~q!#,n>1. ~2.16b! He e h (q) is a eal unc ion de ined by e2i h ~q!52e2iq2 be 2 a /22e2 be 2 a 111e2 a 22e2 a /2cosq eiq2 be 2 a /22e2 be 2 a 111e2 a 22e2 a /2cosq, ~2.17! and h ~0!5 p /2. ~2.18! I has he p ope y h (2q)52 h (q)1 p . The eigen ec o s j n(q) e i y he closu e ela ion pn ~0!1 E 0 p dq j n~q! j m~q! pm ~0!5 d nm .~2.19! By using he abo e equa ion, any ini ial condi ion can be exp essed as a sum o e he eigen ec o s, 55 6345GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . . Dn~ 50![pn~ 50!2pn ~0!5 E 0 p dqg~q! j n~q!, ~2.20! wi h g~q!5( m50 `Dm~ 50! j m~q! pm ~0!.~2.21! Now, i is i ial o w i e he ime e olu ion o he de ia ion om equilib ium Dn( ), Dn~ ![pn~ !2pn ~0!5 E 0 p dqg~q! j n~q!e2 l~q!. ~2.22! This p o ides he gene al solu ion o he mas e equa ion, o he ime independen empe a u e case. Since l(q) is s ic ly posi i e o all q,0<q< p ,D n( ) goes o ze o in he in ini e ime limi , as expec ed. As al eady discussed, a ze o empe a u e he model e- duces o a symme ic andom walk wi h an abso bing bound- a y a n50. The e o e, he p obabili y dis ibu ion ends o a s a iona y s a e wi h pn5 d n,0 . The decay o his s a e is e y slow and aging e ec s occu , e en i he sys em was ini ially in equilib ium a low empe a u es. Since i is easily seen ha a T50 ou model becomes equi alen o ‘‘model B’’ s udied in de ail in Re . 17, we will no discuss he aging e ec s he e. III. RELAXATION PROPERTIES In his sec ion we a e going o s udy he elaxa ion p op- e ies o he model a a gi en cons an empe a u e. A en ion will be ocused on ~a! he ime au oco ela ion unc ion o ene gy in equilib ium and ~b! he linea elaxa ion o ene gy a e a empe a u e pe u ba ion. I mus be s essed ha bo h quan i ies does no coincide, because he ensemble desc ip- ion o he model does no co espond o he canonical one. A. Ene gy ime au oco ela ion unc ion The ime au oco ela ion unc ion o he ene gy in equi- lib ium is gi en by ^ E~0!E~ ! & 05( n50 ` ( m50 ` e n e mp1 u 1~n, u m,0!pm ~0!,~3.1! whe e p1 u 1(n, u m,0) is he condi ional p obabili y o inding he sys em in s a e na ime gi en i was ini ially in s a e m. Le us in oduce he esponse unc ion ~ !5 ^ E~0!E~ ! & 02 ^ E & 0 2 ^ E2 & 02 ^ E & 0 2,~3.2! ha e i ies ~0!51, lim →` ~ !50. ~3.3! The condi ional p obabili y p1 u 1(n, u m,0) is he solu ion o he mas e equa ion ~2.10!wi h he ini ial condi ion p1 u 1~n,0 u m,0!5 d nm .~3.4! By making use o Eq. ~2.19!i is easy o see ha p1 u 1~n, u m,0!5pn ~0!1 E 0 p dq j m~q! pm ~0! j n~q!e2 l~q!, ~3.5! since pn (0) co esponds o he null eigen alue, and j n(q) o l(q). The e o e, i is ^ E~0!E~ ! & 05( n,m50 ` e n e mpn ~0!pm ~0! 1( n,m50 ` e n e m E 0 p dq j n~q! j m~q!e2 l~q! 5 ^ E & 0 21 E 0 p dqa2~q!e2 l~q!,~3.6! whe e we ha e in oduced he unc ion a~q!5( n50 ` e n j n~q!.~3.7! Subs i u ion o Eq. ~3.6!in o Eq. ~3.2!yields ~ !5 * 0 p dqa2~q!e2 l~q! * 0 p dqa2~q!.~3.8! I ollows ha ( ) decays mono onically om i s ini ial alue, (0)51, o ze o. This could ha e been o eseen, since i is a gene al p ope y o equilib ium au oco ela ion unc ions in models whose dynamics is desc ibed by means o mas e equa ions wi h he ansi ion a es e i ying he de ailed balance condi ion. The p oblem has been educed o calcula e he unc ion a(q), de ined by Eq. ~3.7!, ha can be w i en as a~q!5( n51 ` e j n~q!52 e j 0~q!,~3.9! because ( n50 ` j n~q!50, ~3.10! due o he o hogonali y o he eigen ec o s j (q) wi h e- spec o he equilib ium dis ibu ion. F om Eqs. ~3.9!and ~2.16a!we ob ain a~q!}cos h ~q!.~3.11! The p opo ionali y cons an in he abo e ela ion is i el- e an o he calcula ion o he esponse unc ion, gi en by Eq. ~3.8!. The unc ion h (q) de ined in Eq. ~2.17!is a he in ol ed, bu simple exp essions a e de i ed bo h in he lim- i s o sho and long imes. Fo sho imes, !1, i is ~ !;e2 lM,~3.12! whe e lM[ S 215 * 0 p dql~q!a2~q! * 0 p dqa2~q!5e a 21. ~3.13! 6346 55A. PRADOS, J. J. BREY, AND B. SA ´NCHEZ-REY Thus, he elaxa ion in he sho ime egime is exponen ial, as i is he usual case in sys ems desc ibed by mas e equa ions.18 In he limi o long imes, a Laplace’s analysis o Eq. ~3.8!gi es ~ !;e a 21 2 p 1/2e9 a /4~11e2 a /22e2 a !2 e2 ~12e2 a /2!2 ~12e2 a /2!4 3/2 . ~3.14! Aside om slow algeb aic co ec ions, he elaxa ion is again exponen ial, bu wi h a cha ac e is ic ime L5~12e2 a /2!2,~3.15! which is di e en om he one o he sho ime egime. This is also he mos common case in models desc ibed by mas e equa ions. This ac , oge he wi h he mono onic decay o he equilib ium au oco ela ion unc ion, leads o a nonexpo- nen ial elaxa ion egime a in e media e imes.18 This e- gime is expec ed o be mo e ele an as he ime scales sepa- a ion becomes la ge . This is he case when e a →1. Then, bo h cha ac e is ic imes di e ge, bu L@ S.~3.16! Taking in o accoun Eq. ~2.4! o a , i ollows ha e a →1 is equi alen o b →`o T→0. In his limi , bo h Eqs. ~3.12!and ~3.14!become much simple . Fo sho imes i is ~ !;e2 a ,~3.17! whe eas in he long ime egion ~ !;1 p 1/2 e2 a 2 /4 ~ a 2 /4!3/2 .~3.18! This la e equa ion shows ha elaxa ion akes place o e a ime scale s5 a 2 4,~3.19! which is much longe han he de ined by he ini ial expo- nen ial. Thus, sepa a ion o ime scales comes up, and non- exponen ial elaxa ion is o be expec ed in an in e media e ime window. The pic u e we ha e ob ained is simila o he one ound in he low- empe a u e elaxa ion o Glaube ’s Ising model.19–21 The e o e, we make use o he same echniques o de i e he beha io o he co ela ion unc ion in he in- e media e ime egime in he low- empe a u e limi . To be- gin wi h, we ob ain an exp ession which is alid in he ime scale de ined by Eq. ~3.19!. We in oduce a new a iable u h ough q5 a u/2. ~3.20! Then, a simple analysis gi es ~ ![ ¯ ~s!54 p E 0 `du u2 ~11u2!2e2s~11u2!,~3.21! whe e e ms o o de a ha e been neglec ed. Fo e y long imes, s@1, Eq. ~3.18!is o cou se eco e ed. Howe e , in he egion s!1 we do no ge he low- empe a u e e sion o he sho ime beha io , as gi en by Eq. ~3.17!, since i co - esponds o he much sho e ime scale de ined by a 21. O e he ime scale s, ha beha io collapses on o he poin s50. Ac ually, s!1 co esponds o an in e media e ime window whe e is la ge bu s5 a 2 /4 is small ( a !1). I is easy o see ha ln ¯ ~s!;24 p 1/2 s1/2,s!1. ~3.22! The e o e, om Eqs. ~3.19!and ~3.21!we ge ln ~ !;2 S 4 a 2 p D 1/2 ,~3.23! which is a s e ched exponen ial o Kohl ausch-William- Wa s ~KWW! unc ion, ln ~ !52 S D g ,~3.24! wi h g 51/2, ~3.25a! 5 p 4 a 2; p 4e be .~3.25b! Thus, a low empe a u es he elaxa ion ime obeys he A henius law, wi h an ‘‘ac i a ion’’ ene gy e . One may ask himsel which is he physical o igin o his beha io , since he sys em does no ha e o su moun any ene gy ba ie o each he g ound s a e. In ou model, as in he one p oposed by Ri o ,9 he e is an en opy ba ie . A low empe a u es a →0 and a symme ic andom walk is pe o med by he sys em among all he exci ed s a es. The cha ac e is ic elax- a ion ime will be domina ed by he di usion p ocess om he mean posi ion in he exci ed egion o he s a e n50. The mean posi ion in he exci ed s a es n ¯ exc is gi en by n ¯ exc5(n51 `npn ~0! (n51 `pn ~0!5 ^ n & 0 12p0 ~0!5~12e2 a !21,~3.26! whe e we ha e made use o Eq. ~2.5a!. This quan i y mus no be con used wi h he a e age numbe o pa icles in ex- ci ed s a es. In he limi o low empe a u es n ¯ exc; a 21@1. ~3.27! Then, an es ima ion o he ime needed o di use un il n50 will be di 5O~n ¯ exc 2!5O~ a 22!.~3.28! The abo e equa ion can be conside ed as a quali a i e expla- na ion o he elaxa ion ime dependence on he empe a- u e shown by Eq. ~3.25b!, since i is easonable o expec ha 5O( di ), he mean ime aken by he sys em o ge o he ‘‘bo leneck’’ in he con igu a ion space. A simpli ied pic u e o he e olu ion o he equilib ium ime au oco ela ion unc ion ( ) o he ene gy can be gi en in e ms o he h ee ime egimes we ha e ound, 55 6347GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . . ~ !5 H e2 a !1, e2~4 a 2 / p !1/2 1! !4 a 22, p 21/2~ a 2 /4!23/2e2 a 2 /4 @4 a 22. ~3.29! A simila beha io has been p e iously ob ained o elax- a ion in di e en models.18,19,22 I mus be no iced ha he scheme desc ibed by Eq. ~3.29!is consis en wi h bo h em- pi ical and nume ical esul s o glassy sys ems, whe e non- exponen ial elaxa ion and KWW beha io is usually ound o e an in e media e ime window.23 I is possible o es ima e oughly he ange o alidi y o he KWW unc ion. One can de e mine he ime in e sec ions iand o he KWW unc ion wi h he sho and long ime exponen ials, espec i ely. I is ound ha i54/ p and a 2 .6.28. In he ime in e al ( i, ) he KWW unc ion is expec ed o hold, and he elaxa ion unc ion e i ies exp(24 a / p )> ( )>0.06. Al hough his is a e y c ude es i- ma ion, we conclude ha mos o he ele an pa o he elaxa ion o ( ) a low empe a u es is gi en by he s e ched exponen ial in Eq. ~3.29!, because a !1. In Fig. 1 we ha e plo ed ( ) o be 510, which co e- sponds o a 56.731023. The solid line is he KWW unc- ion gi en by Eq. ~3.23!. As discussed in he pa ag aph abo e, i is alid o e an in e media e ime window co e- sponding o he ele an pa o he elaxa ion. Fo e y long imes, elaxa ion is exponen ial, and he KWW unc ion is no a good app oxima ion. Fo e y sho imes, elaxa ion is also exponen ial, bu he di e ence wi h he KWW unc ion is negligible o e he scale o he igu e. Finally, i mus be ema ked once mo e ha he KWW decay ound a in e media e imes ollows om he exis ence o wo exponen ial egimes alid a e y sho and e y long imes wi h a clea sepa a ion o hei espec i e ime scales. A de ailed discussion can be ound in Re . 18 o any sys em whose dynamics is desc ibed by a mas e equa ion. The main poin is whe he mos o he ele an pa o he elaxa ion can be desc ibed by a KWW unc ion as a consequence o a clea ime scale sepa a ion. This happens in ou model be- cause he elaxa ion spec um becomes e y b oad a low empe a u es. This is no a gene al p ope y o all mas e equa ions, and KWW elaxa ion may no show up o a gi en choice o he ansi ion a es, i he elaxa ion spec um as- socia ed o hem emains na ow a low empe a u es. B. Linea elaxa ion o he ene gy The ene gy elaxa ion a e a empe a u e pe u ba ion is cha ac e ized by he esponse unc ion c ~ !5 ^ E~ ! & 2 ^ E & 0 ^ E~0! & 2 ^ E & 0,~3.30! whe e ^ E~ ! & 5( n50 ` e npn~ !5 e @12p0~ !#.~3.31! Using he de ini ion o Dnin Eq. ~2.22!, we ha e c ~ !5D0~ ! D0~0!.~3.32! Subs i u ion o he exac solu ion o he mas e equa ion o cons an empe a u e ob ained in Sec. II, Eqs. ~2.21!and ~2.22!, leads o c ~ !5 * 0 p dqg~q!cos h ~q!e2 l~q! * 0 p dqg~q!cos h ~q!.~3.33! We ha e o calcula e g(q), om he ini ial condi ions Dn(0). We will conside ha he sys em was in equilib ium a a empe a u e b 1D b . Then, he empe a u e was ins an- aneously changed o b a 50. In he linea esponse ap- p oxima ion, Dn~0!5pn ~0!~ b 1D b !2pn ~0!~ b !5dpn ~0! d b D b ,~3.34! and he unc ion g(q) in Eq. ~2.21! eads FIG. 2. Ene gy elaxa ion in he low- empe a u e egion, o a empe a u e alue co esponding o e /kBT510. The diamonds a e he nume ical e alua ion o Eq. ~3.33!, while he solid line co e- sponds o he KWW unc ion o Eq. ~3.43!. In his loga i hmic scale, we ha e es ic ed ou sel es o posi i e alues o he e- sponse unc ion. FIG. 1. Plo o he equilib ium au oco ela ion unc ion o en- e gy, o a empe a u e alue co esponding o e /kBT510. The dia- monds a e he nume ical e alua ion o Eq. ~3.8!, and he solid line is he s e ched exponen ial o Eq. ~3.29!. 6348 55 A. PRADOS, J. J. BREY, AND B. SA ´NCHEZ-REY g~q!5D b ( n50 ` j n~q!d d b lnpn ~0!.~3.35! The exp ession o he equilib ium dis ibu ion, Eq. ~2.5!, is equi alen o lnpn ~0!52 be ~12 d n0!2 a ~n11!,;n>0, ~3.36! and subs i u ion o Eqs. ~2.16!and ~3.36!in o Eq. ~3.35!, oge he wi h he ela ion d a d b 52 e 2 e2 be /2 11e2 be /2 52 e 2e2 a 2 be /2,~3.37! leads, a e some algeb a, o g~q!5 S 2 p D 1/2 e D b F e~ be 2 a !/2cos h ~q!11 2e2 be 22 a cos@q1 h ~q!#22e2 a /2cos h ~q!1e2 a cos@ h ~q!2q# ~11e2 a 22e2 a /2cosq!2 G .~3.38! The abo e exp ession o g(q) is a he in ol ed o a bi- a y empe a u e. In he low- empe a u e limi , a →0, in o- ducing again he ime scale sde ined by Eq. ~3.19!and he a iable uo Eq. ~3.20!, one ge s c ~ ![ c ¯ ~s!58 p E 0 `duu2~u221! ~11u2!3e2s~11u2!.~3.39! The elaxa ion o he ene gy akes place o e a ime scale o o de a 22, as i was he case o he equilib ium ene gy au- oco ela ion. In he s ime scale, he ini ial exponen ial e- laxa ion does no show up, because he sho ime beha io o c ( ) is gi en by c ~ !.e22 sinh a ,~3.40! and a low empe a u es i s cha ac e is ic ime scale (2 a )21collapses on o he poin s50. Fo e y long imes, s@1, a Laplace analysis o Eq. ~3.39!yields c ¯ ~s!;22 p 1/2 e2s s3/2 .~3.41! The p e ious equa ion ells us ha ene gy elaxa ion is no mono onic. In ac , i is p o ed in Appendix B ha E 0 `d c ~ !50, ~3.42! implying ha c ( ) is nega i e in a ime egion. Howe e , in he in e media e ime window s!1 a s e ched exponen ial decay is again ob ained, hough he gene al a gumen de el- oped in Re . 18 canno be di ec ly applied. Fo s!1, i is easy o show om Eq. ~3.39! ha ln c ~ !;2 S 16 a 2 p D 1/2 .~3.43! The e o e, a simpli ied pic u e o he ene gy elaxa ion a low empe a u es is ob ained, which is simila o he one ound be o e o he ene gy au oco ela ion. In e ms o he h ee ele an ime egimes ha ha e a isen in ou discus- sion, c ~ !5 H e22 a !1, e2~16 a 2 / p !1/2 1! !4 a 22 22 p 21/2~ a 2 /4!23/2e2 a 2 /4 @4 a 22. ~3.44! A e y long imes, he elaxa ion unc ion c ( ) c osses he axis and decays o ze o om nega i e alues. This is qui e a small e ec , since a nume ical es ima ion o he minimum o c ( ) gi es c min.20.05. The e o e, he KWW unc ion in Eq. ~3.44!also gi es a ele an in o ma ion abou he ene gy elaxa ion a low em- pe a u es. In pa icula , i s elaxa ion ime E5 p 16 a 22; p 16e be ,~3.45! can be used o cha ac e ize he elaxa ion o ene gy a e a homogenous pe u ba ion in empe a u e. I mus be e- ma ked ha Ealso ollows an A henius law a low em- pe a u es. A quali a i e explana ion o his beha io , in e ms o he di usi e mo ion o he sys em, can be gi en along he same way as in he p e ious sec ion. Ob iously, he s e ched exponen ial app oxima ion is no able o explain he c ossing o he axis ha akes place a e y long imes, bu i accu- a ely i s mos o he ele an pa o ene gy elaxa ion, namely up o c .0.1. In Fig. 2 he ene gy elaxa ion unc ion ob ained nume i- cally is compa ed wi h he KWW unc ion in Eq. ~3.44!. The alue o he pa ame e is he same as in Fig. 1, i.e., be 510 ( a 56.731023). In he a iables used in Fig. 2, exponen ial elaxa ion co esponds o a s aigh line o uni y slope, while KWW elaxa ion is ep esen ed by a s aigh line o slope equal o he pa ame e g in Eq. ~3.24!. The loga i hm scale used ampli ies he disc epancies, especially o sho imes, whe e he di e ence be ween he KWW unc ion and he ini ial exponen ial is in ac negligible. IV. THERMAL CYCLES He e we a e in e es ed in s udying he beha io o he model when i is con inuously cooled down om high o low empe a u es, and a e wa ds ehea ed. This is usually called a he mal cycle. Upon desc ibing i , he sys em may de ia e om equilib ium while being cooled, leading o he kine ic phenomenon known as he labo a o y glass ansi ion. In he hea ing p ocess, equilib ium is app oached again a high 55 6349GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . . empe a u es, bu he sys em ollows a di e en cu e om he cooling one, and hys e esis shows up. The kine ic beha - io jus discussed is shown by a wide class o ma e ials,2,4 and also by some simple models.9,11–13,21,24 Ne e heless, analy ical esul s a e sca ce,13,21 al hough qui e a gene al ex- plana ion o he hys e esis phenomenon has been gi en.12 I can be unde s ood as he mono onic app oach o a ‘‘no mal’’ cu e, di e en om he equilib ium one, cha ac e izing hea ing p ocesses. As he p oo in Re . 12 was made o he canonical ensemble, a gene aliza ion o he case conside ed he e is p esen ed in Sec. IVB. The emainde o his sec ion is o ganized as ollows. Fi s , we s udy cooling p ocesses, and he exis ence o he labo a o y glass ansi ion. Secondly, hea ing p ocesses a e conside ed, paying special a en ion o he appea ance o hys e esis, and ela ing i o he end o he sys em o ap- p oach he no mal cu e. Le us poin ou ha we ha e no been able o sol e ex- ac ly he mas e equa ion o he case o ime-dependen em- pe a u e, ha implies ha he ansi ion a es a e also ime dependen . The p ocedu e de eloped in Re . 13 is alid when he eigen ec o s o he mas e equa ion do no depend on empe a u e. This is no he case he e, since he eigen ec o s o he mas e equa ion, gi en by Eq. ~2.16!, a e empe a u e dependen h ough he unc ion h (q) in Eq. ~2.17!. The e- o e, we ha e pe o med a Mon e Ca lo simula ion o he mas e equa ion, using a gene aliza ion o he Bo z-Kalos- Lebowi z algo i hm25 o mas e equa ions wi h ime- dependen ansi ion a es.26 Ne e heless, some analy ical es ima ions can be done, and hey will be compa ed wi h nume ical esul s. A. Cooling p ocesses and labo a o y glass ansi ion Now we a e going o s udy he con inuous cooling o he sys em o low empe a u es. In o de o analyze he de ia ion om equilib ium alues o he p ope ies o he sys em, we will ollow a easoning simila o ha used in Re . 13. We s a om he elaxa ion spec um o he mas e equa ion, Eq. ~2.15!, and no ice ha he modes ldepend on empe a u e h ough a , and he e o e hey a e ime dependen in a gi en cooling p og am T( ). In his spec um, he elaxa ion a es o he sys em a y wi h hei label q, om he minimum alue, co esponding o q50, l1511e2 a 22e2 a /25~12e2 a /2!2,~4.1! o he maximum one, o q5 p , l2511e2 a 12e2 a /25~11e2 a /2!2.~4.2! Gi en a cooling law, o each o he elaxa ion modes we can associa e a cha ac e is ic ime scale s~q!5 E 0d 8l~q;T8!,~4.3! whe e 0is he ex apola ed ime o which he empe a u e would anish acco ding o he cooling p og am, and T8[T( 8). The ime s(q) is oughly p opo ional o he e - ec i e numbe o ansi ions le o he mode l(q;T) be o e eaching T50. Fo imes longe han he one (q) making s(q)51, one can conside ha he mode will no expe imen any mo e ansi ions. Thus, o empe a u es lowe han he one co esponding o (q) he con ibu ion o he mode will no e ol e in ime and can be conside ed as ‘‘ ozen.’’ In his way, we can de e mine a eezing empe a u e T(q) o each alue o q. Equi alen ly, one can in oduce he no ion o a ‘‘dema ca ion’’ mode qD(T), such ha modes wi h q<qD(T) a e ozen, while modes wi h q.qD(T) a e s ill elaxing a he gi en empe a u e.13,27 The labo a o y glass ansi ion begins a he empe a u e T1[T( 1) gi en by he ela ion E 1 0d 8l1~T8!51~4.4! o qD~T1!50, ~4.5! i.e., only he slowes elaxa ion a e is ozen, and he de ia- ion om equilib ium s a s o . On he o he hand, he sys- em will be comple ely ozen a a empe a u e T2[T( 2) o which he as es elaxa ion mode does no e ol e any mo e, namely, E 2 0d 8l2~T8!51~4.6! o qD~T2!5 p .~4.7! A global image o he eezing phenomenon can be ob ained by means o he ime scale s5 E 0d 81 ~T8!,~4.8! whe e (T) is he ime cha ac e izing he elaxa ion o he p ope y Pwe a e in e es ed in a e a empe a u e pe u ba- ion. Fo ins ance, in ou model would be he KWW elax- a ion ime Ein Eq. ~3.45!, i we wan o desc ibe he ene gy e olu ion du ing he cooling p ocess. An es ima ion o he ‘‘global’’ eezing empe a u e T o he p ope y Pis ob- ained by making s51, i.e., 15 E 0d 1 ~T!,~4.9! and hen T 5T( ). Since he labo a o y glass ansi ion is e y na ow in empe a u e, a leas when he sys em is slowly cooled, an app oxima ion o he ozen alue o he p ope y Punde conside a ion would be P0(T ), i.e., he equilib ium alue a i s eezing empe a u e T . I is impo an o no e ha he empe a u es T1,T2, and T depend bo h on he cooling a e cand he cooling law (T) de ining he cooling p og am, dT d 52 c ~T!.~4.10! This is also he case in o he simple models whose dynamics is desc ibed in e ms o mas e equa ions. Fo some choices o he cooling law (T), he sys em emains in equilib ium a all empe a u es.6,13 We a e no going o discuss his p oblem 6350 55A. PRADOS, J. J. BREY, AND B. SA ´NCHEZ-REY he e, bu ocus ou a en ion on he beha io o he sys em when i is being linea ly cooled, dT d 52 c,~4.11! i.e., (T)51, which is he mos usual cooling p og am in eal expe imen s4and also in heo e ical s udies o model sys ems.9,11,24,28 The elaxa ion modes l1and l2, Eqs. ~4.1!and ~4.2!, and he ime cha ac e izing he ene gy elaxa ion Ea e w i en as unc ions o a , de ined by Eq. ~2.4!. Then, i is use ul o ans o m he ime in eg al in he de ini ion o he sscales in o an in eg al o e a wi h he aid o d a d 52 c~12e2 a !@ln~e a 21!#2,~4.12! whe e Eq. ~4.11!has been aken in o accoun , and c52kB c e ~4.13! is an adimensional cooling a e, gi ing he ime scale o e which a e ol es. As discussed abo e, he beginning o he labo a o y glass ansi ion is es ima ed o ake place a a ime 1such ha T( 1)5T1, being T1 he empe a u e in Eq. ~4.5!, i.e., he one a which he slowes elaxa ion mode eezes. By using Eqs. ~4.1!and ~4.12!, we can w i e 151 c E 0 a 1d a ~12e2 a /2!2 ~12e2 a !@ln~e a 21!#2,~4.14! whe e a 1[ a ( 1). In he limi o slow cooling, c!1, and i ollows ha a 1!1. Fo his case, Eq. ~4.14!simpli ies o 151 4 c E 0 a 1d aa ~ln a !2.~4.15! To sol e his ela ion o a 1, we make he change o a iable a 5 a 1x, E 0 a 1d aa ~ln a !25 a 1 2 ~ln a 1!2 E 0 1dx x @11~lnx/ln a 1!#2 ; a 1 2 2~ln a 1!2.~4.16! The las in eg al can be done by di iding he in e al (0,1) in o he wo subin e als (0, u ln a 1 u 21) and ( u ln a 1 u 21,1). In he i s in e al, he in eg and is bounded by uni y, and he in- eg al is negligible. In he second in e al, i is u lnx u ! u ln a 1 u , gi ing ise o he esul in Eq. ~4.16!. Subs i u ion in o Eq. ~4.15!yields 151 8 c a 1 2 ~ln a 1!2.~4.17! By making use o he slow cooling condi ion, a 1!1, we ha e 2ln a 1;ln~8 c!.~4.18! Now, we ake in o accoun ha a 1;exp(2 b 1 e /2), o ge T1; e kB 1 u ln~8 c! u .~4.19! In o de o calcula e he ic i e empe a u e T , we s a om Eq. ~4.9!, wi h he elaxa ion ime o ene gy Egi en by Eq. ~3.45!, 15 E 0 a d a d d a E 21~ a !516 p c E 0 a d aa 2 ~e a 21!@ln~e a 21!#2. ~4.20! As be o e, a is he alue o a co esponding o T . Fo slow cooling, i is a !1, since a , a 1. Then, he abo e equa ion educes o 1516 p c E 0 a d aa ~ln a !2.~4.21! In his way, we ha e a i ed a an exp ession simila o Eq. ~4.15! o a 1. The e o e, E 0 a d aa ~ln a !2; a 2 2~ln a !2~4.22! and 158 p c a 2 ~ln a !2.~4.23! Again, a easoning along he line o he one abo e Eq. ~4.19! gi es us T ; e kB 1 u ln~ p c/8! u .~4.24! A simila dependence on he cooling a e is ob ained in eal expe imen s,4and has also been ound in Glaube ’s Ising model.13 Taking in o accoun he commen below Eq. ~4.9! one can es ima e he esidual alue o he ene gy, i.e., e [lim T→0 @ ^ E & ~T!2 ^ E & 0~T!#5 ^ E & 0~T !.~4.25! In he limi o slow cooling, c!1, Eq. ~4.25!leads o a po en ial dependence on he cooling a e o he esidual en- e gy, e } c 1/2 . A simila beha io o he esidual p ope ies has been ob ained in some models o glasses.5,13,6 In Fig. 3 he e olu ion o he mean ene gy o he cooling p og am in Eq. ~4.11!, wi h an adimensional cooling a e c50.02, is plo ed. The depa u e om equilib ium oughly begins a he empe a u e ob ained om Eq. ~4.19!, kBT1/ e .0.55. The es ima ion o he eezing empe a u e, ob ained by using Eq. ~4.24!is kBT / e .0.21, in good ag ee- men wi h he nume ical esul . The ozen alue o he en- e gy gi en by he Mon e Ca lo simula ion is ^ E & / e 50.076, while he alue ob ained om T is ^ E & 0(T )/ e 50.081. Again, he app oxima ed heo y p o ides a easonable es i- ma ion o he ac ual alue. 55 6351GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . .