Linear response of vibrated granular systems to sudden changes in the vibration intensity

Linea esponse o ib a ed g anula sys ems o sudden changes in he ib a ion in ensi y J. Ja ie B ey and A. P ados Fı ´sica Teo ´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, 41080, Se illa, Spain 共Recei ed 15 Janua y 2001; published 10 May 2001兲 The sho - e m memo y e ec s ecen ly obse ed in ib a ion-induced compac ion o g anula ma e ials a e s udied. I is shown ha hey can be explained by means o qui e plausible hypo hesis abou he mesoscopic desc ip ion o he e olu ion o he sys em. The exis ence o a c i ical ime sepa a ing egimes o ‘‘anomalous’’ and ‘‘no mal’’ esponses is p edic ed. A simple model i ing in o he gene al amewo k is analyzed in he de ail. The ela ionship be ween his pape and p e ious s udies is discussed. DOI: 10.1103/PhysRe E.63.061301 PACS numbe 共s兲: 45.70.Cc, 61.43.Fs, 81.05.Rm I. INTRODUCTION Expe imen s ha e shown ha when a loose packing o g ains is submi ed o e ical ib a ion o ‘‘ apping,’’ i slowly app oaches a s eady s a e o highe packing ac ion 关1,2兴. The inal s eady densi y is a dec easing unc ion o he dimensionless pa ame e cha ac e izing he ib a ion in en- si y. Mo eo e , he elaxa ion is slowe o smalle ib a ion in ensi y. In he ime e olu ion o he sys em nei he con ec- ion e ec s no oscilla o y beha io a e obse ed. The s udy o he kine ics o compac ion is impo an bo h om a o mal poin o iew and because o i s economical ele ance in many indus ial p ocesses. Mos o he peculia beha io s exhibi ed by g anula ma e ials submi ed o ib a ion o ap- ping p ocesses show a g ea simila i y wi h con en ional s uc u al glasses. This includes slow elaxa ion, annealing p ope ies, and hys e esis e ec s. The i s s udy o he esponse o a g anula sys em o a sudden change in he ib a ion in ensi y we a e awa e o , was ca ied ou by means o nume ical simula ions o a model o compac ion 关3兴, and he da a indica ed he p es- ence o memo y e ec s in he e olu ion o he densi y o he sys em. Ve y ecen ly 关4兴, memo y e ec s we e also di ec ly obse ed in a se ies o expe imen s. The esul s showed ha he sys em has a sho - e m memo y o i s shaking his o y, so ha he esponse in he e olu ion o he densi y o a change in he ib a ion in ensi y a a gi en ime, is no de e mined by he densi y a ha ime. Ma hema ically, his phenomenon implies ha he ime e olu ion o he densi y does no obey a closed o dina y i s -o de di e en ial equa ion. In his pape , we p opose a gene al heo e ical amewo k o unde s and he o igin and cha ac e is ic ea u es o he memo y e ec s seen bo h in simula ions and in expe imen s. Using qui e plausible hypo hesis, we will be able o explain he sho - ime esponse o he sys em o a small change in he ib a ion in ensi y. In pa icula , he heo y p edic s ha a dec ease 共inc ease兲in he in ensi y can lead o an inc ease 共dec ease兲o he compac ion a e on sho - ime scales, in ag eemen wi h expe imen s. Ne e heless, is no necessa ily so. I he change in he in ensi y is made a he ea ly s ages o he compac ion p ocess, he heo y we will de elop leads o a modi ica ion o he compac ion a e ha ing he same sign as he in ensi y change. In ac , he e is a c i ical ime, which depends on he apping in ensi y be o e he change, sepa a - ing he egions o ‘‘no mal’’ and ‘‘anomalous’’ esponses. The exis ence o hese wo di e en egimes has no been e i ied expe imen ally up o now, al hough such a beha io has been nume ically obse ed in a simple model o g anu- la compac ion 关5兴. As an illus a ion o he heo y, we discuss i s applica ion o a model o compac ion in oduced ecen ly 关6,7兴. The model has al eady been shown o ep oduce he quali a i e beha io o g anula ma e ials unde apping. He e we will show ha i also cap u es he same sho - e m memo y e - ec s seen in he expe imen s. Mo eo e , i i s pe ec ly in he gene al scheme de eloped in his pape , he e o e p o id- ing a i s es o alida ion o he ideas in which he heo y is based. We ha e also used he model o in es iga e he elaxa ion o he sys em ollowing a pe u ba ion in he i- b a ion in ensi y o a sho - ime pe iod. This idea also o igi- na es om he expe imen s epo ed in Re . 关4兴. The esul s indica e ha he esponse unc ion is accu a ely desc ibed by a Kohl ausch-Williams-Wa s 共KWW兲o s e ched exponen- ial unc ion. The pape is o ganized as ollows. In he nex sec ion, some gene al p ope ies o he equa ion go e ning he ime e olu ion o he densi y in apping p ocesses o g anula me- dia a e discussed. These p ope ies a e used in Sec. III o analyze he sho - e m memo y e ec s by conside ing he esponse o he sys em o a small change in he ib a ion in ensi y. The heo y is pa icula ized o a simple model o apping in Sec. IV, whe e o he pa e ns o change o he ib a ion in ensi y a e also conside ed. The choices we e o igina ed om he expe imen s epo ed in Re . 关4兴. Finally, Sec. V con ains some addi ional commen s and inal e- ma ks, as well as a ele an discussion o he ela ionship o ou pape o p e ious expe imen al and heo e ical s udies. II. EVOLUTION OF THE DENSITY IN DISCRETE TAPPING PROCESSES Le ⌫deno e he dimensionless pa ame e cha ac e izing he in ensi y o he ib a ion applied o he g anula medium. In ypical expe imen s 关1兴,⌫is de ined as he a io o peak accele a ion o a ap o he g a i y g. Unde e y gene al condi ions, he ime e olu ion o he densi y ␳ in a disc e e apping p ocess will be gi en a a mesoscopic le el by an equa ion o he o m PHYSICAL REVIEW E, VOLUME 63, 061301 1063-651X/2001/63共6兲/061301共9兲/$20.00 ©2001 The Ame ican Physical Socie y63 061301-1 ␳ ˙⬅d ␳ 共 兲 d ⫽ 1共⌫兲 ␮ 1共 兲⫺ 2共⌫兲 ␮ 2共 兲.共1兲 He e he ime is measu ed in uni s o comple e aps in some con inuous limi , 1(⌫) and 2(⌫) a e semide ini e posi i e unc ions o ⌫ha ing dimensions o equency, and ␮ 1( ) and ␮ 2( ) a e posi i e quan i ies depending on he s a e o he sys em, bu hey a e no uni ocally de e mined by he densi y a he same ins an ␳ ( ). The e o e, Eq. 共1兲is no in gene al a closed equa ion and canno be sol ed by i sel . The wo e ms on he igh -hand side o he equa ion desc ibe elemen a y p ocesses inc easing and dec easing he densi y, espec i ely. The s uc u e o Eq. 共1兲as a gain-loss equa ion is consis- en wi h he expe imen al obse a ions in compac ion p o- cesses, as we will discuss in de ail in he ollowing. Also, i he elemen a y e en s aking place in he sys em being i- b a ed can be desc ibed by means o a Mas e equa ion, a o mal equa ion like his ollows di ec ly. This is he case o some simple kine ic models o compac ion in oduced e- cen ly 关6–9兴. Since Eq. 共1兲desc ibes he e olu ion o he densi y as a consequence o apping, he unc ions 1and 2mus anish in he limi o no apping ⌫⫽0, so ha 1共0兲⫽ 2共0兲⫽0. 共2兲 Because o con inui y, i ollows, a leas o small alues o he in ensi y ⌫, ha 1 ⬘共⌫兲⬅d d⌫ 1共⌫兲⬎0, 2 ⬘共⌫兲⬅d d⌫ 2共⌫兲⬎0. 共3兲 We will assume ha he abo e inequali ies hold o a bi a y ⌫. The physical eason o his assump ion is ha we expec he numbe o elemen a y p ocesses aking place in he sys- em o inc ease as ⌫inc eases. O cou se, his does no imply by i sel ha he a e o a ia ion o he densi y also in- c eases. The beha io o ␳ ˙depends on he ne balance be- ween he gain and loss elemen a y e en s, as indica ed by Eq. 共1兲. This pic u e is in ag eemen wi h he quali a i e ole o empe a u e played by he shaking in ensi y in many di - e en aspec s 关2,7,10–13兴. In he long- ime limi o a apping p ocess wi h cons an ⌫, he expe imen s show ha he sys em eaches a s eady s a e wi h a densi y ␳ s, which is a mono onic dec easing unc ion o ⌫, as displayed by he ‘‘ e e sible’’ b anch in cycling expe imen s 关9兴. Le us poin ou ha he elaxa ion p ocess is e y slow, and o e y small alues o ⌫ he s eady densi y is ha d o each wi hin he expe imen al ime scale. The e o e, he unc ion ␳ s(⌫) e i ies ha d ␳ s(⌫)/d⌫⬍0, and i is bounded by he wo o mal limi s ␳ min⫽lim ⌫→⬁ ␳ s共⌫兲, ␳ max⫽lim ⌫→0 ␳ s共⌫兲.共4兲 Pa icula iza ion o Eq. 共1兲 o a s eady s a e yields 1共⌫兲 ␮ 1s⫽ 2共⌫兲 ␮ 2s,共5兲 whe e ␮ 1sand ␮ 2sdeno e he s eady alues o he quan i ies ␮ 1and ␮ 2, espec i ely. As poin ed ou abo e, ␮ 1( ) and ␮ 2( ) a e no expec ed o be simply unc ions o ␳ ( )in gene al. Bu , on he o he hand, i seems sensible o assume ha he s eady s a e eached by a gi en sys em in a apping expe imen is ully de e mined by he in ensi y ⌫o , equi a- len ly, by ␳ s. The e o e, we assume ha ␮ 1sand ␮ 2sa e unc ions o ␳ s, and in he ollowing we a e going o in es- iga e some quali a i e p ope ies o hese unc ions. Fo ␳ s → ␳ min , ␮ 2s( ␳ s) mus anish, since by de ini ion a ␳ s ⫽ ␳ min he e a e no p ocesses dec easing he densi y. The e- o e, i is lim ␳ s→ ␳ min ␮ 2s共 ␳ s兲⫽0, lim ␳ s→ ␳ min ␮ 1s共 ␳ s兲⬎0. 共6兲 The second ela ionship exp esses ha s a ing om he den- si y ␳ min , any apping p ocess o a bi a y in ensi y ⌫can only p oduce an inc ease o he densi y. Wha happens in he s eady high-densi y limi ? A simila a gumen o he one ca ied ou abo e would lead o lim ␳ s→ ␳ max ␮ 1s共 ␳ s兲⫽0, lim ␳ s→ ␳ max ␮ 2s共 ␳ s兲⬎0. 共7兲 Ne e heless, some ca e is equi ed when analyzing his limi . Simple models o disc e e apping lead o an abso - ben s eady s a e in he high-densi y limi 关6,7兴. Tha means ha he sys em will no be able o lea e his s a e when submi ed o apping o a bi a y in ensi y. This is equi alen o saying ha ␮ 2s( ␳ s) also anishes o ␳ s→ ␳ max .Asa consequence, and in o de o include such a possibili y in ou o mula ion, ins ead o Eq. 共7兲we will assume he mo e gen- e al and p ecise condi ion lim ␳ s→ ␳ max ␮ 1s共 ␳ s兲 ␮ 2s共 ␳ s兲⫽0, 共8兲 i.e., ␮ 2sⰇ ␮ 1swhen ␳ s→ ␳ max , and he densi y loss e m is dominan in ha limi . Le us no e ha Eq. 共6兲yield lim ␳ s→ ␳ min ␮ 1s共 ␳ s兲 ␮ 2s共 ␳ s兲⫽⬁.共9兲 The simples beha io ha is consis en wi h Eqs. 共8兲and 共9兲is ha he a io ␮ 1s/ ␮ 2sbe a mono onic dec easing unc- ion o he s eady densi y ␳ sgoing om in ini y o ze o. Since he e is no any physical eason o expec a mo e com- plica ed densi y dependence, we assume his is he case in ou o malism. F om he s eady condi ion gi en by Eq. 共5兲,i ollows ha ␮ 1s ␮ 2s ⫽ 2共⌫兲 1共⌫兲⬅g共⌫兲.共10兲 The unc ion g(⌫) is a measu e o he a e o he decompac- ion p ocesses wi h espec o he a e o he compac ion ones. Because o Eqs. 共8兲and 共9兲,i is J. JAVIER BREY AND A. PRADOS PHYSICAL REVIEW E 63 061301 061301-2 lim ⌫→⬁ g共⌫兲⫽⬁, lim ⌫→0g共⌫兲⫽0. 共11兲 Taking he de i a i e wi h espec o he in ensi y ⌫in Eq. 共10兲, we ob ain: dg共⌫兲 d⌫⫽d ␳ s d⌫ d d ␳ s 冉 ␮ 1s ␮ 2s 冊 ⬎0, 共12兲 whe e we ha e aken in o accoun he mono onically de- c easing densi y dependence o ␮ 1s/ ␮ 2sassumed abo e. The physical meaning o Eq. 共12兲is e iden ; he a e o he decompac ion p ocesses g ows as e wi h ⌫ han he a e o he compac ion p ocesses. Also his implica ion o ou as- sump ions seems physically plausible. In summa y, we can w i e he equa ion o he ime e o- lu ion o he densi y in disc e e apping p ocesses as d ␳ 共 兲 d ⫽ 1共⌫兲关 ␮ 1共 兲⫺g共⌫兲 ␮ 2共 兲兴,共13兲 wi h 1and gbeing posi i e inc easing unc ions o ⌫, bo h o hem anishing in he limi ⌫→0. The quan i ies ␮ 1( ) and ␮ 2( ) a e some momen s o he comple e dis ibu ion unc ion o he sys em, and hey con ain he in luence o co ela ions on he e olu ion o he densi y. As a conse- quence, Eq. 共13兲is no a closed equa ion. Because g(⌫) anishes o ⌫→0, i a apping expe imen wi h low enough in ensi y ⌫is ca ied ou , he decompac ion e m 1(⌫)g(⌫) ␮ 2( ) will be negligible in he i s s ages o he p ocess, i.e., ␮ 1共 兲 g共⌫兲 ␮ 2共 兲Ⰷ1, 共14兲 and he e olu ion o he sys em will be app oxima ely de- sc ibed by d ␳ 共 兲 d ⯝ 1共⌫兲 ␮ 1共 兲.共15兲 A much la e imes, when ␳ ( ) is close enough o he asymp o ic s eady alue, he decompac ion con ibu ion in Eq. 共13兲plays a decisi e ole, leading o a s eady densi y ␳ s⬍ ␳ max , and i is ␮ 1共 兲 g共⌫兲 ␮ 2共 兲⫽O共1兲.共16兲 The obse ed beha io ha he sys em ends owa ds a s eady s a e and, he e o e, a egime whe e ␮ 1, ␮ 2, and g e i y a ela ionship o he o m gi en in Eq. 共16兲, can be unde s ood i ␮ 1( ) dec eases in ime while ␮ 2inc eases. Qui e in e es ingly, his is consis en wi h a mean- ield ap- p oxima ion in which ␮ 1( ) is eplaced by ␮ 1s关 ␳ ( )兴and ␮ 2( )by ␮ 2s关 ␳ ( )兴. Since ␳ ( ) inc eases mono onically in ime, and ␮ 1s/ ␮ 2sis a mono onic dec easing unc ion o he densi y, i ollows ha he le -hand side o Eq. 共16兲will decay in ime. O cou se, as long as Eq. 共15兲is accu a e, he la ge he ⌫ he as e he compac ion o he sys em, in ag eemen wi h expe imen s. O e a la ge ime scale, he comple e Eq. 共15兲, including he decompac ion e m, is needed in o de o ex- plain he dependence o he s eady densi y ␳ son ⌫, and also he exis ence o a a slow long- ime ail in he elaxa ion o he densi y, once ␳ s⫺ ␳ ( ) is e y small. In his con ex , he p esence o an ‘‘anomalous’’ densi y elaxa ion, ollowing an in e se loga i hm law, would be associa ed o some spe- ci ic dynamical p ope ies o he compac ion e m ␮ 1when he sys em is submi ed o ‘‘nonlinea ’’ apping p ocesses 关1,2兴. We use he e m ‘‘nonlinea ’’ he e in he sense ha , in he expe imen s, he ini ial alue o he densi y is no e y close o he s eady densi y. La e on, we will show ha an e olu ion equa ion like Eq. 共13兲applies in he case o a simple model ecen ly in o- duced o desc ibe disc e e apping 关6,7,10兴. Ano he simila equa ion is ob ained o he ‘‘pa king’’ model 关8,9,14,15兴, al hough his la e e e s o con inuous ib a ion p ocesses, in which he sys em is no allowed o elax o a me as able con igu a ion be ween e e y wo ib a ion cycles. In he pa king model, he s a e o maximum densi y ␳ max is no o ally abso ben , bu his possibili y has been included in ou heo y, as discussed below Eq. 共7兲. By iden i ying he in ensi y o apping ⌫wi h he a io be ween he deso p ion and adso p ion a es in he pa king model, i is i ial o check by using he exp essions in Re . 关8兴 ha he quan i ies co esponding o ␮ 1sand ␮ 2s e i y ha hei a io is an inc easing unc ion o ␳ s, and also he limi ing beha io gi en in Eqs. 共8兲and 共9兲. Mo eo e , he s eady densi y is an inc easing unc ion o he quan i y playing he ole o he ib a ion in ensi y. In conclusion, he pa king model belong o he gene al class o sys ems we ha e conside ed. III. RESPONSE TO SMALL VIBRATION INTENSITY JUMPS In his sec ion we will in es iga e whe he Eq. 共13兲, which has been buil unde e y gene al a gumen s and is expec ed o ha e a wide ange o applicabili y, is able o p edic he memo y e ec s ecen ly obse ed in ib a ion-induced com- pac ion in g anula ma e ials 关4兴. The ac ha he equa ion is no closed o he densi y, implies ha i s ime e olu ion in a gi en expe imen wi h cons an ⌫is no de e mined by i s he ini ial alue. S a ing om he same alue ␳ 0, di e en ime e olu ions a e possible depending on he way in which he sys em was p epa ed. Ou aim is o analyze some pa - icula ele an mani es a ions o his gene al s a emen . Conside ha , s a ing om a gi en con igu a ion, he sys- em is apped wi h an in ensi y ⌫. A a ce ain ime w, he in ensi y is ins an aneously changed o ⌫⫹⌬⌫. Qui e pecu- lia ly, i has been obse ed in he expe imen s ha he change in he compac ion a e has opposi e sign ha o ⌬⌫ on sho ime scales, hough in he long- ime egime he e- laxa ion is slowe o smalle alues o he in ensi y o i- b a ion ⌫. The same kind o e ec has also been p e iously ound nume ically in some models o compac ion 关5,16,3兴, al hough i only shows up when he ime in e al wis no oo sho . I ⌫is changed a he beginning o he compac ion LINEAR RESPONSE OF VIBRATED GRANULAR . . . PHYSICAL REVIEW E 63 061301 061301-3 p ocess, he a ia ion o he compac ion a e has he same sign as ⌬⌫ 关5,17兴. Applica ion o Eq. 共13兲 o he ins an w ⫺, jus be o e he change in he in ensi y o he ib a ion, yields w⬅ ␳ ˙共 w ⫺兲⫽ 1共⌫兲关 ␮ 1w ⫺⫺g共⌫兲 ␮ 2w ⫺兴,共17兲 whe e ␮ 1w ⫺⫽ ␮ 1( w ⫺) and ␮ 2w ⫺⫽ ␮ 2( w ⫺). When he in ensi y o ib a ion is changed in o ⌫⫹⌬⌫, he compac ion a e becomes w ⬘⬅ ␳ ˙共 w ⫹兲⫽ 1共⌫⫹⌬⌫兲关 ␮ 1w ⫹⫺g共⌫⫹⌬⌫兲 ␮ 2w ⫹兴.共18兲 The con inui y o he dis ibu ion unc ion o he sys em im- plies ha ␮ 1 ⫺⫽ ␮ 1 ⫹and ␮ 2 ⫺⫽ ␮ 2 ⫹ o an ins an aneous jump o ⌫, al hough he e is a discon inui y ⌬ w⫽ w ⬘⫺ win he compac ion a e. Fo ⌬⌫ small we can app oxima e ⌬ w ⌬⌫ ⫽ 1 ⬘共⌫兲关 ␮ 1w⫺g共⌫兲 ␮ 2w兴⫺ 1共⌫兲g⬘共⌫兲 ␮ 2w⫽ 1 ⬘共⌫兲 1共⌫兲 w ⫺ 1共⌫兲g⬘共⌫兲 ␮ 2w.共19兲 The e o e, i o e he compac ion cu e co esponding o in ensi y ⌫, we de ine he unc ion ␭共 兲⫽ 1 ⬘共⌫兲 1共⌫兲 共 兲⫺ 1共⌫兲g⬘共⌫兲 ␮ 2共 兲,共20兲 he sign o his unc ion a he ime wwhen he in ensi y is changed will de e mine he ela i e beha io o ⌬ wwi h espec o ⌬⌫, o in ini esimal changes o he la e . I ␭w ⬅␭( w)⬍0, he anomalous esponse obse ed in he expe i- men s will ollow, while i ␭w⬎0 he compac ion a e will change in he same di ec ion as ⌬⌫. Le us analyze he sign o he unc ion ␭( ). In he long- ime limi , o mally w →⬁, he sys em is known o each he asymp o ic s eady densi y, so ha w→0 and, consequen ly, ␭⬁⫽lim →⬁ ␭共 兲⫽⫺ 1共⌫兲g⬘共⌫兲 ␮ 2s共⌫兲⬍0, 共21兲 whe e i has been aken in o accoun ha bo h 1(⌫) and g1(⌫) a e posi i e inc easing unc ions o ⌫and ha ␮ 2s(⌫)⬎0. On he o he hand, i he ini ial densi y in he expe imen is he minimum possible densi y o he sys em a es ␳ min , co esponding o he andom loose packing con igu a ion, i ollows om he p ope ies o ␮ 2 ha lim →0 ␭共 兲⫽ 1 ⬘共⌫兲 1共⌫兲 共0兲⬎0. 共22兲 E en hough we ha e conside ed in ou discussion ha ␳ ( ⫽0)⫽ ␳ min in o de o de i e he abo e inequali y, he same esul will apply i he ini ial densi y is close enough o i , so ha he i s e m on he igh -hand side o Eq. 共20兲domi- na es he second one in he ini ial egime. Then, we conclude ha o sho imes, ⌬ wand ⌬⌫ ha e he same sign, while o la ge imes hei signs a e opposi e. This ende s compa ible and explains wha is seen in he expe imen s and also in nume ical s udies o simple models. F om ou analysis i ollows ha he e is 共a leas 兲a ime c, which depends on he alue o ⌫, such ha he esponse o he sys em o a small a ia ion o he in ensi y o apping is quali a i ely di e en o ⬍ cand ⬎ c. The s udy ca ied ou in his sec ion has been es ic ed o small ins an aneous changes in ⌫, allowing he use o a lin- ea analysis o Eq. 共13兲. Whe he he beha io o he sys em emains he same when submi ed o a ini e change in he shaking in ensi y, i canno be in e ed om ou analysis. In his case, nonlinea e ec s can modi y d ama ically he e- sponse o he sys em. Mo e will be said abou his in he nex sec ion o he pape . IV. APPLICATION TO A SIMPLE MODEL FOR COMPACTION The gene al scena io de eloped in he p e ious sec ions will be pa icula ized he e o a one-dimensional la ice model o compac ion 关7,10兴. In he model, each si e ican be ei he emp y o occupied by a pa icle. A a iable miis de- ined, being mi⫽1 in he o me case and mi⫽0 in he la e . A con igu a ion o he sys em is ully speci ied by gi ing he alues o all he a iables m⬅ 兵 mi 其 . As usual, we will e e o he emp y si es as being occupied by a hole. Le us desc ibe he dynamics o he sys em when submi - ed o a disc e e apping p ocess. Mechanical s abili y e- qui es ha all he holes be isola ed, i.e., su ounded by wo pa icles, a he end o e e y ap. The ime e olu ion o he sys em is de ined as a Ma ko p ocess, and o mula ed by means o a Mas e equa ion o he p obabili y dis ibu ion o he sys em 关7,10兴. The equa ion con ains he ansi ion a es W(m 兩 m⬘) om s a e m⬘ o s a e m. The e a e h ee kinds o possible ansi ions. Indica ing only he a iables associa ed o he si es in ol ed in he ansi ions, he non anishing an- si ion a es a e 共1兲Elemen a y di usi e e en s conse ing he numbe o pa icles, W共010 兩 100兲⫽W共010 兩 001兲⫽ ␣ 2,共23兲 共2兲T ansi ions inc easing he numbe o pa icles, W共010 兩 101兲⫽ ␣ 2,共24a兲 W共001 兩 101兲⫽W共100 兩 101兲⫽ ␣ 4.共24b兲 共3兲T ansi ions inc easing he numbe o holes, i.e., dec eas- ing he numbe o pa icles, W共01010 兩 00100兲⫽ ␣ 2 2,共25a兲 W共01010 兩 01000兲⫽W共01010 兩 00010兲⫽ ␣ 2 4.共25b兲 J. JAVIER BREY AND A. PRADOS PHYSICAL REVIEW E 63 061301 061301-4 In he abo e equa ions, ␣ is a posi i e cons an , cha ac e iz- ing he apping p ocess comple ely, and playing in he model a ole simila o he in ensi y o ib a ion ⌫in eal expe i- men s. Fo ␣ ⫽0, he sys em e ol es om any a bi a y ini- ial con igu a ion o a inal s eady s a e wi h densi y ␳ s共 ␣ 兲⫽1 2关1⫹共1⫹4 ␣ 兲⫺1/2兴.共26兲 F om he e i ollows ha lim ␣ →0 ␳ s⫽1⬅ ␳ max , lim ␣ →⬁ ␳ s⫽1 2⬅ ␳ min ,共27兲 being d ␳ s d ␣ ⬍0, 共28兲 o all ␣ . The e o e he densi y in he model has he same kind o dependence on he in ensi y ␣ as assumed in he gene al discussion in Sec. II. The ime e olu ion o ␳ is ob ained om he Mas e equa ion o he model, and eads 关18兴 ␳ ˙⫽ ␣ x101共 兲⫺ ␣ 2 2 冋 x00100共 兲⫹1 2x01000共 兲⫹1 2x00010共 兲 册 , 共29兲 whe e x010 is he concen a ion o h ee-si e clus e s o he o m hole-pa icle-hole, x00100 is he concen a ion o i e- si e clus e s o med by a hole be ween wo pai s o pa icles, and so on. Compa ison o Eqs. 共1兲and 共29兲allows o iden i y 1共 ␣ 兲⫽ ␣ , 2共 ␣ 兲⫽ ␣ 2,共30兲 ␮ 1共 兲⫽x101共 兲, ␮ 2共 兲⫽1 2x00100共 兲⫹1 4关x01000共 兲⫹x00010共 兲兴.共31兲 In he s eady s a e, he only co ela ions in he sys em a e hose o bidding o ha e wo nea es -neighbo holes 关18兴. Then, i is a simple ma e o compu e he s eady alues o he se e al clus e concen a ions appea ing in Eq. 共31兲wi h he esul ␮ 1s⫽共1⫺ ␳ s兲2 ␳ s,共32兲 ␮ 2s⫽共1⫺ ␳ s兲共2 ␳ s⫺1兲2 ␳ s 2.共33兲 In he limi ␳ s→ ␳ min⫽1/2, ␮ 1s→ 1 2, ␮ 2s→0, 共34兲 while in he high-densi y limi ␳ s→ ␳ max⫽1, bo h ␮ 1sand ␮ 2s anish, as a consequence o he abso ben cha ac e o he s a e wi h all he si es occupied by pa icles. The a io ␮ 1s共 ␳ s兲 ␮ 2s共 ␳ s兲⫽ ␳ s共 ␳ s⫺1兲 共2 ␳ s⫺1兲2共35兲 anishes in his la e limi . Equa ions 共34兲and 共35兲a e in ag eemen wi h Eqs. 共6兲and 共8兲. Mo eo e , ␮ 1s( ␳ s)/ ␮ 2s( ␳ s) is a mono onic dec easing unc ion o ␳ sand, consis en ly 关see Eq. 共10兲兴, g共 ␣ 兲⫽ 2共 ␣ 兲 1共 ␣ 兲⫽ ␣ 共36兲 is an inc easing unc ion o ␣ , anishing in he limi ␣ →0. We conclude ha his model o compac ion i s pe ec ly he gene al pic u e de eloped in he p e ious sec ions. Equa- ion 共13兲pa icula ized o he model is d ␳ 共 兲 d ⫽ ␣ 共 ␮ 1⫺ ␣␮ 2兲,共37兲 wi h ␮ 1and ␮ 2de ined in Eq. 共31兲. To sol e Eq. 共37兲we would need some 共app oxima e兲exp essions o he clus e concen a ions as unc ions o he densi y. I we submi he sys em o he apping expe imen de- sc ibed in Sec. III, he e ec o he in ensi y change ⌬ ␣ a ⫽ won he compac a e will be gi en by ⌬ w ⌬ ␣ ⫽ w ␣ ⫺ ␣␮ 2w,共38兲 in he limi o small ⌬ ␣ . The e o e, he unc ion de e mining whe he he esponse o he sys em will be ‘‘no mal’’ o ‘‘anomalous’’ is FIG. 1. Time e olu ion o he unc ion ␭de ined in Eq. 共39兲, o a ib a ion in ensi y ␣ ⫽0.15 共solid line兲. Also plo ed is he mean- ield app oxima ion o ␭共dashed line兲. LINEAR RESPONSE OF VIBRATED GRANULAR . . . PHYSICAL REVIEW E 63 061301 061301-5 ␭共 兲⫽ 共 兲 ␣ ⫺ ␣␮ 2共 兲.共39兲 In Fig. 1 his unc ion is plo ed o ␣ ⫽0.15. The cu e has been ob ained by Mon e Ca lo simula ion o he Mas e equa ion o he sys em. The da a ep esen an a e age o e 10 di e en uns. The ini ial s a e was he one co esponding o he s eady minimum densi y. Fo his pa icula alue o he in ensi y ␣ ,␭( ) changes sign be ween aps 19 and 20, i.e., 19⭐ c⭐20. Fo compa ison pu poses, we ha e also plo ed he mean- ield app oxima ion o ␭( )共dashed line兲. The la e has been cons uc ed by subs i u ing in Eq. 共39兲 ␮ 1( ) and ␮ 2( )by ␮ 1s关 ␳ ( )兴and ␮ 2s关 ␳ ( )兴, espec i ely, and using o he densi y he simula ion esul s. I is seen ha he mean- ield app oxima ion also changes sign, bu o la ge imes, and i is always abo e he ‘‘exac ’’ Mon e Ca lo cu e. This is consis en wi h he mean- ield app oxima ion gi ing a as e app oach o he s eady s a e han he ac ual elaxa ion o he sys em 关18兴. Acco ding o he esul s de i ed in his pape , ⌬ wis expec ed o ha e a di e en sign han ha ⌬ ␣ o w⬎ cand he same o w⬍ c. In o de o check his heo e ical p e- dic ion, we ha e ca ied ou se ies o Mon e Ca lo simula- ions, all o hem s a ing in he minimum densi y con igu a- ion, wi h ␣ ⫽0.15. A w⫽50, he alue o he in ensi y ␣ was ins an aneously changed o ␣ ⬘. The esul s o ou di - e en alues o ␣ ⬘a e epo ed in Fig. 2, namely, 0.1,0.125,0.15,0.175, and 0.2, om op o bo om. The cen- al alue co esponds o no change. Since in hese simula- ions i is w⬎ c, he compac ion a e is obse ed o dec ease as he alue o ␣ ⬘inc eases. I is also seen ha he ampli ude o he jump in he compac ion a e is la ge o ␣ ⬘⫽0.2 (⌬ ␣ ⫽0.05) han o ␣ ⬘⫽0.1 (⌬ ␣ ⫽⫺0.05). This ea u e canno be explained by Eq. 共38兲, and i is due o nonlinea e ec s ha ha e been neglec ed in he linea app oxima ion used he e. This will be analyzed below. In Fig. 3 he same se o expe imen s is ca ied ou , wi h he only di e ence ha in his case he in ensi y ␣ is modi- ied a w⫽10⬍ c. The se e al cu es co espond o he same alues as in Fig. 2, bu now hey a e o de ed om bo om o op. As p edic ed by he heo y, he a ia ion o he compac ion a e has he same sign as he change in ␣ . Mo e- o e , he same kind o nonlinea e ec s as in Fig. 2 a e p esen . Now, we will b ie ly discuss he nonlinea co ec ions in ⌬ ␣ o he change in he compac ion a e. I is easy o show ha ⌬ w⫽⌬ ␣ ␭w⫺共⌬ ␣ 兲2 ␮ 2w.共40兲 The second e m on he igh -hand side o Eq. 共40兲is ne- glec ed in he linea app oxima ion. In his simple model, he nonlinea co ec ion is always nega i e, so i can modi y d ama ically he esponse o he sys em o he jump ⌬ ␣ i he linea e m ⌬ ␣ ␭w⬎0. In pa icula , he e is a c i ical alue ⌬ ␣ c⫽␭w ␮ 2w共41兲 such ha ⌬ w⫽0. Fo smalle jumps, 兩 ⌬ ␣ 兩 ⬍ 兩 ⌬ ␣ c 兩 , he sign o ⌬ wis he one p edic ed by he linea app oxima ion, bu o la ge jumps, 兩 ⌬ ␣ 兩 ⬎ 兩 ⌬ ␣ c 兩 , he sign o ⌬ wis he oppo- si e o he p edic ion o he linea app oxima ion. Fo he sake o conc e eness, in Fig. 4 we ha e epea ed he nume i- cal expe imen o Fig. 2, bu wi h la ge in ensi y jumps. F om he Mon e Ca lo simula ion, we ob ain he c i ical FIG. 2. E olu ion o he densi y ␳ , as a unc ion o he numbe o aps . Fi e nume ical expe imen s a e shown, he ib a ion in- ensi y ␣ was changed a w⫽50, whe e ␭⬍0, om 0.15 o 0.1, 0.125, 0.15, 0.175, and 0.2, om op o bo om. Thus, he cen al cu e co esponds o no change in he apping in ensi y ␣ 共solid line兲, while he do ed and dashed lines co espond o a dec ease and an inc ease in ␣ , espec i ely. The ‘‘anomalous’’ esponse ex- pe imen ally obse ed shows up. FIG. 3. The same expe imen o Fig. 2, bu he change in in en- si y is in oduced a an ea lie ime w⫽10, a which ␭⬎0. The cu es co espond o he same alues o ␣ ⬘as in Fig. 2, bu now a e o de ed om bo om o op. In his egion he esponse is ‘‘no - mal,’’ i.e., he compac ion a e inc eases wi h he ib a ion in en- si y. J. JAVIER BREY AND A. PRADOS PHYSICAL REVIEW E 63 061301 061301-6 alue ⌬ ␣ c⯝⫺0.1 o ␣ ⫽0.15 and w⫽50. Then, a w⫽50 we change he ib a ion in ensi y om ␣ ⫽0.15 o ␣ ⬘⫽ ␣ ⫹⌬ ␣ c⫽⫺0.05, inding ha he compac ion a e does no change in he sho - ime limi ⫺ w→0. Mo eo e , i he ib a ion in ensi y is u he dec eased, ␣ ⬘⫽0.03, he com- pac ion a e also dec eases, while he linea app oxima ion p edic ed an inc ease o he compac ion a e i ␣ ⬘⬍ ␣ , since ␭w⬍0. Following Re . 关4兴, we ha e also conside ed ano he se- ies o nume ical expe imen s whe e he sys em was apped up o he same densi y wi h h ee di e en in ensi ies, ␣ ⫽0.2, 0.15, and 0.1, espec i ely. A e wa ds, he sys em was always apped wi h he same in ensi y ␣ ⬘⫽0.15. The ime e olu ion o he densi y is shown in Fig. 5, whe e he ime o igin o each expe imen has been aken a he ime when he sys em eached he p esc ibed densi y, namely, ␳ ⫽0.8. The igu e clea ly shows ha he e olu ion o he den- si y o ⬎0 s ongly depends on he p e ious apping his- o y, indica ing he ele ance o sho - e m memo y e ec s. Ma hema ically, his is equi alen o say ha ␮ 1( ) and ␮ 2( ) in Eq. 共1兲a e no de e mined uni ocally by he densi y a he same ime, so ha i is no in ac a closed i s -o de o dina y di e en ial equa ion. No e ha in all he plo ed cu es he jump in he compac ion a e has opposi e sign han he a ia ion o he in ensi y. We ha e e i ied ha ␭( ) is nega i e a he ime in which he in ensi y is modi ied in all cases, he beha io being hen consis en wi h he heo y. V. DISCUSSION Along his pape , we ha e s udied he nonequilib ium lin- ea esponse o a ib a ed g anula sys em o an ins an a- neous change in he in ensi y o he aps. In he i s pa , a gene al heo y was de eloped on he basis o some plausible hypo hesis abou he mesoscopic dynamics o he sys em. The esul s a e in quali a i e ag eemen wi h he expe imen- al obse a ions. In pa icula , he p esence o sho - e m memo y e ec s appea s as co ela ed wi h he elaxa ion p ope ies o he sys em a cons an in ensi y. An impo an heo e ical p edic ion, no obse ed in he expe imen s ye , is he exis ence o a c i ical ime c. Fo imes ⬍ c he e- sponse o he sys em o a change in he in ensi y is ‘‘no - mal,’’ in he sense ha an inc ease in he in ensi y p oduces a posi i e jump in he compac ion a e. On he o he hand, o ⬎ c, an ‘‘anomalous’’ esponse is p oduced. The change in he compac ion a e has opposi e sign han ha o he modi ica ion o he ib a ion in ensi y, in con as wi h he long- ime beha io ound in expe imen s, whe e he e- laxa ion is as e o la ge ib a ion in ensi y. In he second pa o he pape , a simple model o com- pac ion has been conside ed. I is shown o i pe ec ly in o he gene al scheme de eloped be o e, allowing a de ailed quan i a i e analysis o he heo e ical p edic ions. This is no a peculia i y o his model, since he ‘‘pa king’’ model 关9兴also e i ies all he condi ions assumed in he heo e ical amewo k. In ac , his is no su p ising because his la e model has a ma hema ical s uc u e e y simila o he one conside ed in his pape 关6兴. In Sec. IV we ha e shown ha ou model ep oduces he expe imen ally obse ed beha io o he sys em when sub- mi ed o changes in he ib a ion in ensi y unde di e en condi ions 关4兴. Now we will e e o a mo e complica ed pa e n o changes in he in ensi y ha a e also discussed in Re . 关4兴. Fi s , he sys em is shaken wi h an in ensi y ⌫0( ␣ 0 in he model no a ion兲 o a long pe iod o ime, so ha he sys em p ac ically eaches a s eady densi y. A e wa ds, a a ime aken as ⫽0, he in ensi y is swi ched o ⌫⬎⌫0 o a FIG. 4. Time e olu ion o he densi y ␳ when he ib a ion in ensi y ␣ is changed a w⫽50, whe e ␭⬍0, om ␣ ⫽0.15 o ␣ ⬘⫽0.05 共ci cles兲and 0.03 共squa es兲, espec i ely. The cu e co - esponding o a cons an ib a ion in ensi y ␣ ⫽0.15 is plo ed o e e ence 共solid line兲. Fo such la ge jumps, he linea app oxima- ion is no alid, and he compac ion a e does no inc ease. In ac , ␣ ⬘⫽0.05 co esponds o he c i ical alue ⌬ ␣ c, o which no change in he compac ion a e is obse ed o sho imes. FIG. 5. Time e olu ion o he densi y o a sys em, which was apped up o he same densi y ␳ ⫽0.8 using h ee di e en in ensi- ies, ␣ ⫽0.2 共ci cles兲, 0.15 共squa es兲, and 0.1 共 iangles兲. A e - wa ds, he sys em was always apped wi h ␣ ⬘⫽0.15. The ime o i- gin o each expe imen has been aken a he ime when he sys em eached he p esc ibed densi y, namely, ␳ ⫽0.8. The e olu ion o ⬎0 s ongly depends on he p ehis o y o he sys em. LINEAR RESPONSE OF VIBRATED GRANULAR . . . PHYSICAL REVIEW E 63 061301 061301-7 gi en pe iod o ime 0and, inally, he sys em is apped again wi h he o iginal in ensi y ⌫0, and he subsequen e- laxa ion o he sys em is s udied. Expe imen ally i was ound ha he elaxa ion is slowe he la ge he 0; he sys- em ‘‘ages.’’ Mo eo e , on he basis o a simple wo-s a e model, i was p oposed ha ␳ 共 兲⫺ ␳ s⬃exp共⫺ ␬ 0 兲 ,共42兲 o ⫺ 0Ⰷ1. In he abo e exp ession, ␬ 0is a dec easing unc ion o 0. Josse and e al. 关4兴also epo ed ha he e- laxa ion can be i ed by a supe posi ion o exponen ials, all o hem wi h he same ampli ude. We ha e ca ied ou nu- me ically his kind o expe imen s in ou model. In Fig. 6 we p esen he esul s ob ained wi h ␣ 0⫽0.3, ␣ ⫽0.5, and ou di e en alues o 0, namely, 0⫽1,2,4, and 8 om bo om o op. The plo ed esponse unc ion ␾ ( ) is de ined as ␾ 共 兲⫽ ␳ s共 ␣ 0兲⫺ ␳ 共 兲 ␳ s共 ␣ 0兲⫺ ␳ 共 0兲,共43兲 whe e ␳ ( 0) is he densi y o he sys em a he ime in which he he in ensi y is swi ched back o ␣ 0. Fo he model, he s eady alues o he densi y can be compu ed analy ically 关6兴, and i is ␳ s( ␣ ⫽0.3)⯝0.8371. Fo a gi en ime in e al ⫺ 0, ␾ ( ) inc eases wi h he ‘‘wai ing’’ ime 0. Thus, he elaxa ion is slowe o la ge 0, consis en ly wi h he ex- pe imen al obse a ion 关4兴. Also, we ha e i ed 共solid lines兲 he da a o a s e ched exponen ial o KWW unc ion 关19兴, ␳ KWW共 兲⫽ ␳ s共 ␣ 0兲⫺ 冋 关 ␳ s共 ␣ 0兲⫺ ␳ 共 0兲兴exp 冋 ⫺ 冉 ⫺ 0 ␶ 冊 ␤ 册 , 共44兲 wi h ␶ and ␤ being i ing pa ame e s. As obse ed in he igu e, he i is qui e sa is ac o y, excep o imes e y close o 0and, p obably, o e y la ge imes. The pa ame e ␤ in Eq. 共44兲measu es he wid h o he elaxa ion- ime dis ibu- ion. The alues we ha e ound go om ␤ ⫽0.366 o 0 ⫽1 o ␤ ⫽0.478 o 0⫽8. The la e is close o he alue 1/2, cha ac e is ic o sys ems whose dynamics a e domina ed by one-dimensional di usi e p ocesses. Howe e , he KWW elaxa ion is no equi alen o a supe posi ion o exponen- ials wi h he same ampli ude, as p oposed in Re . 关4兴. The e- o e, his poin dese es mo e wo k in he u u e, bo h heo- e ically and expe imen ally. Wi h espec o he long- ime beha io p edic ed by Eq. 共42兲, we could no each a de ini e answe . Al hough he nume ical da a seems o be compa ible wi h i , he noise is oo la ge and u he high-p ecision s ud- ies would be equi ed. Finally, a c ucial poin in he analysis p esen ed in his pape is he small ampli ude o he pe u ba ion in he ib a- ion in ensi y ⌬⌫. As poin ed ou a he end o Sec. III, he beha io ollowing a la ge change in he in ensi y may be di e en . In he model conside ed in Sec. IV he nonlinea co ec ions a e e y simple, leading always o a dec ease in he compac ion a e and o he appea ance o a c i ical alue o he in ensi y jump, such ha no change in he ib a ion in ensi y is obse ed in he sho - ime egime. Mo eo e , o jumps la ge han he c i ical one, he sign o he change in he compac ion a e is e e sed as compa ed wi h he p edic- ion o he linea app oxima ion. We hink ha i is wo h looking o his kind o beha io in o he models o com- pac ion, and also in expe imen s wi h eal g anula sys ems. ACKNOWLEDGMENTS This esea ch has been pa ially suppo ed by he Di ec- cio ´n Gene al de In es igacio ´n Cien ı ´ icayTe ´cnica 共Spain兲 h ough G an No. PB98-1124. 关1兴J.B. Knigh , C.G. Fa ndich, C.N. Lau, H.M. Jaege , and S.R. Nagel, Phys. Re . E 51, 3957 共1995兲. 关2兴E.R. Nowak, J.B. Knigh , E. Ben-Naim, H.M. Jaege , and S.R. Nagel, Phys. Re . E 57, 1971 共1998兲. 关3兴M. Nicodemi, Phys. Re . Le . 82, 3734 共1999兲. 关4兴C. Josse and, A. Tkachenko, D.M. Mue h, and H.M. Jaege , Phys. Re . Le . 85, 3632 共2000兲. 关5兴D.A. Head, Phys. Re . E 62, 2439 共2000兲. 关6兴J.J. B ey, A. P ados, and B. Sa ´nchez-Rey, Phys. Re . E 60, 5685 共1999兲. 关7兴J.J. B ey, A. P ados, and B. Sa ´nchez-Rey, Physica A 275, 310 共2000兲. 关8兴P.L. K api sky and E. Ben-Naim, J. Chem. Phys. 100, 6778 共1994兲. 关9兴E. Ben-Naim, J.B. Knigh , E.R. Nowak, H.M. Jaege , and S.R. Nagel, Physica D 123, 380 共1998兲. FIG. 6. Relaxa ion unc ion ␾ ( ), de ined in Eq. 共43兲,o he model, when i is p epa ed by apping o a long ime wi h ␣ ⫽0.3, and a e wa ds apped o 0⫽1, 2, 4,and 8 共 om bo om o op兲wi h a la ge in ensi y ␣ ⬘⫽0.5. Finally, he in ensi y is u ned back o he o iginal in ensi y ␣ ⫽0.3. All he cu es end o ze o in he in ini e ime limi , and he solid lines a e he bes nume ical i s o a KWW unc ion. J. JAVIER BREY AND A. PRADOS PHYSICAL REVIEW E 63 061301 061301-8 关10兴A. P ados, J.J. B ey, and B. Sa ´nchez-Rey, Physica A 284, 277 共2000兲. 关11兴S.F. Edwa ds and R.B.S. Oakesho , Physica A 157, 1080 共1989兲; A. Meh a and S.F. Edwa ds, ibid. 157, 1091 共1989兲. 关12兴S.F. Edwa ds and D.V. G ine , Phys. Re . E 58, 4758 共1998兲. 关13兴A. Ba a , J. Ku chan, V. Lo e o, and M. Selli o, Phys. Re . Le . 85, 5034 共2000兲; e-p in cond-ma /0011492; S. Ba a and V. Lo e o, J. Phys. A 336, 4401 共2000兲. 关14兴A.J. Kolan, E.R. Nowak, and A.V. Tkachenko, Phys. Re . E 59, 3094 共1999兲. 关15兴J. Talbo , G. Ta jus, and P. Vio , Phys. Re . E 61, 5429 共2000兲. 关16兴J. Talbo , G. Ta jus, and P. Vio , e-p in cond-ma /0008183. 关17兴In he lowe g aph in Fig. 2 o Re . 关3兴, i is obse ed ha o ⫺ w→0, when he in ensi y o he ib a ion is inc eased, he heigh o he sys em dec eases, i.e., he densi y inc eases. This g aph co esponds o he lowes in ensi y conside ed, so ha he wai ing ime wis much smalle han he cha ac e is ic elaxa ion ime. This explains he change in beha io as com- pa ed wi h he o he wo plo s in he same igu e. 关18兴J. J. B ey and A. P ados 共unpublished兲. 关19兴G. Williams and D.C. Wa s, T ans. Fa aday Soc. 66,80 共1970兲. LINEAR RESPONSE OF VIBRATED GRANULAR . . . PHYSICAL REVIEW E 63 061301 061301-9