Closed model for granular compaction under weak tapping

Closed model o g anula compac ion unde weak apping 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, E-41080 Se illa, Spain 共Recei ed 25 June 2003; published 20 No embe 2003兲 A one-dimensional la ice model is o mula ed o s udy apping dynamics and he long ime s eady dis i- bu ion in g anula media. The dynamics conse es he numbe o pa icles in he sys em, and densi y changes a e associa ed wi h he c ea ion and des uc ion o emp y si es. The model is shown o be consis en wi h Edwa ds’ he modynamics heo y o powde s. The ela ionship wi h la ice models in which he numbe o pa icles is no conse ed is discussed. DOI: 10.1103/PhysRe E.68.051302 PACS numbe 共s兲: 45.70.Cc, 05.50.⫹q, 81.05.Rm I. INTRODUCTION In he las ew yea s, a g ea deal o e o is being made ying o unde s and he physical mechanisms leading o compac ion in weakly ib a ed g anula sys ems, and he p ope ies o he s eady s a e e en ually eached in he long ime limi . This has been p omp ed and s imula ed by he seminal pape s o he Chicago g oup epo ing expe imen al esul s o compac ion 关1–3兴. G anula compac ion consis s o he inc ease in he densi y, s a ing om an ini ial low- densi y s a e, as a consequence o ex e nal exci a ions, usu- ally e ical shakes o aps. E e y ap is ollowed by a ee elaxa ion, so ha in he p ocess he sys em goes h ough a se ies o blocked con igu a ions. S a ing om an ‘‘e godic hypo hesis’’ o powde s, based on he ex ensi e, global, cha ac e o he dynamics induced by shaking, Edwa ds and co-wo ke s 关4兴ha e o mula ed a mic oscopic heo y o he s eady s a e o ib a ed g anula media ha is simila o con en ional s a is ical mechanics. Mo eo e , hey assume ha he s eady s a e is ully cha ac- e ized by he olume o he sys em, which hen plays a ole analogous o ha o he ene gy in he usual he mal sys ems. This p o ides he ‘‘mic ocanonical’’ desc ip ion. The associ- a ed ‘‘canonical’’ p obabili y dis ibu ion is ob ained by maximizing he s a is ical en opy unde he condi ion ha he a e age olume is gi en. O cou se, he p obabili y o a gi en con igu a ion depends only on i s olume. The pa am- e e conjuga ed o he olume, simila o he he mal em- pe a u e, was named compac i i y by he au ho s in Re . 关4兴. Up o now, he e has been no de ini e expe imen al es o he abo e he modynamic heo y o powde s. The measu e o he compac i i y, o he en opy, o a g anula sys em seems a a he di icul ask no only in eal expe imen s bu also in ealis ic models, al hough some p ocedu es ha e been p oposed. They a e based on he de e mina ion o he a e - age olume and i s luc ua ions as a unc ion o he con ol pa ame e o he shaking, e.g., he ib a ion in ensi y 关1,5兴. F om hese wo unc ions, he compac i i y can be ob ained, in p inciple, by in eg a ion, al hough his p og am is ha d o ca y ou in p ac ice due o he unce ain y o he measu e- men s. Ano he al e na i e way, his one based on he gene - aliza ion o he Eins ein ela ion be ween di usi i y and mo- bili y, has been ecen ly discussed and analyzed in a sys em o inelas ic ha d sphe es by means o molecula dynamics simula ions 关6兴. On he o he hand, he alidi y o Edwa ds’ heo y has been s udied in he con ex o se e al simple models, wi h di e en unde lying physical mechanisms. I has been ound ha he esul s o Te is and spin-glass models 关7–9兴a e consis en wi h he heo y. In hese sys ems, he numbe o pa icles is ixed bu mos o he esul s a e nume ical, due o he complexi y o he models used. One-dimensional Ising models, wi h o wi hou kine ic cons ains, ha e also been conside ed 关10–14兴, because hey a e simple enough as o allow a de ailed analy ical s udy in many cases. Fo weak apping, ag eemen wi h Edwa ds’ heo y was ound again, al hough disc epancies show up in he limi o s ong ap- ping. Qui e in e es ingly, all he Ising-like models in Re s. 关10–14兴ha e been o mula ed as open sys ems. The numbe o pa icles does no emain cons an , bu i changes along he compac ion p ocess, as a consequence o adso p ion- deso p ion e en s om a heo e ical pa icle ese oi in con- ac wi h he sys em. Ins ead, i is he olume ha is kep ixed, in his way leading o he a ia ion o he densi y. Then, al hough i is ue ha he s eady dis ibu ion o hese models can be conside ed as a ‘‘g and canonical’’ ensemble gene aliza ion o he heo y, i is also clea ha i is no cha ac e ized by he compac i i y 共 empe a u e兲bu by an- o he pa ame e playing he ole o he chemical po en ial. This di e ence is e iden ly ele an when ying o ela e any o hem wi h he cha ac e is ics o he ib a ion p ocess, e.g., i s in ensi y. Beyond ha , he dis inc ion migh become concep ually c ucial when dealing wi h g anula mix u es and seg ega ion phenomena. In ha case, each o he di e - en species is going o ha e i s own analogous o he chemi- cal po en ial pa ame e 关15兴. Whe he o no i is also needed o conside di e en compac i i ies 共 empe a u es兲,asi has been sugges ed ecen ly 关16兴, is a di e en ques ion. The aim o his pape is o p esen a closed, cons an numbe o pa icles, one-dimensional la ice model o com- pac ion. Again he model is simple enough as o be analy i- cally ac able. Du ing he apping p ocess, pa icles di use and also emp y si es 共holes兲a e c ea ed and des oyed in he sys em, acco ding o well de ined ules. The la e a e chosen o mimic, in a c ude way, wha happens in eal compac ion expe imen s. Mo e p ecisely, he model ies o ep esen a *Elec onic add ess: [email p o ec ed] †Elec onic add ess: [email p o ec ed] PHYSICAL REVIEW E 68, 051302 共2003兲 1063-651X/2003/68共5兲/051302共8兲/$20.00 ©2003 The Ame ican Physical Socie y68 051302-1 e ical sec ion o a ib a ed wo-dimensional sys em. In a shake, he leng h o he sec ion can inc ease because emp y egions 共holes兲a e c ea ed be ween he pa icles. These e- gions can be used o he pa icles o di use. A e wa ds, once he shake has ended, he sys em ies o compac due o he ac ion o g a i y. This is accomplished in he model by means o he elimina ion o holes. Bu , because o he geo- me ical cons ains ollowing om excluded olume e ec s o he ha d pa icles in he neighbo ing e ical sec ions, no all he holes can be des oyed in he ee e olu ion. Only la ge enough emp y egions can be educed. As a conse- quence o he combina ion o a ap and he nex ee elax- a ion, he leng h can inc ease in some egions o he sys em and dec ease in o he s. The ne balance de e mines he global beha io o he sys em in he compac ion p ocess. The plan o he pape is as ollows. In Sec. II, he model is o mula ed a he mesoscopic le el o desc ip ion by means o a mas e equa ion o he ansi ion p obabili y. This equa ion is exac ly sol ed o he s eady dis ibu ion in Sec. III, and he associa ed mac oscopic desc ip ion is dis- cussed in Sec. IV, whe e i is shown o be in ag eemen wi h Edwa ds’ he modynamic desc ip ion. The compac i i y is iden i ied in e ms o he pa ame e s cha ac e izing he me- soscopic dynamics o he model. Also, he dis ibu ion o domains is de i ed he e. Sec ion V con ains a de ailed dis- cussion o he ela ionship be ween closed and open models, and be ween he compac i i y and ugaci y pa ame e s. The pape ends wi h a sho summa y and some addi ional dis- cussions. II. THE MODEL We conside a one-dimensional la ice ha ing N⫹1 pa - icles. The numbe o si es in he la ice is a iable and changes wi h ime acco dingly wi h he ules o be speci ied below. Those si es ha a e no occupied by a pa icle a e said o be emp y o , equi alen ly, being a hole. The dynamics o he sys em is de ined ying o mimic apping expe imen s o he s udy o compac ion in g anula media 关1–3兴. These expe imen s ypically in ol e wo di e - en se ies o p ocesses o qui e di e en na u e. The sys em is submi ed o aps o pulses sepa a ed by ime in e als o which he sys em is allowed o elax eely, un il being apped in a me as able con igu a ion. The e o e, each ap s a s in he me as able con igu a ion eached in he p e ious ee elaxa ion. The aps a e cha ac e ized by hei du a ion and hei ampli ude. Physically, he e ec o he aps is o dec ease he local densi y in some egions o he sys em, mo ing g ains om hei me as able posi ions, and allowing a pos e io eo de - ing in he ee elaxa ion. We will speci y i s he dynamics du ing he elaxa ion p ocesses, since i leads o iden i ying he possible me as able, o blocked, con igu a ions o he model. I will be assumed ha in he ee elaxa ion, he sys em ies o educe i s leng h by elimina ing some o he emp y si es o he la ice. Mo e p ecisely, whene e he e is a g oup o nea es neighbo holes, all excep one a e elimi- na ed. These a e he only p ocesses aking place in he ee elaxa ion, and ha e p obabili y one. The e o e, he numbe o pa icles is conse ed in he elaxa ion, bu he leng h o he sys em, measu ed by he o al numbe o si es, is in gen- e al educed. As a consequence o he abo e dynamics o elaxa ion, he me as able con igu a ions o he model a e cha ac e ized by ha ing all he holes su ounded by wo pa icles, i.e., he holes a e isola ed. In o de o displace he sys em om one o hese s a es, i has o be ex e nally pe u bed, o ins ance by means o a ap. To comple e he desc ip ion o he dynam- ics o ou model in a compac ion expe imen , he possible ansi ions aking place du ing a ap and s a ing a a me a- s able con igu a ion, ha e o be iden i ied and hei p obabili- ies speci ied. Two kinds o elemen a y p ocesses will be conside ed. Each o hem will be discussed sepa a ely in he ollowing. Fi s , a pa icle can be ansien ly deso bed om he la - ice and pos e io ly adso bed in an emp y si e in he neigh- bo hood o i s p e ious posi ion. This p ocess is es ic ed by he ollowing ule. A pa icle can be deso bed om a si e du ing he ap only i a leas one o i s nea es neighbo si es is emp y. Mo e p ecisely, he p obabili y o hese e en s is p opo ional o he numbe o nea es neighbo holes o he pa icle being deso bed. This es ic ion ies o nai ely model he sho ange cons ains making di icul s uc u al ea angemen s in g anula ma e ials. Then, du ing a ap, he p obabili y o deso p ion o a pa icle ha ing only one nea - es neighbo hole is ␣ , while i is 2 ␣ i i is su ounded by wo holes. A e wa ds, he pa icle is eabso bed ei he in i s own o iginal si e o in any o he nea es neighbo holes, wi h a p obabili y ha is p opo ional o he numbe o holes nex o he si e conside ed. In Fig. 1, cases 共a兲and 共b兲in- ol e p ocesses s a ing wi h he ansi o y deso p ion o a pa icle. Pa icles and holes a e ep esen ed by ci cles and c osses, espec i ely. In he case e e ed o as 共b兲in he igu e, he elimina ion o a hole happening in he nex ee e olu ion has been also indica ed. I is seen ha he ne esul o he se ies o e en s aking place du ing he ap and he ee elaxa ion is, in his case, he des uc ion o a la ice si e, wi h he consequen dec ease o he la ice leng h. Du ing he ap, he c ea ion o an emp y si e o hole is also possible, bu only be ween wo nea es neighbo pa - icles loca ed a one o he ends o a domain o a leas wo pa icles. The p obabili y o he co esponding elemen a y e en s, e e ed o as case 共c兲in Fig. 1, is ␤ . No e ha hese p ocesses o hole c ea ion a e jus he in e se o hose p o- O X O O OOXO O X O X Oβ O O X O O X O O O X X O O X O O O X X O O O X O O X O X O O X X O O O O X X O O X O O O O X O 1/2 1/2 1/4 1/4 (a) (b) α α 2α O X X X O (c) FIG. 1. Elemen a y ea angemen s o he sys em in a single ap and he ollowing ee elaxa ion in he weak ib a ion limi . The ajec o ies leading o a inal s a e iden ical o he ini ial one a e no shown. J. J. BREY AND A. PRADOS PHYSICAL REVIEW E 68, 051302 共2003兲 051302-2 ducing he des uc ion o a hole. Fu he mo e, i will be assumed ha only one ansi ion akes place a he mos in e e y egion on he sys em du ing each ap, i.e., no si e is in ol ed in o wo di e en p ocesses in he same pe u ba ion o he sys em. Physically, his hy- po hesis implies o conside he limi o weak and sho aps 关10,14兴. In summa y, he dynamics o he model in he shak- ing p ocess is de ined by he e ec i e ansi ions gi en in Table I, desc ibing he combined e en s associa ed wi h a ap and he nex ee elaxa ion. The ansi ions only a ec he clus e s shown, and hei p obabili ies a e independen o he con igu a ion o he emainde o he sys em. To o mula e he model in a mo e ma hema ical language, and also o cha ac e ize he me as able con igu a ions, i is con enien o de ine a se o a iables n⬅ 兵 n1,...,nN 其 . The a iable ni akes he alue uni y i he e is a hole nex o he igh o pa icle i, while i anishes i he e is no hole, i.e., i pa icles iand i⫹1 a e in nea es neighbo si es. By de ini- ion, i is assumed ha he e is no hole o he le o pa icle 1 no o he igh o pa icle N⫹1. Bo h pa icles de ine he bounda ies o he sys em. I is easily ealized ha his p op- e y is p ese ed by he dynamics o he sys em unde ap- ping, as de ined abo e. Then, we ha e es ablished a one o one ela ionship be ween a se o N a iables aking alues 0 and 1 and he me as able con igu a ions o he model. The ansi ion p obabili ies in Table I can be exp essed in e ms o he ni a iables. Deno ing Rin ⬅ 兵 n1,...,Rini,...,nN 其 , wi h Rini⫽1⫺ni, he p obabil- i y W(n⬘ 兩 n) o he se e al e en s going om con igu a ion n o con igu a ion n⬘in he e ec i e dynamics desc ibing shaking p ocess a e W共RiRi⫹1n 兩 n兲⫽ ␣ 2关ni共1⫺ni⫹1兲⫹共1⫺ni兲ni⫹1兴,共1兲 W共Rin 兩 n兲⫽ ␣ 2共ni⫺1ni⫹nini⫹1兲⫹ ␤ 关ni⫺1共1⫺ni兲 ⫹共1⫺ni兲ni⫹1兴.共2兲 Equa ion 共1兲co esponds o di usi e e en s, while he i s and second e ms on he igh -hand side 共 hs兲o Eq. 共2兲 co espond o he des uc ion and c ea ion o holes, espec- i ely. The Ma ko p ocess de ined by hese ansi ion p ob- abili ies is i educible, i.e., all he me as able con igu a ions o he la ice a e connec ed by a chain o ansi ions wi h nonze o p obabili y. To e i y his p ope y, we begin by no ing ha any me as able con igu a ion can be connec ed wi h he con igu a ion cha ac e ized by ha ing jus a hole loca ed nex o he igh o a gi en ixed pa icle. This is because holes can be mo ed h ough pa icles by means o di usi e e en s, so ha he wo consecu i e holes can al- ways be loca ed on bo h sides o he same pa icle. A e - wa ds, one o he holes can be elimina ed by a ype 共b兲e en o Fig. 1. This p ocedu e can be epea ed un il he e is only a hole in he la ice, which can hen be di used o he desi ed si e. This p o es he abo e s a emen . Bu , since each e ec- i e ansi ion ha e i s in e se also wi h nonze o p obabili y, he abo e pa hs can also be e e sed, concluding ha all he me as able con igu a ions a e connec ed. The i educibili y p ope y o he Ma ko p ocess implies ha he e is a unique s eady p obabili y dis ibu ion o he p ocess 关17兴. This dis- ibu ion will be explici ly ob ained in he ollowing sec ion. In Fig. 2, he elaxa ion o he pa icle densi y is shown as a unc ion o he ‘‘scaled ime’’ ␶ ⫽ ␣ n, whe e nis he num- be o aps be o e measu ing he densi y o di e en alues o ␣ and ␤ . The ini ial s a e o all he cu es was he leas dense me as able con igu a ion, ␳ ⫽0.5, in which he e is a hole be ween e e y wo pa icles. In all he epo ed cases ␤ Ⰶ ␣ , so ha p ocesses dec easing he densi y o pa icles a e only ele an when he sys em is nea he mos compac s a e, ␳ ⫽1. Mo eo e , as ␤ Ⰶ ␣ Ⰶ1, he e is a uni e sal beha - io up o a e y la ge numbe o aps ⫽O( ␣ ⫺1), i.e., ␣ n ⫽O(1). Fo longe imes, when p ocesses dec easing he densi y become ele an , nⲏO( ␤ ⫺1), he sys em ap- p oaches a s eady s a e cha ac e ized by he a io ␤ / ␣ . The obse ed uni e sal scaled cu e is e y well desc ibed by he ou -pa ame e empi ical law ␳ 共 兲⫽ ␳ ⬁⫺ ␦ ␳ ⬁ 1⫹Bln 冉 1⫹ ␶ ␶ c 冊 ,共3兲 TABLE I. T ansi ion p obabili ies o he elemen a y ea ange- men s aking place in a single ap in he weak ib a ion egime. Ini ial s a e Final s a e P obabili y OOXO OXOO ␣ /2 OXOO OOXO ␣ /2 OXOXO OXOO ␣ /2 OXOXO OOXO ␣ /2 OXOO OXOXO ␤ OOXO OXOXO ␤ 103102101100101102103104105 αn 0.5 0.6 0.7 0.8 0.9 1 ρ FIG. 2. E olu ion o he densi y o pa icles, as a unc ion o he scaled ime de ined in he ex . The cu es co espond o he pai s o alues ␣ ⫽10⫺3, ␤ ⫽10⫺5共ci cles兲, ␣ ⫽10⫺2, ␤ ⫽5⫻10⫺5 共squa es兲, and ␣ ⫽10⫺2, ␤ ⫽5⫻10⫺6共diamonds兲, while he solid line is he bes in e se loga i hmic i , Eq. 共3兲, wi h he pa ame e s gi en in he ex . CLOSED MODEL FOR GRANULAR COMPACTION UNDER... PHYSICAL REVIEW E 68, 051302 共2003兲 051302-3 wi h ␳ ⬁⫽1.04, ␦ ␳ ⬁⫽0.54, B⫽1.17, and ␶ c⫽2.63. As i is he case wi h he expe imen al da a 关1兴and also wi h nume i- cal esul s om o he simple models 关10,18兴, he loga i hmic i is no expec ed o gi e he co ec asymp o ic densi y o pa icles. In ac , in ou case i is ␳ ⬁⬎1, which is clea ly unphysical. A simila beha io o ␳ ⬁was ound in Re . 关10兴. Also, alues o ␳ ⬁la ge han he andom close packing limi ha e been epo ed om he i o expe imen al da a 关1兴. III. THE STEADY DISTRIBUTION To ind he s eady dis ibu ion o he Ma ko p ocess de- sc ibing he e ec i e dynamics o he model, we a e going o assume i e i ies de ailed balance. O cou se, his has o be jus i ied a pos e io i by showing ha such a dis ibu ion ex- is s. The e o e, we look o a ime-independen dis ibu ion p(s)(n) ha ing he p ope y W共n⬘ 兩 n兲p(s)共n兲⫽W共n 兩 n⬘兲p(s)共n⬘兲共4兲 o all con igu a ions nand n⬘. A di ec i s consequence o his equa ion is ha all he me as able con igu a ions wi h he same numbe o holes ha e he same p obabili y in he s eady s a e. This ollows om he ac ha hey a e con- nec ed by di usi e e en s and di usion is iso opic in he e ec i e dynamics, as seen in Table I. The e o e, he dis i- bu ion unc ion e i ying Eq. 共4兲can only depend on he numbe o holes NH⫽兺 i N ni,共5兲 in he con igu a ion, bu no on hei spa ial dis ibu ion. So, we can w i e pNH (s)共n兲⫽ 共NH兲 Z,共6兲 whe e he numbe o holes, NH, in he con igu a ion nhas been made explici in he no a ion, (NH) is a unc ion o be de e mined, and Zdeno es a no maliza ion cons an , Z⫽兺 n 共NH兲.共7兲 S ill emains o be analyzed, i Eq. 共6兲can be made compa - ible wi h Eq. 共4兲, when pa icula ized o e ec i e e en s inc easing 共and dec easing兲 he numbe o holes. The la e eads ␤ pNH (s)共n兲⫽ ␣ 2pNH⫹1 (s)共n⬘兲.共8兲 He e n⬘is a con igu a ion di e ing om nby he c ea ion o a hole. Use o Eq. 共6兲gi es 共NH⫹1兲 共NH兲⫽2 ␤ ␣ 共9兲 and, by i e a ion, 共NH兲⫽C 冉 2 ␤ ␣ 冊 NH,共10兲 o NH⭓1, Cbeing an a bi a y cons an ha will be aken equal o uni y. In his way, we ha e p o en ha he sys em has he p ope y o de ailed balance and ha i s s eady dis- ibu ion is gi en by pNH (s)共n兲⫽ ␥ ⫺NH Z,共11兲 Z⫽兺 n ␥ ⫺NH⫽兺 NH⫽1 N ⍀N共NH兲 ␥ ⫺NH,共12兲 whe e ␥ ⫽ ␣ /2 ␤ and ⍀N(NH) is he numbe o me as able con igu a ions o he la ice ha ing NHholes and, o cou se, N⫹1 pa icles. I is wo h ema king ha no app oxima ion has been done in o de o de i e he s eady dis ibu ion, Eq. 共11兲, i.e, i is alid o any alue o ␥ . The s eady a e age numbe o holes and i s dispe sion can be e alua ed om Z by 具 NH 典 s⬅兺 nNHpNH (s)共n兲⫽⫺ ⳵ lnZ ⳵ ln ␥ ,共13兲 具 共⌬NH兲2 典 s⬅ 具 NH 2 典 s⫺ 具 NH 典 s 2⫽ ⳵ 2lnZ ⳵ 共ln ␥ 兲2⫽⫺ ⳵ 具 NH 典 s ⳵ ln ␥ . 共14兲 A simple combina o ial a gumen gi es ⍀N共NH兲⫽N! NH!共N⫺NH兲!共15兲 and subs i u ion o his exp ession in o Eq. 共12兲yields Z⫽ 冉 1⫹1 ␥ 冊 N ⫺1. 共16兲 The e o e, i ollows by applica ion o Eq. 共13兲 ha , in he limi o la ge N, 具 NH 典 s⫽N 1⫹ ␥ .共17兲 The igh -hand side o Eq. 共17兲is a mono onic dec easing unc ion o ␥ o ixed numbe o pa icles N, i.e., he leng h o he sys em dec eases as ␥ inc eases. The e o e, ␥ ⫺1plays a ole simila o he ib a ion in ensi y in eal g anula ex- pe imen s o compac ion in he model. I ollows ha , in he physical image depic ed by he p esen model, he p obabil- i y o di usion p ocess ␣ is expec ed o g ow as e wi h he ib a ion in ensi y han he p obabili y o c ea ion o holes ␤ , a leas in he weak apping limi . The leng h 共 olume兲o a con igu a ion is gi en by L⫽NH⫹N,共18兲 and he a e age pa icle densi y is J. J. BREY AND A. PRADOS PHYSICAL REVIEW E 68, 051302 共2003兲 051302-4 ␳ (s)⫽N 具 L 典 s ⫽N 具 NH 典 s⫹N⫽1⫹ ␥ 2⫹ ␥ .共19兲 In Fig. 3, he nume ical alues o he s eady densi y o pa - icles, ob ained by Mon e Ca lo simula ion o he model, a e compa ed wi h he heo e ical p edic ion, Eq. 共19兲, and an excellen ag eemen is ound. The speci ic leng h pe pa - icle, in si e uni s, is he in e se o he pa icle densi y, 具 l 典 s⬅N⫹ 具 NH 典 s N⫽2⫹ ␥ 1⫹ ␥ .共20兲 I s dispe sion is ob ained om Eqs. 共14兲and 共16兲, N 具 共⌬l兲2 典 s⫽共2⫺ 具 l 典 s兲共 具 l 典 s⫺1兲,共21兲 p esen ing a maximum o 具 l 典 s⫽3/2, i.e., when he a e age numbe o holes is N/2 and he densi y o pa icles ␳ (s) ⫽2/3, i.e., ␥ ⫽1. The nume ical e alua ion o he leng h luc- ua ions is compa ed wi h he heo e ical p edic ion, as gi en by Eq. 共21兲, in Fig. 4. Again, a e y good ag eemen is ob- se ed o he ange o ‘‘ ib a ion in ensi ies’’ ␥ ⫺1plo ed. Ou side his window o ib a ion in ensi ies, he leng h luc- ua ions a e e y small and, he e o e, a he ha d o measu e in he simula ions. IV. THERMODYNAMIC DESCRIPTION Following Edwa ds and co-wo ke s ideas 关4兴, he s eady dis ibu ion 共11兲can be exp essed in he canonical o m pNH (s)共n兲⫽e⫺NH/X Z,Z⫽兺 NH ⍀N共NH兲e⫺NH/X,共22兲 whe e X⫽共ln ␥ 兲⫺1共23兲 is he so-called compac i i y. I is he conjuga ed he mody- namic pa ame e o he olume in gen ly ib a ed g anula sys ems, in an analogous way as he empe a u e is he con- juga e o he ene gy in usual he mal sys ems. No e ha , in Eq. 共22兲, he a io NH/Xcan be eplaced in bo h he nume a- o and he denomina o by L/X, whe e Lis he leng h 共 ol- ume兲o he con igu a ion, as de ined in Eq. 共18兲. The s uc- u e o he abo e s eady dis ibu ion is consis en wi h he wo main ing edien s o Edwa ds’ heo y, namely, ha he measu e o e me as able con igu a ions is la , and ha he e is a unique pa ame e he olume, cha ac e izing he mac o- scopic s a e o he sys em. Le us poin ou ha , e y e- cen ly, he heo y has been ex ended o include se e al mac- oscopic con ol pa ame e s, in an e o o explain he disc epancies obse ed in some models wi h s ong apping 关12,19兴, and also seg ega ion pa e ns in bina y models 关16兴. I is clea ha such an ex ension does no apply o ou model, which is designed o desc ibe compac ion in one-componen sys ems unde weak apping. In he same con ex , he exp es- sion o he compac i i y in Eq. 共23兲dese es some com- men s. Al hough Xcan be o mally nega i e, o alues ␥ ⬍1, i is qui e doub ul ha his ac be physically ele an , since his ange o alues o ␥ co esponds o s ong apping, lead- ing o low s a iona y densi ies, namely, wi h an a e age num- be o holes 具 NH 典 s⬎N/2. The possibili y o nega i e alues o he compac i i y has been also ound in o he simple mod- els 关10,20兴, and i is associa ed wi h he exis ence o a maxi- mum leng h o he me as able con igu a ions. In he limi ␥ →⬁, i.e., asymp o ically weak apping, he s eady concen a ion o pa icles, ␳ (s), gi en by Eq. 共19兲can be app oxima ed by ␳ (s)⯝1⫺1 ␥ ,共24兲 and, using he de ini ion in Eq. 共23兲, X⫺1⫽⫺ln关1⫺ ␳ (s)兴.共25兲 102101100101 γ−1 0.5 0.6 0.7 0.8 0.9 1 ρ(s) FIG. 3. Compa ison be ween he nume ical alues o he s eady densi y o pa icles and he heo e ical p edic ion gi en by Eq. 共19兲. 101100101 γ−1 0 0.05 0.1 0.15 0.2 0.25 0.3 N <(∆l)2>s FIG. 4. Compa ison be ween he nume ical e alua ion o he leng h luc ua ions and he heo e ical p edic ion gi en by Eq. 共21兲. CLOSED MODEL FOR GRANULAR COMPACTION UNDER... PHYSICAL REVIEW E 68, 051302 共2003兲 051302-5 This ela ion be ween he compac i i y and he s eady den- si y has also been ound in a model wi h acili a ed dynamics ha ing a a iable numbe o pa icles and ixed olume 关10兴, and a simila beha io has been epo ed om he analysis o expe imen al da a 关3兴. An ‘‘en opy’’ Sassocia ed wi h he dis ibu ion p(s)can be de ined in he usual way, S⫽⫺兺 np(s)共n兲ln p(s)共n兲,共26兲 and use o Eq. 共22兲gi es S⫽ 具 NH 典 s X⫹lnZ.共27兲 Taking in o accoun Eq. 共13兲, i is easily e i ied ha ⳵ S ⳵ 具 L 典 s ⫽ ⳵ S ⳵ 具 NH 典 s ⫽1 X,共28兲 consis en ly wi h he physical meaning o he compac i i y as discussed abo e. Gi en ha he mac oscopic s a e o he sys em is cha ac e ized by a single pa ame e , i is possible o exp ess he en opy in e ms o only he densi y o pa icles, o he in ensi y pa ame e ␥ , o he compac i i y. Then, o ins ance, in he limi o la ge N he en opy can be w i en as S N⫽1 X共1⫹e1/X兲 ⫹ln1⫹e1/X e1/X.共29兲 In addi ion o he global p ope ies conside ed up o now, i is also possible o ob ain in o ma ion abou he domain s uc u e o he s eady con igu a ions. In pa icula , we a e going o de i e he e he p obabili y dis ibu ion o he num- be o pa icles in a domain. A domain o size is de ined as a clus e o pa icles, i.e., wo holes wi h pa icles in be ween. Fi s , we conside he p obabili y F (s)o inding a local domain o size , F (s)⬅ 具 nk共1⫺nk⫹1兲•••共1⫺nk⫹ ⫺1兲nk⫹ 典 s,共30兲 wi h ⭐1. Use o Eqs. 共11兲and 共15兲yields F (s)⫽1 Z兺 NH⫽0 N⫺ ⫺1 ⍀N⫺ 共NH兲 ␥ ⫺2⫺NH⫽ ␥ ⫺2 冉 ␥ 1⫹ ␥ 冊 ⫹1 , 共31兲 whe e he limi o la ge Nhas been conside ed once again. Then, he p obabili y o a domain o size ,P( ), is gi en by he condi ional p obabili y o inding a clus e o consecu- i e pa icles plus a hole o he igh o a gi en hole, i.e., P共 兲⫽F (s) 具 nk 典 s ⫽N 具 NH 典 sF (s)⫽ ␥ ⫺1 共1⫹ ␥ 兲 .共32兲 This dis ibu ion is co ec ly no malized: 兺 ⫽1 ⬁ P共 兲⫽1. 共33兲 I is ins uc i e o exp ess he dis ibu ion o domain sizes in e ms o he a e age leng h pe pa icle, 具 l 典 s. This is easily accomplished by means o Eq. 共20兲, ob aining P共 兲⫽共2⫺ 具 l 典 s兲 ⫺1共 具 l 典 s⫺1兲.共34兲 V. RELATIONSHIP BETWEEN CLOSED AND OPEN MODELS FOR COMPACTION In he p eceeding sec ion, we ha e in oduced he com- pac i i y X om he canonical o m o he s eady p obabili y dis ibu ion, Eq. 共22兲. In he Edwa ds and co-wo ke s o mu- la ion o he g anula he modynamic heo y 关4兴, he compac- i i y was de ined by X⫺1⫽ 冉 ⳵ S ⳵ V 冊 N ,共35兲 whe e he en opy Sis gi en by S⫽ln⍀N,共36兲 ⍀Nbeing he numbe o blocked con igu a ions o , in he language used in his pape , me as able s a es. In Eq. 共35兲, he numbe o pa icles in he sys em is kep cons an . The quan i y ⍀N o he model conside ed in his pape is gi en by Eq. 共15兲and o NⰇ1, NHⰇ1, Eq. 共35兲leads o X⫺1⫽lnN⫺NH NH.共37兲 This is he mic ocanonical 共cons an olume兲 e sion o he canonical 共cons an compac i i y兲exp essions 共17兲and 共23兲. In ac , combina ion o hese wo la e exp essions gi es X⫺1⫽lnN⫺ 具 NH 典 s 具 NH 典 s.共38兲 The exp ession equi alen o Eq. 共36兲in he canonical en- semble is Eq. 共26兲. I is e iden ha , in he limi o la ge sys ems, i is consis en wi h he de ini ion o Xin Eq. 共35兲. In se e al p oposed models o compac ion, la ices wi h a ixed numbe o si es, i.e., ixed leng h, ha e been consid- e ed. The dynamics is de ined in ol ing elemen a y p o- cesses associa ed wi h he adso p ion and deso p ion o pa - icles in such a way ha he numbe o pa icles in he la ice changes along he shaking expe imen . This is he mecha- nism o which he densi y in he sys em a ies wi h ime. In pa icula , se e al models leading o simila kind o me a- s able con igu a ions as in he model in his pape ha e been discussed in de ail 关10–12,19,14兴. Then, aside om de ails ha a e i ele an o he ollowing analysis, he numbe o blocked con igu a ions is gi en by Eq. 共15兲, which we e- w i e in he o m ⍀L共NH兲⫽共L⫺NH兲! NH!共L⫺2NH兲!,共39兲 J. J. BREY AND A. PRADOS PHYSICAL REVIEW E 68, 051302 共2003兲 051302-6 whe e he numbe o si es o he la ice Lis now conside ed as ixed and L/2⭓NH⭓1. Mo eo e he s eady dis ibu ion, in he weak apping limi was ound o ha e he o m p(s)⬘共n兲⫽ ␩ ⫺NH Z⬘,共40兲 wi h Z⬘⫽兺 NH⫽1 L/2 ⍀N共NH兲 ␩ ⫺NH,共41兲 and ␩ is a gi en pa ame e , depending on he speci ic model, and cha ac e izing he dynamical e en s in he sys em unde shake. Then, om Eq. 共40兲a compac i i y X⬘was iden i ied as X⬘⫽共ln ␩ 兲⫺1.共42兲 This de ini ion is equi alen o X⬘⫺1⫽ 冉 ln⍀L共NH兲 ⳵ NH 冊 L ⫽⫺ 冉 ⳵ S ⳵ N 冊 L ,共43兲 o , using Eq. 共39兲, X⬘⫺1⫽ln共N⫺NH兲2 NHN.共44兲 This exp ession di e s om Eq. 共37兲excep in he limi o high densi y NH/NⰆ1, in which bo h educe o ln(N/NH), bu his only indica es ha he same densi y is ob ained in his limi i X⫽X⬘. Ne e heless, i mus be s essed ha Xis associa ed o apping p ocesses a cons an numbe o pa - icles, while X⬘desc ibes p ocesses a cons an olume. Equi alen ly, Xcha ac e izes ensembles wi h ixed N, and X⬘ ensembles wi h ixed L. In his con ex , hei physical na u e is a he di e en . The pa ame e Xis he compac i i y in- oduced by Edwa ds and, on he o he hand, ␩ ⫺1, ela ed wi h X⬘by Eq. 共42兲, plays he ole o a ugaci y o he pa icles. In e ms o he en opy, Xand X⬘a e ela ed by X⫺1⫽X⬘⫺1⫹ 冉 ⳵ S ⳵ N 冊 NH .共45兲 VI. CONCLUSION In his pape , a one-dimensional model o compac ion in g anula media has been p esen ed. One o i s main ea u es, as compa ed wi h p e ious Ising-like models, is ha he ime e olu ion unde apping conse es he numbe o pa icles in he sys em, while i is he olume ha changes in he com- pac ion p ocess. This is in ac wha happens in compac ion expe imen s. Consequen ly, he s eady dis ibu ion is cha ac- e ized by he compac i i y ins ead o a gene alized ugaci y. The s eady dis ibu ion unc ion has been de i ed and he compac i i y iden i ied in e ms o he pa ame e s de ining he mesoscopic dynamics o he model. I has been ound ha he esul s a e consis en wi h Edwa ds’ he modynami- cal heo y o powde s. Ne e heless, since he model is o - mula ed in he con ex o weak and sho apping, i is in ac qui e doub ul ha he same conclusions we e eached om a gene aliza ion o s onge apping p ocesses. Le us poin ou ha his would equi e o modi y he o mula ion o ou model by including he possibili y ha a la ice egion would expe imen se e al elemen a y exci a ions du ing he same ap. The ela ionship be ween closed and open models, and be ween compac i i y and ugaci y, has been discussed. A he mesoscopic le el o desc ip ion used in his pape , he exp ession o one o hem in e ms o he ansi ion a es canno be in e ed om he exp ession o he o he . Ne e - heless, i is ue ha hey co espond o di e en de i a i es o he same en opy unc ion, like in usual he mal sys ems. The model p esen ed he e can be easily gene alized o mix u es o se e al kinds o g ains, hen allowing he s udy o seg ega ion phenomena. 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