Glasslike dynamical beha io in hie a chical models submi ed o con inuous cooling and hea ing
p ocesses
A. P ados and J. J. B ey
Fı
´sica Teo
´ ica, Facul ad de Fı
´sica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080 Se illa, Spain
共Recei ed 15 Ma ch 2001; published 24 Sep embe 2001兲
The dynamical beha io o a kind o models wi h hie a chically cons ained dynamics is in es iga ed. The
models exhibi many p ope ies esembling eal s uc u al glasses. In pa icula , we ocus on he s udy o
ime-dependen empe a u e p ocesses. In cooling p ocesses, a phenomenon analogous o he labo a o y glass
ansi ion appea s. The esidual p ope ies a e analy ically e alua ed, and he concep o ic i e empe a u e is
discussed on a physical basis. The e olu ion o he sys em in hea ing p ocesses is go e ned by he exis ence o
a no mal solu ion o he e olu ion equa ions, which is app oached by all he o he solu ions. This end o he
sys em is di ec ly ela ed o he glassy hys e esis e ec s shown by hese sys ems. The exis ence o he no mal
solu ion is no es ic ed o he linea egime a ound equilib ium, bu i is de ined o any a bi a y, a - om-
equilib ium, si ua ion.
DOI: 10.1103/PhysRe E.64.041505 PACS numbe 共s兲: 64.70.P , 05.70.Ln, 45.70.⫺n
I. INTRODUCTION
In ecen yea s, he e has been qui e a conside able
amoun o wo k in models in which glassy beha io is gen-
e a ed no by quenched diso de , bu by kine ic cons ain s.
The kine ic es ic ions a e esponsible o he slow elax-
a ion, since he s a e o a pa icle o a g oup o pa icles can
only change i some condi ion o i s en i onmen is ul illed.
In pa icula , ‘‘ acili a ed’’ models ha e been conside ed,
bo h o s uc u al glasses 关1–5兴and o g anula sys ems
关6兴. The cha ac e is ic ea u e o acili a ed models is ha a
pa icle 共spin兲can only change i s s a e i a ce ain numbe o
i s neighbo s is in an exci ed s a e. Also, hie a chically con-
s ained models ha e been used o s udy s e ched exponen-
ial elaxa ion in glasses 关7兴. In hese models, he sys em is
s uc u ed in le els and a pa icle in a gi en le el can only
make a ansi ion i a gi en clus e o pa icles in he lowe
le el is in a ce ain subse o con igu a ions. Then, he dy-
namics o he se e al le els a e coupled, and he cha ac e -
is ic elaxa ion imes inc ease wi h he le el index. Hie a -
chically cons ained dynamics may be ele an o hose
complex sys ems in which he ime e olu ion o he slowes
modes is con olled by he elaxa ion o he as es ones. This
quali a i e pic u e is adequa e o desc ibe, among o he p ob-
lems, p o eins elaxa ion 关8兴and he densi ica ion o powde s
and s uc u al glasses a high p essu e 关9,10兴. Ve y ecen ly,
a kind o hie a chically cons ained dynamics has been
shown o exhibi , in qui e a na u al way, loga i hmic elax-
a ion 关11兴. This kind o ‘‘anomalous,’’ highly nonexponen-
ial, decay is obse ed in a wide a ie y o complex sys ems,
including spin glasses 关12,13兴, g anula ma e ials
关9,10,14,15兴, s uc u al glasses 关16–18兴, and p o ein models
关8,19兴.
The aim o his wo k is o s udy he dynamical beha io
o a gene al class o hie a chically cons ained models when
submi ed o mo e complica ed p ocesses. In pa icula , we
a e in e es ed in he beha io o a sys em wi h hie a chical
cons ain s when he empe a u e changes in ime, which
makes he coupling be ween he le els ime dependen . The
conside a ion o ime-dependen empe a u e p ocesses e-
qui es an ex ension o he o iginal model as o mula ed by
Palme e al. 关7兴. This will be done in a e y simple, bu
na u al, way: he coupling be ween he le els a ies in ime
because he p obabili y o a clus e con igu a ion allowing
elaxa ion o a pa icle in he nex le el depends on he em-
pe a u e. On he o he hand, nei he he numbe o pa icles
in a gi en le el no he leng h o he clus e s ‘‘ acili a ing’’
he elaxa ion depends on he empe a u e. They a e consid-
e ed as quan i ies de ined in he coa se-g ained desc ip ion o
he sys em in oduced o model he physical p oblem a
hand.
Le us no e ha hie a chical models can also be applied o
he analysis o non he mal sys ems, such as g anula ma e i-
als in he dense egime. Fo hose ma e ials, he mal ene gy
is no enough o make he sys em explo e he phase space o
con igu a ions. Then, he sys em mus be ex e nally exci ed
— o ins ance, ib a ing i — in o de o be able o e ol e.
In hese si ua ions, he ole o empe a u e is played by he
in ensi y o he ex e nal d i ing. I he s a iona y s a e
eached by he sys em in he long- ime limi can be desc ibed
by Edwa d’s heo y 关20,21兴, he compac i i y X, which is
analogy o he empe a u e in he mal sys ems, will be a
unc ion o he in ensi y o he ex e nal o ce. Then, by ex-
ploi ing he analogies o Edwa d’s heo y, i.e., subs i u ing
olume by ene gy and compac i i y by empe a u e, i is
possible o inco po a e non he mal sys ems in ou o mula-
ion.
P ocesses in which he empe a u e is ime dependen a e
physically ele an because hey can be used o s udy some
cha ac e is ic dynamical aspec s o glasses. Fo ins ance,
when a supe cooled liquid is cooled down o e y low em-
pe a u es, a labo a o y glass ansi ion is obse ed. A d a-
ma ic change in he beha io o he sys em akes place, and i
depa s om he equilib ium cu e, ge ing ‘‘ ozen’’ in a
a - om-equilib ium s a e. This ansi ion appea s as a con-
sequence o he as inc ease o he elaxa ion ime wi h de-
c easing empe a u e. In o de o cha ac e ize he cooling
p ocess, expe imen al physicis s o en use he esidual alue
o he ele an physical p ope ies, i.e., he di e ence be-
ween hei ac ual alues o e he cooling cu e and he
alue ob ained by ex apola ion o he equilib ium cu e o
PHYSICAL REVIEW E, VOLUME 64, 041505
1063-651X/2001/64共4兲/041505共15兲/$20.00 ©2001 The Ame ican Physical Socie y64 041505-1
he e y-low- empe a u e egion 关22,23兴. I he sys em is e-
hea ed om he nonequilib ium s a e, i e u ns o equilib-
ium o high empe a u es, bu i ollows a di e en cu e
om he cooling one, gi ing ise o hys e esis e ec s. This
phenomenon is ela ed o he ‘‘nonlinea i y’’ o glassy elax-
a ion: he app oach owa ds he equilib ium cu e depends
on he con igu a ion o he sys em, measu ed by he so-called
ic i e empe a u e 关22–24兴. Na ayanaswami’s heo y p o-
ides a phenomenological explana ion o his beha io 关22–
25兴. In e es ingly, a simila beha io has been ound in i-
b a ed g anula ma e ials when he apping in ensi y is a ied
in a cyclic way 关26兴, al hough he hys e esis e ec s a e mo e
e iden when he hea ing p ocess begins in a loosely packed
s a e, e e ed as o he ‘‘i e e sible’’ b anch in he expe i-
men s.
We will s a om a e y gene al hie a chical spin model,
in which pseudospins a e o ganized in o le els, labeled by an
index n. The pseudospins a e assumed o co espond o some
coa se-g ained desc ip ion o he sys em. They can ake only
wo alues, ep esen ing, o ins ance, wo possible densi ies
o a ce ain small sub olume o he sys em. One pseudospin
in le el n⫹1 can only lip be ween he wo possible alues i
a clus e o
␮
nspins in le el nis in a gi en subse o con-
igu a ions. This is he basic cha ac e is ic o hie a chically
cons ained models as in oduced in Re . 关7兴, and i slows
down he elaxa ion in le el n⫹1, as compa ed wi h ha o
le el n. He e we will conside he simple choice ha all he
spins in he clus e mus be in he up 共exci ed兲s a e. The
exac dynamical equa ion o he e olu ion o he pseu-
dospins in ol es e y complica ed momen s o he p obabil-
i y dis ibu ion. To ge an exac ly sol able model, a ‘‘mean-
ield’’ app oxima ion will be in oduced. Then, he
cha ac e is ic elaxa ion ime
␶
no le el nis seen o inc ease
bo h wi h he index label n, due o he hie a chical cons ain ,
and also wi h dec easing empe a u e, since he con igu a-
ions allowing he sys em o elax become less p obable
when he empe a u e is lowe ed.
Some exac dynamical esul s o sys ems desc ibed wi h
mas e equa ions wi h ime-dependen ansi ion a es a e
known. In pa icula , he exis ence o a ‘‘no mal’’ solu ion,
i.e., a solu ion o he mas e equa ion ha is app oached by
all he o he s, has been p o ed on a e y gene al basis 关27兴.
The main equi ed condi ions a e he i educibili y o he
Ma ko p ocess o long enough imes and ha he ansi ion
a es be ex e nally con olled, so ha hey do no depend on
he p obabili y dis ibu ion o he sys em. Mo eo e , i has
been es ablished ha he no mal solu ion ends o he equi-
lib ium cu e o e y high empe a u es in con inuous hea -
ing p ocesses 关27兴. The linea co ec ion o he no mal solu-
ion wi h espec o he equilib ium cu e has been compu ed
using Hilbe ’s me hod 关28兴. I is no e iden whe he hese
esul s s ill hold when app oxima ions a e in oduced in he
dynamics o he sys em. In pa icula , in ‘‘mean- ield’’- ype
app oxima ions, he demons a ions in Re s. 关27,28兴a e no
alid, since he ansi ion a es become unc ionals o he
p obabili y dis ibu ion unc ion. Ne e heless, we will show
ha all he abo e men ioned p ope ies apply o ou simpli-
ied model. This is a good es o he plausibili y o he
app oxima ions ca ied ou and pe haps an indica ion ha he
esul s ob ained he e a e mo e gene al han he de i a ions in
Re s. 关27,28兴.
The o ganiza ion o he pape is as ollows. In Sec. II he
hie a chical model is in oduced, and he exac e olu ion
equa ion o he a e age spin is ob ained. By in oducing a
mean- ield app oxima ion, his equa ion can be closed. A -
e wa ds, a speci ic, bu qui e gene al, choice o he unc-
ions de ining he model is made. This allows us o in oduce
a con inuous limi in which he elaxa ion o he sys em a
cons an empe a u e is sol ed in Sec. III. Time-dependen
empe a u e p ocesses a e conside ed in Sec. IV, whe e he
gene al solu ion o he e olu ion o he p obabili y dis ibu-
ion and he a e age ene gy a e ob ained. The gene al solu-
ion is simila o he exp ession p oposed by Na ayanaswami
on a phenomenological basis 关22,24,25兴. Sec ion IV is de-
o ed o he analysis o Hilbe ’s expansion, which is alid in
he e y-high- empe a u es egime. Cooling p ocesses a e
add essed in Sec. V whe e, o he sake o simplici y, a con-
c e e cooling law is s udied, o which he esidual p ope ies
a e analy ically calcula ed. A quali a i e analysis o he
glasslike ansi ion is p esen ed in Sec. V. I allows us o gi e
e y good es ima es o he esidual p ope ies and leads o
he in oduc ion o he concep o ic i e empe a u e in a
e y na u al way. The beha io o he sys em when i is
ehea ed om low empe a u es is conside ed in Sec. VI. The
main ole played by he no mal solu ion o he unde s and-
ing o he hys e esis e ec s shows up. Mo eo e , he analysis
clea ly indica es ha he ele ance o he no mal solu ion is
no es ic ed o nea equilib ium si ua ions. Finally, a dis-
cussion o he main poin s in his wo k is gi en in Sec. VII.
II. DYNAMICS OF HIERARCHICALLY CONSTRAINED
MODELS
In his sec ion a gene al kind o spin model wi h hie a -
chically cons ained dynamics will be in oduced. We will
ocus on he e olu ion o he a e age alue o he spin,
which is supposed o be he ele an a iable. Fo ins ance,
in a he mal sys em i will be di ec ly ela ed o he mean
ene gy. Then, le us conside a sys em whose deg ees o
eedom can be classi ied in o le els, labeled by an index n
⫽0,1,2,...,nmax . The deg ees o eedom in le el nwill be
ep esen ed by Nnpseudospins,
␴
i
(n)⫽⫾1, i⫽1,2,...,Nn.
The Hamil onian o he sys em is assumed o ha e he o m
H⫽h兺
n⫽0
nmax 兺
i⫽1
Nn
mi
(n),mi
(n)⫽1⫹
␴
i
(n)
2.共2.1兲
No e ha mi
(n)is he occupa ion numbe o he ‘‘up’’ 共⫹1兲
s a e o he co esponding si e. In o de o w i e Eq. 共2.1兲we
ha e supposed ha he e is no in e ac ion be ween he pseu-
dospins, bu he e is an ‘‘ex e nal ield’’ h. Fo a he mal
sys em, Hgi es he ene gy o a gi en mic os a e o he
sys em, while o a non he mal sys em, like a powde , i
could be in e p e ed as he olume o a gi en, mechanically
s able, con igu a ion o ‘‘g ains’’ 关20,21兴. Using he e mi-
nology o he mal sys ems, he a e age alue o he dimen-
sionless ene gy pe spin o e he ensemble o sys ems con-
side ed is
A. PRADOS AND J. J. BREY PHYSICAL REVIEW E 64 041505
041505-2
␧⫽
具
H
典
Nh ⫽1
N兺
n⫽0
nmax 兺
i⫽1
Nn
pi
(n),共2.2兲
whe e
pi
(n)⫽
具
mi
(n)
典
⫽1⫹
具
␴
i
(n)
典
2共2.3兲
is he p obabili y ha he i h spin o le el be in he up s a e,
N⫽兺n⫽0
nmaxNnis he o al numbe o pseudospins, and he an-
gula b acke s deno e s a is ical ensemble a e age.
The sys em is conside ed o be in con ac wi h a hea ba h
a empe a u e T, so ha he equilib ium a e age alue o he
pseudospins does no depend ei he on ino on n, and i is
gi en by
具
␴
典
e⬅
具
␴
i
(n)
典
e⫽⫺ anh
冉
1
T*
冊
.共2.4兲
He e T*is a dimensionless empe a u e, and T*⫽2kBT/h,
kBbeing Bol zmann’s cons an . Fo he sake o concision,
we will d op he as e isk in he ollowing. The abo e a e age
alue o spin ollows om he equilib ium p obabili y o he
‘‘up’’ s a e o any spin,
pe⬅pi,e
(n)⫽e⫺1/T
e⫺1/T⫹e1/T.共2.5兲
In he limi o in ini e empe a u e o ze o ex e nal ield, bo h
s a es o he pseudospins a e equip obable, pe⫽1/2, and
具
␴
典
e⫽0. F om Eq. 共2.2兲, i ollows ha he equilib ium
alue o ␧is
␧e⫽pe.共2.6兲
Fo g anula ma e ials, he ole o he empe a u e Tis
played by he compac i i y 关20,21兴, which is linked o he
in ensi y o he pe u ba ion, allowing he sys em o explo e
he con igu a ion space.
The dynamics o he model is o mula ed by means o a
mas e equa ion wi h single-spin- lip Glaube ansi ion a es
关29兴. Le us conside he lip o a gi en spin
␴
i
(n). This an-
si ion connec s a gi en con igu a ion
␴
o he whole sys em
wi h he con igu a ion Ri
(n)
␴
, whe e Ri
(n)is he ope a o
which o a es he spin
␴
i
(n), keeping all he o he spins he
same. The ansi ion a e o he lip o he spin
␴
i
(n)in con-
igu a ion
␴
is
Wi
(n)共
␴
兲⫽1
2
␣
i
(n)共
␴
兲
冋
1⫹
␴
i
(n) anh
冉
1
T
冊
册
.共2.7兲
The cha ac e is ic elaxa ion a e
␣
i
(n)o he spin
␴
i
(n)de-
pends on he con igu a ion
␴
o he sys em h ough he hi-
e a chical cons ain
␣
i
(n)共
␴
兲⫽关
␣
ki
(n⫺1)共
␴
兲
␣
ki⫹1
(n⫺1)共
␴
兲•••
␣
ki⫹
␮
n⫺1⫺1
(n⫺1)
⫻共
␴
兲兴1/
␮
n⫺1
兿
j⫽ki
ki⫹
␮
n⫺1⫺1
␦
␴
j
(n⫺1) ,⫹1,共2.8兲
whe e
␦
ij is he K onecke del a. This exp ession implies
ha he spin
␴
i
(n)needs, in o de o lip, ha all he spins
belonging o a clus e o leng h
␮
n⫺1s a ing a a gi en spin
kio le el n⫺1 mus be in he up 共⫹1兲s a e. Besides, he
cha ac e is ic lip a e o he spin is he a e age o he cha -
ac e is ic lip a es o he spins belonging o he clus e de-
e mining i s possibili y o change. This es ic ing condi ion
is schema ically depic ed in Fig. 1. No e ha he possibili y
o a gi en spin in le el n o lip is es ic ed by he s a e o a
se o clus e s in all le els n⬘⬍n, he numbe o clus e s
in ol ed in each le el inc easing as n⬘dec eases. The hie -
a chical cons ain implies ha he con igu a ion wi h all he
spins in he down (⫺1) s a e is comple ely abso ben ; i.e.,
he sys em does no e ol e in ime om ha con igu a ion.
We a e in e es ed in he ime e olu ion o he a e age spin
␴
i
(n), which is gi en in Glaube dynamics by 关29兴
d
d
具
␴
i
(n)
典
⫽⫺2
具
␴
i
(n)Wi
(n)共
␴
兲
典
,共2.9兲
and subs i u ion o Eqs. 共2.7兲and 共2.8兲in o his exp ession
yields
d
d
具
␴
i
(n)
典
⫽⫺
冓
关
␣
ki
(n⫺1)共
␴
兲
␣
ki⫹1
(n⫺1)共
␴
兲•••
⫻
␣
ki⫹
␮
n⫺1⫺1
(n⫺1) 共
␴
兲兴1/
␮
n⫺1共
␴
i
(n)⫺
具
␴
典
e兲
⫻
兿
j⫽ki
ki⫹
␮
n⫺1⫺1
␦
␴
j
(n⫺1) ,⫹1
冔
,共2.10兲
whe e we ha e aken in o accoun ha (
␴
i
(n))2⫽⫹1 o all i,
n. This equa ion is a he in ol ed, since i couples he e o-
lu ion o
具
␴
i
(n)
典
o momen s o he p obabili y dis ibu ion
con aining an inc easing numbe o spins o all he le els n⬘
such ha 0⭐n⬘⬍n. The le els 0⭐n⬘⭐n⫺2 en e in o he
equa ion h ough he a es
␣
j
(n⫺1)(
␴
). Then, we in oduce a
his s age a so o ‘‘mean- ield’’ app oxima ion o he an-
FIG. 1. The amed spin in le el n,
␴
i
(n), has a non anishing
p obabili y o changing i s s a e only i he amed clus e o spins
in le el n⫺1 a e in he s a e shown in he igu e, i.e. all o hem up
共⫹1兲. In his example, we ha e aken ki⫽i⫺2 and
␮
n⫺1⫽5.
GLASSLIKE DYNAMICAL BEHAVIOR IN . . . PHYSICAL REVIEW E 64 041505
041505-3
si ion a es, which is closely ela ed, in spi i , o he seminal
wo k o Palme e al. 关7兴. Wi hin his coa sening dynamics,
i is easonable o expec ha spins in le el ne ol e o e a
ime scale qui e la ge han ha cha ac e is ic o le el n⫺1.
This means ha spins in le el n⫺1 change many imes hei
s a e be o e a ansi ion in le el n akes place. Thus, we
eplace he p oduc o he K onecke del as in Eq. 共2.8兲by i s
a e age alue, i.e., by he p obabili y P(n⫺1)(
␴
ki
(n⫺1)
⫽⫹1,...,
␴
ki⫹
␮
n⫺1⫺1
(n⫺1) ⫽⫹1) ha all
␮
n⫺1spins o he
gi en clus e a e in he up s a e,
␣
i
(n)共
␴
兲⫽关
␣
ki
(n⫺1)共
␴
兲
␣
ki⫹1
(n⫺1)共
␴
兲•••
␣
ki⫹
␮
n⫺1⫺1
(n⫺1) 共
␴
兲兴1/
␮
n⫺1
⫻P(n⫺1)共
␴
ki
(n⫺1)
⫽⫹1,...,
␴
ki⫹
␮
n⫺1⫺1
(n⫺1) ⫽⫹1兲.共2.11兲
Mo eo e , we will es ic ou sel es o si ua ions whe e he e
is spa ial homogenei y wi hin each o he le els, so ha he
dependence on he speci ic si e conside ed in a gi en le el
can be d opped, ob aining
␣
(n)共
␴
兲⫽
␣
(n⫺1)共
␴
兲P(n⫺1)共
␴
1
(n⫺1)⫽⫹1,...,
␴
␮
n⫺1
(n⫺1)⫽
⫹1兲.共2.12兲
I e a ion o he abo e ela ion gi es
␣
(n)共
␴
兲⫽
␣
(0)
兿
j⫽0
n⫺1
P(j)共
␴
1
(j)⫽⫹1,...,
␴
␮
j
(j)⫽⫹1兲,
共2.13兲
wi h
␣
(0) being a cons an ha cha ac e izes he elaxa ion
a e o he spins belonging o le el n⫽0, whose dynamics is
no cons ained. As
␣
(0) de e mines he basic ime scale,
which is a bi a y, we will ake
␣
(0)⫽1 in he ollowing. In
he mean- ield app oxima ion jus in oduced, he ime e o-
lu ion o he a e age alue o he spin, Eq. 共2.10兲, akes he
o m
d
d
具
␴
(n)
典
⫽⫺
兿
j⫽0
n⫺1
P共
␴
1
(j)⫽⫹1,...,
␴
␮
j
(j)⫽⫹1兲共
具
␴
(n)
典
⫺
具
␴
典
e兲.共2.14兲
Now, a new app oxima ion will be made. The p obabili y
P(j)in Eq. 共2.14兲will be subs i u ed by i s equilib ium alue.
This would be exac in linea esponse a ound equilib ium,
bu i will be aken he e as an app oxima ion leading o he
basic equa ion o ou hie a chically cons ained model,
namely,
d
d
具
␴
(n)
典
⫽⫺
兿
j⫽0
n⫺1
pe
␮
j共
具
␴
(n)
典
⫺
具
␴
典
e兲,共2.15兲
whe e peis he equilib ium p obabili y o any spin being in
he up s a e, gi en by Eq. 共2.5兲. Again, on physical g ounds,
his is a sensible app oxima ion due o he sepa a ion o he
cha ac e is ic ime scales o he di e en le els. Because o
he hie a chically cons ained dynamics, spins in le el n⫺1
each equilib ium o e a ime scale in which spins in le el n
ha e no begun o e ol e. Howe e , as a consequence o he
las app oxima ion, he con igu a ion wi h all he spins in he
down s a e is no longe abso ben .
F om Eq. 共2.15兲, an equi alen equa ion can be w i en o
he e olu ion o he p obabili y p(n)o he up s a e in le el n,
de ined in Eq. 共2.3兲, i.e.,
d
d p(n)⫽⫺
␣
n共p(n)⫺pe兲,共2.16兲
␣
nbeing he cha ac e is ic elaxa ion a e o le el n,
␣
n⫽pe
gn,gn⫽兺
j⫽0
n⫺1
␮
j.共2.17兲
Equa ion 共2.16兲implies ha , due o he hie a chical con-
s ain s, he spin elaxa ion slows down wi h inc easing le el
n, since
␣
nis a dec easing unc ion o n, because pe⬍1.
Equa ion 共2.16兲is he main esul in his sec ion. In he ol-
lowing, we will explo e i s implica ions, conside ing i s
p ocesses a cons an empe a u e in he nex sec ion, and
cooling and hea ing p ocesses in he emainde o he pape .
III. RELAXATION AT CONSTANT TEMPERATURE
Fo he case o cons an empe a u e T, and he e o e con-
s an
␣
n, Eq. 共2.16兲is easily sol ed,
p(n)共 兲⫺pe⫽关p(n)共0兲⫺pe兴e⫺
␣
n .共3.1兲
Then, each spin elaxes exponen ially o equilib ium wi h he
a e cha ac e is ic o i s le el.
Fo he homogenous si ua ions wi hin each le el we a e
conside ing, he dimensionless mean ene gy pe spin 关30兴
de ined in Eq. 共2.2兲simpli ies o
␧共 兲⫽兺
n⫽0
nmax
wnp(n)共 兲,共3.2兲
whe e wn⫽Nn/Nis he ac ion o spins in le el n, e i ying
兺n⫽0
nmaxwn⫽1.
Pu ing Eq. 共3.1兲in o Eq. 共3.2兲yields
␧共 兲⫽␧e共T兲⫹兺
n⫽0
nmax
wn关p(n)共0兲⫺pe兴e⫺
␣
n ,共3.3兲
o he elaxa ion o he ene gy a cons an empe a u e. In
o de o p oceed, we will conside he simple case in which
he ini ial p obabili y dis ibu ion p(n)(0) does no depend on
he index le el n. This will be he si ua ion, o ins ance,
when he ini ial s a e co esponds o equilib ium a a di e -
en empe a u e T⫹⌬T. Thus, he elaxa ion unc ion o he
physical p ope y desc ibed by he Hamil onian o he sys-
em is gi en by
␾
共 兲⬅␧共 兲⫺␧e
␧共0兲⫺␧e
⫽兺
n⫽0
nmax
wne⫺
␣
n .共3.4兲
A. PRADOS AND J. J. BREY PHYSICAL REVIEW E 64 041505
041505-4
This equa ion is a gene aliza ion o he esul de i ed by
Palme e al. in hei pionee ing wo k in hie a chically con-
s ained dynamics 关7兴, which co esponds o he choice pe
⫽1/2. This is o mally equi alen o he pa icula iza ion o
Eq. 共3.4兲 o T→⬁. I o he posi i e alues o he empe a-
u e a e conside ed, he e ec is an inc ease o he elaxa ion
imes
␶
n⫽
␣
n
⫺1,共3.5兲
since peis a dec easing unc ion o he empe a u e. A main
ad an age o he o mula ion o he hie a chical models as
p esen ed he e, aside om i s la ge gene ali y, is ha i al-
lows analysis o p ocesses in which he empe a u e o a
he mal sys em 共o he ib a ion in ensi y in a g anula sys-
em兲changes in ime. This kind o p ocesses will be ad-
d essed in he nex sec ion. A mean elaxa ion ime
␶
can be
de ined as
␶
⫽
冕
0
⬁d
␾
共 兲⫽兺
n⫽0
nmax
wn
␶
n,共3.6兲
p o iding a quan i a i e measu e o he ime i akes he sys-
em o elax o equilib ium a empe a u e T.
Le us conside ha he ac ion o spins in le el n,wn,
and he numbe o ‘‘ acili a ing’’ spins in le el n,
␮
n, de-
pend e y smoo hly on n; i.e., hey can be exp essed as unc-
ions o he o m
wn⫽w共n
␩
兲
␮
n⫽
␮
共n
␩
兲,共3.7兲
whe e
␩
Ⰶ1. These seem o be sensible condi ions when
modeling a eal sys em, in which he in oduc ion o he
le els and he pseudospins is associa ed o some coa se-
g ained desc ip ion. By de ining
xn⫽n
␩
,共3.8兲
which is a con inuous a iable in he limi
␩
→0, he sums
o e ncan be eplaced by in eg als. The elaxa ion a e o
le el n, gi en by Eq. 共2.17兲, becomes a unc ion o he con-
inuous a iable x,
␣
共x兲⫽pe
g(x),共3.9兲
wi h
g共x兲⫽
兺
k⫽0
n⫺1
␮
k
␩
兺
k⫽0
nmax
wk
␩
⫽
冕
0
xdx⬘
␮
共x⬘兲
冕
0
xmaxdx⬘w共x⬘兲
,共3.10兲
whe e xmax⫽nmax
␩
, and he no maliza ion o he weigh s wn
has been used. The e o e, he elaxa ion unc ion
␾
( ), gi en
by Eq. 共3.4兲, becomes
␾
共 兲⫽
兺
n⫽0
nmax
␩
wne⫺
␣
n
兺
n⫽0
nmax
␩
wn
⫽
冕
0
xmaxdxw共x兲e⫺
␣
(x)
冕
0
xmaxdxw共x兲
,
共3.11兲
in he con inuous limi .
In gene al, Eq. 共3.11兲is ma hema ically a he in ol ed,
since i depends bo h on he unc ions
␮
(x) and w(x). The
simples possibili y appea s o be
␮
(x) p opo ional o w(x),
i.e.,
␮
np opo ional o wno , equi alen ly, o Nn. In o he
wo ds, he simples kind o hie a chically cons ained mod-
els shows up when he numbe o ‘‘ acili a ing’’ spins a a
le el is an ex ensi e unc ion o he numbe o spins a he
same le el 关11兴. This condi ion is exp essed as
␮
共x兲⫽aw共x兲,共3.12兲
wi h abeing a cons an , independen o x. In his case, i is
use ul o de ine he new a iable
u⫽
冕
0
xdx⬘w共x⬘兲
冕
0
xmaxdx⬘w共x⬘兲
,共3.13兲
measu ing he ac ion o spins belonging o le els up o n
⫽x/
␩
. In e ms o u, he elaxa ion a es o Eq. 共3.9兲a e
gi en by
␣
共u兲⫽pe
g(u),g共u兲⫽au,共3.14兲
he elaxa ion unc ion is exp essed as
␾
共 兲⫽
冕
0
1due⫺
␣
(u) ,共3.15兲
and he mean elaxa ion ime eads
␶
⫽
冕
0
⬁d
␾
共 兲⫽
冕
0
1du
␶
共u兲⫽pe
⫺a⫺1
a
兩
ln pe
兩
,共3.16兲
wi h
␶
(u)⫽
␣
⫺1(u). I is in e es ing o conside si ua ions
o which pe
aⰆ1, so ha he minimum elaxa ion a e
␣
(1)
is much smalle han he maximum one
␣
(0)⫽1 in ou di-
mensionless ime scale. In his case, he elaxa ion unc ion
␾
( ) is linea in ln o e an in e media e ime window, 1
Ⰶ Ⰶpe
⫺a, namely 关11兴,
␾
共 兲⬃1⫺1
a
兩
ln pe
兩
共
␥
⫹ln 兲,共3.17兲
whe e
␥
s ands o Eule ’s cons an ,
␥
⯝0.577. This kind o
linea loga i hmic beha io is cha ac e is ic o a g ea a ie y
o complex sys ems, including spin glasses 关12,13兴, g anula
ma e ials 关9,10,14,15兴, s uc u al glasses 关16–18兴, and p o-
ein models 关8,19兴. In he p esen con ex , he condi ion pe
a
Ⰶ1 co esponds o a ‘‘low’’- empe a u e limi , in which he
GLASSLIKE DYNAMICAL BEHAVIOR IN . . . PHYSICAL REVIEW E 64 041505
041505-5

mean elaxa ion ime, gi en by Eq. 共3.16兲, is e y la ge; i.e.,
he elaxa ion o he sys em becomes e y slow.
I is wo h no ing ha , in e ms o he u a iable, he
con inuous limi is o mally ob ained by changing he unc-
ions o he index le el nby he co esponding unc ions o
he a iable uand by making he eplacemen
兺
n⫽0
nmax
wn→
冕
0
1du.共3.18兲
I ollows ha , in he con inuous limi , he dynamical beha -
io o he sys em does no depend explici ly on he le el
popula ions Nn, bu only on he elaxa ion a es
␣
exp essed
as unc ions o u, as gi en by Eq. 共3.14兲.
IV. TIME-DEPENDENT TEMPERATURE PROCESSES
In his sec ion p ocesses in which he empe a u e changes
in ime will be s udied. The e olu ion equa ion 共2.16兲is now
d
d p(n)共 兲⫽⫺
␣
n共T兲关p(n)共 兲⫺pe共T兲兴,共4.1兲
whe e T⫽T( ). The gene al solu ion o Eq. 共4.1兲is
p(n)共 兲⫽关p(n)共 0兲⫺pe共T0兲兴
␹
n共 , 0兲⫹pe共T兲
⫺
冕
0
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
␹
n共 , ⬘兲.共4.2兲
He e T0⫽T( 0) is he ini ial alue o he empe a u e, T
⫽T( ), T⬘⫽T( ⬘), T⬙⫽T( ⬙), and we ha e in oduced he
unc ion
␹
n共 1, 2兲⫽exp
冉
⫺
冕
2
1d
␣
n共 兲
冊
.共4.3兲
The abo e equa ion is alid o any law o a ia ion o he
empe a u e. Taking in o accoun Eq. 共3.2兲, he a e age en-
e gy pe spin, ␧, is gi en by 关30兴
␧共 兲⫽兺
n⫽0
nmax
wn关p(n)共 0兲⫺pe共T0兲兴
␹
n共 , 0兲⫹␧e共T兲
⫺
冕
0
d ⬘d␧e共T⬘兲
dT⬘
dT⬘
d ⬘兺
n⫽0
nmax
wn
␹
n共 , ⬘兲,共4.4兲
whe e ␧eis he a e age equilib ium ene gy, de ined in Eq.
共2.6兲.
Le us assume ha he sys em is ini ially a equilib ium
wi h T⫽T0. Then he i s e m on he igh -hand side 共RHS兲
o Eq. 共4.4兲 anishes and
␧共 兲⫽␧e共T兲⫺
冕
0
d ⬘d␧e共T⬘兲
dT⬘
dT⬘
d ⬘M共 , ⬘兲,共4.5兲
whe e
M共 , ⬘兲⫽兺
n⫽0
nmax
wn
␹
n共 , ⬘兲共4.6兲
is a memo y unc ion. In he case o cons an empe a u e,
M( , ⬘) is equal o he elaxa ion unc ion
␾
( ⫺ ⬘), as seen
by compa ing Eq. 共4.6兲wi h Eq. 共3.4兲. The s uc u e o Eq.
共4.5兲is he same as ha o Na ayanaswami’s phenomeno-
logical heo y o glasses 关22–25兴. A simila esul was ob-
ained some yea s ago o he one-dimensional Ising model
wi h Glaube dynamics 关31兴.
High- empe a u e limi : Hilbe ’s me hod
We a e going o look o a solu ion o Eq. 共4.1兲by means
o Hilbe ’s me hod. A special solu ion
pH
(n)共 兲⫽兺
k⫽0
⬁
pH
(n),k共 兲共4.7兲
is cons uc ed in an i e a i e way as ollows. We ake
pH
(n),0共 兲⫽pe共T兲,共4.8兲
while o k⭓1
pH
(n),k共 兲⫽⫺
␣
n
⫺1共T兲dp(n),k⫺1共 兲
d .共4.9兲
Equa ion 共4.8兲shows ha Hilbe ’s expansion ag ees wi h
he equilib ium dis ibu ion o he lowes o de . Besides, o
k⫽1 we ge om Eq. 共4.9兲
pH
(n),1共 兲⫽⫺
␶
n共T兲dpe共T兲
dT
dT
d .共4.10兲
This equa ion indica es he main limi a ion o Hilbe ’s
me hod. Due o he di e gence o he elaxa ion imes
␶
n
⫽
␣
n
⫺1in he low- empe a u e limi 关see Eq. 共2.17兲兴, also
p(n),1 di e ges in ha limi . As a consequence, Hilbe ’s so-
lu ion is only accu a e in he high- empe a u e egime, in
which an expansion a ound equilib ium p o ides a good ap-
p oxima ion. Res ic ing ou sel es o high empe a u es, we
app oxima e
pH
(n)共T兲⯝pe共T兲⫺
␶
n共T兲dpe共T兲
dT
dT
d 共4.11兲
and, om Eq. 共3.2兲,
␧H共T兲⯝␧e共T兲⫺d␧e共T兲
dT
dT
d 兺
n⫽0
nmax
wn
␶
n共T兲.共4.12兲
Taking in o accoun he de ini ion o he a e age elaxa ion
ime
␶
, Eq. 共3.6兲, he abo e exp ession is seen o be equi a-
len o
␧H共T兲⯝␧e共T兲⫺d␧e共T兲
dT
dT
d
␶
共T兲,共4.13兲
A. PRADOS AND J. J. BREY PHYSICAL REVIEW E 64 041505
041505-6
which ag ees wi h he high- empe a u e beha io o Eq.
共4.5兲. In Eqs. 共4.11兲and 共4.13兲,pH
(n)and ␧Hdepend on ime
only h ough he empe a u e T( ). Thus, Hilbe ’s me hod
p o ides a ‘‘no mal’’ solu ion, in he sense o en used in
kine ic heo y. The alidi y o a exp ession iden ical o Eq.
共4.13兲, also in he high- empe a u e limi , has been es ab-
lished o a qui e gene al class o sys ems whose dynamics is
desc ibed by a mas e equa ion 关28兴. Al hough we ha e made
he e se e al d as ic app oxima ions in o de o ge a closed
equa ion o he a e age spin, he high- empe a u e limi o
he solu ion, gi en by Hilbe ’s me hod, emains o mally he
same as ha o he exac solu ion o he o iginal model.
Ce ainly, his is a good p ope y o hose app oxima ions.
Wha is he physical meaning o he ailu e o Hilbe ’s
expansion o low empe a u es? Due o he di e gence o
he cha ac e is ic elaxa ion imes, he sys em does no ha e
enough ime o elax o he equilib ium cu e a e y low
empe a u es, and i ge s ‘‘ ozen’’ in a a - om-equilib ium
s a e. Since Hilbe ’s me hod is an expansion a ound equilib-
ium, i ails in he low- empe a u e egion. In ac , he RHS
o Eq. 共4.11兲becomes nega i e o low enough empe a u es.
On he o he hand, Hilbe ’s expansion is use ul o es ima e
he alues o he physical p ope ies in he ‘‘ ozen’’ s a e
关28兴. Also, Hilbe ’s me hod p o ides a quali a i e unde -
s anding o he hys e esis e ec s appea ing in he mal cycles
共cooling and ehea ing兲. In cooling p ocesses (dT/d ⬍0), i
is ␧H⭓␧e, while in hea ing p ocesses (dT/d ⬎0), i is ␧H
⭓␧e. Then, ␧Hlies o opposi e sides o he equilib ium
cu e o cooling and hea ing p ocesses, and hys e esis e -
ec s show up in he mal cycling expe imen s, as will be
discussed in mo e de ail in Sec. VI.
V. COOLING PROCESSES
Nex , we a e going o s udy he con inuous cooling o he
sys em down o e y low empe a u es. The o igin o ime is
aken a he beginning o he cooling p ocess. The ini ial
condi ion will be he equilib ium con igu a ion a a ‘‘high’’
empe a u e T0, i.e.,
p(n)共0兲⫽pe共T0兲.共5.1兲
Then, he i s co ec ion in Hilbe ’s expansion, p(n),1( ), is
e y small as compa ed wi h pe(T0) o T→T0. Pa icula -
iza ion o Eq. 共4.2兲 o he abo e ini ial condi ion gi es
p(n)共 兲⫽pe共T兲⫺
冕
0
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
␹
n共 , ⬘兲.共5.2兲
Since pe(T) is an inc easing unc ion o he empe a u e, o
con inuous cooling p ocesses i is
p(n)共 兲⭓pe共T兲 o all and n.共5.3兲
The possible de ia ions om he equilib ium dis ibu ion al-
ways lead o an inc ease o he p obabili y o he spin being
in he exci ed s a e. Mo eo e , Eq. 共5.2兲di ec ly implies ha
␧共 兲⫽␧e共T兲⫺
冕
0
d ⬘d␧e共T⬘兲
dT⬘
dT⬘
d ⬘兺
n⫽0
nmax
wn
␹
n共 , ⬘兲⭓␧e共T兲,
共5.4兲
whe e we ha e used Eq. 共3.2兲. This inequali y has been ex-
pe imen ally obse ed in glass- o ming liquids 关22,23兴.
Since he epo ed expe imen s we e made a cons an p es-
su e, he quan i y ␧conside ed he e mus be in e p e ed as
he en halpy in ha con ex .
In o de o p oceed u he in ou analysis, he con inuous
limi in oduced in he s udy o he elaxa ion a cons an
empe a u e in Sec. III will be conside ed. As al eady men-
ioned, his con inuous limi is expec ed o be close o he
desc ip ion o eal sys ems han he disc e e le el pic u e.
Besides, o he sake o conc e eness, we will es ic ou -
sel es o hose models e i ying Eq. 共3.12兲. The index le el
nis subs i u ed by he con inuous a iable u, de ined in Eq.
共3.13兲, ep esen ing he ac ion o he o al numbe o spins
up o le el n. Wi h an ob ious change o no a ion, Eq. 共5.2兲
becomes
p共 ;u兲⫽pe共T兲⫺
冕
0
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
␹
共 , ⬘;u兲,共5.5兲
whe e
␹
共 , ⬘;u兲⫽exp
冉
⫺
冕
⬘
d ⬙
␣
共T⬙;u兲
冊
,共5.6兲
wi h
␣
(T;u) gi en by Eq. 共3.14兲, i.e.,
␣
共T;u兲⫽pe共T兲au.共5.7兲
Also, using Eq. 共3.18兲, i is ound ha
␧共 兲⫽␧e共T兲⫺
冕
0
d ⬘d␧e共T⬘兲
dT⬘
dT⬘
d ⬘
冕
0
1du
␹
共 , ⬘;u兲.
共5.8兲
The ime e olu ion o he p obabili y p( ;u) depends on
he explici o m o he cooling law. In expe imen s, linea
cooling is usually employed,
dT
d ⫽⫺ c,共5.9兲
whe e c⬎0 is he cooling a e de e mining he ime scale
c
⫺1o e which he empe a u e changes. Linea cooling im-
plies ha pe(T) depends on ime in a a he in ol ed way.
F om Eqs. 共2.5兲and 共5.9兲one ge s
dpe
d ⫽⫺ 1
2 cpe共1⫺pe兲
冉
ln pe
1⫺pe
冊
2
.共5.10兲
We a e in e es ed in cooling p ocesses o which he em-
pe a u e changes slowly in ime, cⰆ1, so ha he sys em
depa s om he equilib ium cu e o e y low empe a-
u es, whe e peⰆ1. Then, a law equi alen o linea cooling,
aside om loga i hmic co ec ions, is
GLASSLIKE DYNAMICAL BEHAVIOR IN . . . PHYSICAL REVIEW E 64 041505
041505-7
dpe
d ⫽⫺ cpe,共5.11兲
ha ing he ad an age ha analy ical calcula ions a e much
mo e simple wi h Eq. 共5.12兲关31–33兴. In he ollowing, we
will use he no a ion
pe共T兲⫽pe,pe共T⬘兲⫽pe
⬘,pe共T0兲⫽pe0.共5.12兲
The ime in eg als in Eq. 共5.5兲can be ans o med in o in e-
g als o e peby means o he cooling law 共5.11兲, wi h he
esul
p共 ;u兲⫽pe⫹
冕
pe
pe0dpe
⬘exp
冉
⫺pe
⬘au⫺pe
au
cau
冊
.共5.13兲
The second e m in his exp ession is dominan in he low-
empe a u e egion, whe e peis e y small. The e o e, i ol-
lows ha spins in any le el u all in a nonequilib ium s a e
o low enough empe a u es. The de ails o his glasslike
ansi ion will be analyzed below, in a sepa a e subsec ion.
One o he main quan i ies cha ac e izing a gi en cooling
p ocess is he esidual alue es o a ele an p ope y . The
esidual alue measu es he excess wi h espec o he equi-
lib ium cu e, ex apola ed o e y low empe a u es,
es⫽lim
T→0
共 ⫺ e兲.共5.14兲
In pa icula , o he p obabili y p( ;u) we ha e om Eq.
共5.13兲
p es共u兲⫽
冕
0
pe0dpe
⬘exp
冉
⫺pe
⬘au
cau
冊
,共5.15兲
which is easily ans o med in o
p es共u兲⫽1
au共 cau兲1
au
冕
0
x0dxx 1
au ⫺1e⫺x,共5.16兲
whe e x0⫽pe0
au/( cau). The slow cooling limi is de ined by
he esidual p ope ies being independen o he ini ial con-
di ions and de e mined uni ocally by he cooling a e 关31–
34兴. In ou case, slow cooling means ha he uppe in eg a-
ion limi in Eq. 共5.16兲can be subs i u ed by in ini y o all u,
i.e.,
pe0
au
cauⰇ1. 共5.17兲
As he u a iable a ies in he in e al 0⭐u⭐1, he slow
cooling condi ion is
caⰆpe0
a⬍1. 共5.18兲
Then, wi h an exponen ially small e o ,
p es共u兲⬃1
au共 cau兲1/au⌫
冉
1
au
冊
⫽共 cau兲1/au⌫
冉
1⫹1
au
冊
.
共5.19兲
This exp ession gi es he p obabili y ha he spins in le el u
be in he up s a e a e y low empe a u es. In Fig. 2, he
esidual p obabili y p es(u⫽1) is plo ed as a unc ion o he
cooling a e c, o a⫽1. The asymp o ic esul , gi en by
Eq. 共5.19兲, is compa ed wi h he nume ical in eg a ion o Eq.
共5.16兲wi h pe0⫽1/2; i.e., he sys em is aken ini ially a
in ini e empe a u e. The ag eemen is qui e good up o c
⯝0.1, which is no e y small.
The e olu ion o he a e age ene gy, o he cooling law
共5.11兲, is gi en by
␧共 兲⫺␧e共T兲⫽
冕
0
1du关p共 ;u兲⫺pe兴
⫽
冕
0
1du
冕
pe
pe0dpe
⬘exp
冉
⫺pe
⬘au⫺pe
au
cau
冊
.
共5.20兲
The esidual ene gy can be easily compu ed by pa icula iz-
ing his exp ession o T→0,
␧ es⫽
冕
0
1du p es共u兲.共5.21兲
The in eg and p es(u), gi en by Eq. 共5.19兲, anishes expo-
nen ially in he limi u→0. A s anda d Laplace analysis can
be made, wi h he esul
␧ es⬃⌫
冉
1⫹1
a
冊
共 ca兲1/a
兩
ln共 ca兲1/a
兩
.共5.22兲
The leading beha io is po en ial wi h c, since
ln␧ es⬃1
aln共 ca兲,共5.23兲
FIG. 2. Residual p obabili y o he slowes modes p es(u⫽1) as
a unc ion o he dimensionless cooling a e cde ined in he main
ex . The ci cles co espond o he nume ical in eg a ion o Eq.
共5.16兲, while he solid line is he p edic ion o he asymp o ic cal-
cula ion, Eq. 共5.19兲. A good ag eemen is obse ed up o c⯝0.1.
A. PRADOS AND J. J. BREY PHYSICAL REVIEW E 64 041505
041505-8
which comes om he uppe limi o in eg a ion, u⫽1, co -
esponding o he la ges elaxa ion ime. The in eg al o e
he whole dis ibu ion unc ion p es(u) gi es a loga i hmic
co ec ion
兩
ln( ca)
兩
⫺1, which makes he esidual ene gy
smalle han he dominan e m ( ca)1/a. This is due o he
inc easing beha io o p es(u) wi h u,p es(u)⭐p es(u⫽1).
In Fig. 3 he esidual alue o he ene gy is plo ed. The
asymp o ic exp ession, Eq. 共5.22兲, is compa ed wi h he nu-
me ical esul s om Eqs. 共5.21兲and 共5.16兲. Good ag eemen
is ound up o c⫽0.01. I is wo h no ing ha , o he cool-
ing law conside ed, he loga i hmic co ec ion o he po en-
ial in cbeha io is no p esen o o he simple models o
s uc u al glasses p e iously s udied 关31–35兴.
Dema ca ion mode, ic i e empe a u e, and glass ansi ion
The exis ence o non anishing esidual p ope ies is an
indica ion o he depa u e o he sys em om equilib ium a
low empe a u es. Due o he di e gence o he cha ac e is ic
elaxa ion imes
␶
n, o low enough empe a u es he sys em
does no ha e enough ime o elax owa ds equilib ium, and
a kine ic phenomenon esembling he labo a o y glass an-
si ion 关22–24兴shows up. Nex , we will y o unde s and he
physical o igin o his kine ic ansi ion.
Le us conside again he ime e olu ion o he p obabili y
dis ibu ion p( ;u) in a cooling p ocess, as gi en by Eqs.
共5.5兲and 共5.6兲. The in eg al in
␹
( , ⬘;u),
I共 , ⬘;u兲⫽
冕
⬘
d ⬙
␣
共T⬙;u兲,共5.24兲
is a measu e o he a e age numbe o ansi ions occu ing
in le el uin he ime in e al be ween ⬘and . Conse-
quen ly, a ma hema ical de ini ion o he limi o slow cool-
ing is ha he condi ion
I共 *,0;u兲Ⰷ1共5.25兲
hold o all u, whe e *is he ime o which he empe a u e
anishes i ex apola ed acco dingly o he p esc ibed cool-
ing law, i.e., T( *)⫽0. Equa ion 共5.25兲gua an ees ha he
sys em expe imen s a la ge numbe o ansi ions be o e ge -
ing e en ually ozen, so ha i has enough ime o o ge
he de ails o he ini ial condi ion. Fo he cooling law de-
ined in Eq. 共5.11兲i is easily e i ied ha Eq. 共5.25兲is
equi alen o Eq. 共5.17兲.
I we a e dealing wi h a slow cooling p ocess, Eq. 共5.25兲
implies ha he e is a ime window o e which
I共 ,0;u兲Ⰷ1. 共5.26兲
This is he ime egime we a e in e es ed in. Le us analyze
he beha io o
␹
( , ⬘;u) as a unc ion o ⬘, 0⭐ ⬘⭐ , o a
gi en ime such ha Eq. 共5.26兲holds. The unc ion
I( , ⬘;u) changes om a e y la ge alue o ze o when ⬘
goes om 0 o . Consequen ly,
␹
( , ⬘;u) inc eases om
p ac ically ze o o uni y when ⬘mo es in he abo e ime
in e al. Le us de ine a ime ( ;u), p io o ,by
I共 , ;u兲⫽1, 共5.27兲
so ha he a e age numbe o ansi ions aking place in
le el uin he ime in e al be ween and equals uni y.
Then,
␹
( , ;u)⫽e⫺1, and he unc ion
␹
( , ⬘;u) changes
om ze o o uni y in a ce ain ime in e al a ound .In
o de o p oceed, we will assume ha his change akes place
in he icini y o e y apidly, as compa ed wi h he a ia-
ion o he es o he in eg and o Eq. 共5.5兲. Decomposing
Eq. 共5.5兲in he o m
p共 ;u兲⫽pe共T兲⫺
冕
0
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
␹
共 , ⬘;u兲
⫺
冕
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
␹
共 , ⬘;u兲,共5.28兲
he i s in eg al is subdominan wi h espec o he second
one and, mo eo e ,
冕
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
␹
共 , ⬘;u兲⯝
冕
d ⬘dpe共T⬘兲
dT⬘
dT⬘
d ⬘
⫽pe共T兲⫺pe关T 共 ;u兲兴,
共5.29兲
whe e T ( ;u) is he empe a u e o he sys em a ime
( ;u), i.e.,
T 共 ;u兲⫽T关 共 ;u兲兴.共5.30兲
No e ha T ( ;u)⬎T, since he ime ins an ( ;u)⬍ . Sub-
s i u ion o Eq. 共5.29兲in o Eq. 共5.28兲and use o he abo e
app oxima ions yields
p共 ;u兲⫽pe关T 共 ;u兲兴.共5.31兲
The a gumen s leading om Eq. 共5.5兲 o Eq. 共5.31兲a e o -
mally equi alen o assume ha
FIG. 3. Dimensionless esidual ene gy ␧ es as a unc ion o he
cooling a e c. As in Fig. 2, he ci cles a e om he nume ical
in eg a ion o Eq. 共5.16兲and he solid line is he p edic ion o he
asymp o ic analysis, gi en by Eq. 共5.22兲. The ag eemen is good o
cⱗ0.01.
GLASSLIKE DYNAMICAL BEHAVIOR IN . . . PHYSICAL REVIEW E 64 041505
041505-9