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