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. Also, i can be use ul o in es i-
ga e he alidi y o he Edwa ds heo y in his case, and
e en ually i s possible gene aliza ions, o ins ance, by ex-
ending he numbe o pa ame e s needed o cha ac e ize he
s eady s a e o he mix u e, as has been ecen ly p oposed
关16,15兴.
ACKNOWLEDGMENTS
We acknowledge suppo om he Minis e io de Ciencia
y Tecnologı
´a共Spain兲 h ough G an No. BFM2002-00303
共pa ially inanced by FEDER unds兲.
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