Glassy beha io in a simple model wi h en opy ba ie s
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
B. Sa
´nchez-Rey
Escuela Poli e
´cnica Supe io , Uni e sidad de Huel a, E-21819 La Ra
´bida, Huel a, Spain
~Recei ed 13 Sep embe 1996!
We s udy he dynamical beha io o a sys em wi h a a iable numbe o pa icles n. The emp y s a e
n50 is he g ound s a e, while all he o he s a es n.0 a e degene a e in ene gy. In equilib ium, he mean
numbe o pa icles is equal o uni y, independen ly o he empe a u e. The s a ic p ope ies a e he same as o
he Backgammon model ecen ly p oposed by Ri o @Phys. Re . Le . 75, 1190 ~1995!#, while a a ia ion o
he kine ics is conside ed. The elemen a y dynamical p ocesses a e he a i al and depa u e o a pa icle. The
a e o he depa u e p ocess is cons an , while he a i al a e is ob ained om he de ailed balance condi ion.
Thus, he e is no ene gy ba ie sepa a ing he g ound s a e n50. Ne e heless, glassy beha io appea s due o
he p esence o e ec i e en opy ba ie s. A low empe a u es, he esponse unc ions a e shown o obey
( ).exp@2( /
)
g
#. In he mal cycles o cooling and ehea ing om low empe a u es, he sys em shows
hys e esis, which ollows om he end o he sys em o app oach he no mal cu e cha ac e izing he hea ing
p og am. @S0163-1829~97!03710-7#
I. INTRODUCTION
The s udy o glassy beha io has been qui e an ac i e ield
in ecen yea s. A e iew o he main ea u es obse ed in
eal glasses, and se e al mic oscopic models showing simi-
la i y wi h hem, can be ound in Re s. 1 and 2. In elaxa ion
expe imen s, he linea esponse unc ions show nonexpo-
nen ial beha io . In pa icula , a Kohl ausch-Williams-Wa s
~KWW!decay is usually ound. In cooling expe imen s a
labo a o y glass ansi ion, in which he p ope ies de ining
he s a e o he sys em become ozen, is obse ed. The an-
si ion is associa ed o a as inc ease o he elaxa ion ime as
he empe a u e is lowe ed. Du ing ehea ing, hys e esis e -
ec s show up, wi h he sys em e u ning o equilib ium ol-
lowing a pa h which is di e en om he cooling one. A
mo e de ailed discussion o he ich phenomenology o
glasses is a ailable in Re s. 3 and 4.
The e is a g ea a ie y o models ying o explain glassy
beha io . The simples one is a wo-le el sys em ~TLS!,
whe e an ene gy ba ie mus be su passed in o de o go
om he exci ed o he g ound s a e.5,6 In some models, he
inc ease o he elaxa ion ime is associa ed o he in oduc-
ion o coope a i i y in he dynamics o he sys em,7bu
he e is also an ene gy ba ie sepa a ing he g ound s a e
om he exci ed ones. This ba ie plays an essen ial ole in
he di e gence o he elaxa ion ime a low empe a u es. On
he o he hand, en opy is known o play an impo an ole in
he desc ip ion o glassy beha io since he pionee ing wo k
by Adam and Gibbs.8
Recen ly, Ri o 9,10 has p oposed a model wi hou ene gy
ba ie s, in he sense ha he sys em can always each he
g ound s a e wi hou any ene gy-ac i a ed p ocess. The dy-
namical s udy o he model has ocused on he mal cycles o
cooling and ehea ing, and ze o- empe a u e p ope ies as ag-
ing. The sys em displays glassylike beha io , despi e he ab-
sence o ene gy ba ie s, and he di e gence o he elaxa ion
ime is due o he en opic con ibu ion o ee ene gy ba i-
e s. These appea because o he small numbe o di ec ions
in phase space along which he ene gy dec eases. Slow e-
laxa ion shows up because he sys em has o explo e a wide
phase space egion be o e eaching he g ound s a e.
Because o he ules go e ning i s dynamics, he model
has been e e ed o as he Backgammon ~BG!model. I can
be isualized in se e al di e en , al hough equi alen , ways.
He e we p esen one o hem, while ano he one is discussed
in he inal sec ion. Suppose we ha e a wo-dimensional la -
ice wi h a pa icle a each si e. Then, an ex e nal mechanism
is in oduced such ha pa icles end o agg ega e in he di-
ec ion pe pendicula o he la ice. Pa icles emaining on
he la ice ha e a la ge ene gy han hose which a e agg e-
ga e o hem, so ha he minimum ene gy is eached when
all pa icles o m a unique agg ega e a a gi en si e. All si es
and pa icles being equi alen , his s a e has a degene a ion
gi en by he numbe o si es ~o pa icles!. The dynamics o
he sys em is de ined by means o a Ma ko p ocess in which
each pa icle can mo e o any o he si e, wi h ansi ions
a es gi en by Me opolis dynamics. Since he spa ial a -
angemen o he si es in he plane plays no ole a all, he
model is o a mean- ield ype. Mean- ield app oxima ions a e
no accu a e o desc ibe elaxa ion h ough ene gy ba ie s in
eal s uc u al glasses, because o he nuclea ion p ocesses
aking place in hem. Ne e heless, as poin ed ou by Ri o ,9
he e ec o en opy ba ie s should no depend e y s ongly
on he ange o he in e ac ions and he in o ma ion ob ained
om his kind o model is expec ed o be ele an also in he
case o sho - anged in e ac ions.
In his wo k we in oduce a model ha keeps he main
cha ac e is ic o Ri o ’s model, namely he absence o en-
e gy ba ie s o ansi ions o he g ound s a e, and allows an
analy ical ea men o he dynamics. We conside a sys em
wi h a a iable numbe o pa icles n, in which he g ound
s a e has no pa icles, n50, while all he s a es wi h n.0
PHYSICAL REVIEW B 1 MARCH 1997-IIVOLUME 55, NUMBER 10
55
0163-1829/97/55~10!/6343~13!/$10.00 6343 © 1997 The Ame ican Physical Socie y
a e degene a e. The s a ics o his sys em is equi alen o he
BG model wi h indis inguishable pa icles.11 The dynamics
is o mula ed by means o a mas e equa ion wi h ansi ion
a es e i ying he de ailed balance condi ion. The equa ion
can be exac ly sol ed o cons an empe a u e p ocesses,
allowing he iden i ica ion o he mechanisms leading o non-
exponen ial elaxa ion and o he di e gence o he elaxa ion
ime. Fo cooling p ocesses we show he ele ance o he
elaxa ion modes o he mas e equa ion and o he ene gy
elaxa ion ime o cha ac e ize he labo a o y glass ansi ion
and he eezing empe a u e, espec i ely. Along hea ing,
he dynamical beha io o he model is unde s ood om he
end o he sys em owa ds a ‘‘no mal’’ cu e.12 In pa icu-
la , he hys e esis e ec , which is so cha ac e is ic o glasses,
is di ec ly ela ed o he app oach o his no mal cu e. The
exis ence o such a cu e is a qui e s ong p edic ion o mod-
els bases on a mas e equa ion o mula ion o he dynamics.
Whe he he e is a no mal cu e also o eal s uc u al
glasses emains an open ques ion.
The esul s ob ained will be compa ed o o he p e iously
conside ed models and, in pa icula , he one-dimensional
Ising model wi h Glaube dynamics.13 Le us men ion ha
he Ising model may be ele an in he con ex o s uc u al
glasses, since i has been p o ed o accu a ely desc ibe he
e olu ion o he con igu a ion o a one-dimensional sys em
o pa icles wi h anha monic and compe ing in e ac ions.14
Al hough ene gy ba ie s exis in he model s udied in Re .
13, glassy beha io appea s in bo h cases o simila easons.
P obably, his is also he case o any model showing glassy
beha io , as long as i s dynamics is desc ibed by a mas e
equa ion.
The plan o he pape is he ollowing. In Sec. II he
model is o mula ed, and he mas e equa ion desc ibing i s
dynamics is sol ed o he cons an empe a u e case. Relax-
a ion p ope ies a e conside ed in Sec. III, ocusing on he
s e ched exponen ial decay ound a low empe a u es. Sec-
ion IIIA is de o ed o he s udy o he equilib ium ime
au oco ela ion unc ion o he ene gy, while linea elax-
a ion a e a empe a u e pe u ba ion is he subjec o Sec.
IIIB. The mal cycles a e s udied in Sec. IV, and cooling
p ocesses a e conside ed in Sec. IVA, whe e he labo a o y
glass ansi ion is analyzed in de ail. Sec ion IVB deals wi h
hea ing p ocesses. The no mal cu e associa ed wi h a gi en
hea ing p og am is de ined, and i s ela ion o he obse ed
hys e esis e ec is discussed. Finally, he main conclusions
o he pape a e summa ized in Sec. V.
II. THE MODEL
The model we conside has a a iable numbe o pa icles
n. This numbe comple ely speci ies he s a e o he sys em.
The emp y s a e, n50, has ze o ene gy,
e
050, and all he
s a es wi h n.0 a e degene a e, wi h ene gy
e
n5
e
. The sys-
em is in con ac wi h a hea and pa icle ba h cha ac e ized
by a empe a u e Tand ugaci y
z
[exp(2
a
). The e o e, he
equilib ium p obabili y o inding he sys em in s a e nis
pn
~0!5Ce2
be
ne2
a
n,~2.1!
whe e
b
5(kBT)21,kBbeing Bol zmann’s cons an . The
cons an Cis de e mined om he no maliza ion condi ion,
and i is gi en by
C512e2
a
12e2
a
~12e2
be
!.~2.2!
Now, we assume ha he ba h is such ha he equilib ium
a e age numbe o pa icles is uni y, independen ly o he
empe a u e, i.e.,
^
n
&
05(
n50
`
npn
~0!51. ~2.3!
This p o ides a ela ionship be ween he ugaci y and he
empe a u e, namely
a
5ln~11e2
be
/2!.~2.4!
We no ice ha exp essions o his kind a e ypical when
passing om a canonical desc ip ion o a g and-canonical
one, and he la e is equi ed o co ec ly ep oduce he num-
be o pa icles in he sys em. Using his ela ion, Eq. ~2.2!
educes o C5exp(2
a
), and he equilib ium dis ibu ion can
be w i en
p0
~0!5e2
a
,~2.5a!
pn
~0!5e2
be
2
a
~n11!,n>1. ~2.5b!
The in oduc ion o a ba h e i ying Eq. ~2.4!has been
s imula ed by he wo k ca ied ou in Re s. 9–11, whe e wo
a ia ions o he BG model a e s udied. In hese models, N
pa icles can occupy Ndi e en ‘‘abacuses’’ 51,...,N.
While in one o he models9,10 he pa icles a e conside ed as
dis inguishable, in he o he one11 hey a e ea ed as indis-
inguishable. This is he only di e ence be ween bo h mod-
els. Excep o an addi i e cons an , he ene gy o a gi en
con igu a ion is p opo ional o he numbe o occupied aba-
cuses. The e is no limi a ion in he numbe o pa icles n
being in a pa icula abacus , excep he one ollowing om
he o al numbe o pa icles, ( n 5N. Since all he aba-
cuses a e equi alen , he a e age numbe o pa icles in each
o hem mus be uni y a equilib ium.
The model desc ibed abo e mimics he equilib ium p op-
e ies o he BG model wi h indis inguishable pa icles. A
b ie discussion o his is gi en in Appendix A. The idea is o
ocus on one o he abacuses, conside ing he emainde o
hem as a ba h in he limi N→`. The condi ion gi en by
Eq. ~2.4!gua an ees ha his limi is aken keeping he same
bo h he numbe o abacuses and he numbe o pa icles.
F om Eq. ~2.5!i is s aigh o wa d o ob ain he equilib-
ium p ope ies o he sys em, as unc ions o he empe a-
u e. The a e age ene gy is
^
E
&
05(
n50
`
e
npn
~0!5
e
~12p0
~0!!5
e
e2
be
/2
11e2
be
/2 ,~2.6!
and i s luc ua ions a e gi en by
s
E
25
^
E2
&
02
^
E
&
0
25
e
2e2
be
/2
~11e2
be
/2!2.~2.7!
Fluc ua ions in he numbe o pa icles a e
s
N
25
^
n2
&
02
^
n
&
0
252e
be
/2.~2.8!
6344 55A. PRADOS, J. J. BREY, AND B. SA
´NCHEZ-REY
This quan i y di e ges in he low- empe a u e limi , whe e
mos o he p obabili y co esponds o he g ound s a e. Due
o he condi ion o he mean numbe o pa icles being equal
o uni y, he p obabili y dis ibu ion has a long ail as a unc-
ion o n. The e o e, he e is an e ec i e co ela ion leng h
associa ed o he di e gence o he luc ua ions o he numbe
o pa icles. Finally, he equilib ium en opy eads
S
kB
52 (
n50
`
pn
~0!lnpn
~0!52ln~11e2
be
/2!1
be
e2
be
/2
11e2
be
/2 .
~2.9!
This exp ession coincides wi h he en opy pe abacus in he
BG model wi h indis inguishable pa icles. I is ee o he
pa hological beha io shown by he en opy in he case o
conside ing he pa icles as dis inguishable, whe e i be-
comes nega i e a low empe a u es.9,11
Nex , we p oceed o o mula e he kine ics o he model.
The elemen a y dynamical p ocesses we will conside a e he
a i al o he depa u e o one pa icle, and he e o e he
dynamical e olu ion o he sys em will be gi en by a one-
s ep p ocess15 mas e equa ion,
dpn
d 5 n11pn111gn21pn212~ n1gn!pn,~2.10!
whe e pn( ) is he p obabili y ha he sys em has npa icles
a ime , nis he ansi ion a e om s a e n o s a e n21
~loss o one pa icle!, and gnis he ansi ion a e om s a e
n o s a e n11~gain o a pa icle!. O cou se, he s a e
n50 is a e lec ing bounda y,
050. ~2.11!
As we do no wan o in oduce any ene gy ba ie ob-
s uc ing he elaxa ion o he sys em owa ds he g ound
s a e, we will ake
n5
n
,n.0, ~2.12!
whe e
n
is a cons an pa ame e wi h dimensions o e-
quency. The ansi ion a es gna e chosen in o de o e i y
he de ailed balance condi ion, i.e.,
g05
n
e2
be
2
a
,~2.13a!
gn5
n
e2
a
,n.0. ~2.13b!
Since he g ound s a e can be eached a any empe a u e
om any o he s a e wi hou su moun ing any ene gy ba ie ,
possible di e gence o he cha ac e is ic elaxa ion ime and
glassy beha io can only appea in he model due o he
p esence o en opy ba ie s. In ac , glassy beha io is o be
expec ed, because a low empe a u es
a
→0 acco ding o
Eq. ~2.4!, and he leading beha io o he ansi ion a es is
gi en by
gn. n5
n
,n.0, ~2.14a!
g0.
n
e2
be
.~2.14b!
One can a gue whe e he model, as o mula ed he e, inco -
po a es he en opy ba ie s which a e so e iden in he o igi-
nal BG model. A low empe a u es, he p obabili y o ind-
ing he sys em in an exci ed s a e a om n51, which is he
only one om which he ene gy can dec ease, is o he same
o de as p1up o n5O(
a
21). This e lec s he equi alence
o all he abacuses in he o iginal BG model. The elaxa ion
slows down because he andom walk pe o med by namong
he exci ed s a es con ibu ing o he ene gy is symme ic,
and i akes a e y la ge ime o he sys em o elax om
s a es n5O(
a
21).
Ve y ecen ly, some andom walk models ha e been p o-
posed o mimic he ze o- empe a u e dynamics o he BG
model.16,17 To pu ou wo k in a p ope con ex , i is impo -
an o no e ha , i s , we will s udy he e he ini e empe a-
u e kine ics o ou model and, secondly, ha we a e using a
g and-canonical ensemble desc ip ion. Fo his eason ou
andom walk is no symme ic, excep in he limi T→0. Ou
aim is no o p opose a model ha exac ly ep oduces he
dynamics o he BG model. Ins ead, we wan o e ain i s
main ea u es in a sol able model, in o de o iden i y he
ele an mechanisms leading om en opy ba ie s o glassy
beha io .
The solu ion o he mas e equa ion in he case o ime
independen empe a u e can be ob ained by using s anda d
p ocedu es.15 The cons an
n
in he ansi ion a es will be
used o se up he ime scale, and hus i will be aken equal
o uni y in he ollowing. We look o he eigen alues and
eigen ec o s o he p oblem. The o me a e gi en by
l~q!511e2
a
22e2
a
/2cosq,~2.15!
whe e q uns in he in e al @0,
p
#. Besides, he e is he
eigen alue l50, whose eigen ec o is he equilib ium dis-
ibu ion gi en by Eq. ~2.5!. All he eigen alues lo he han
l50 a e s ic ly posi i e, as i mus be he case o a mas e
equa ion wi h ansi ion a es e i ying de ailed balance. The
eigen ec o associa ed o l(q)is
j
0
~
q
!
5
S
2
p
D
1/2
e~
be
2
a
!/2cos
h
~q!,~2.16a!
j
n~q!5
S
2
p
D
1/2
e2[
be
1
a
~11n!]/2cos@nq1
h
~q!#,n>1.
~2.16b!
He e
h
(q) is a eal unc ion de ined by
e2i
h
~q!52e2iq2
be
2
a
/22e2
be
2
a
111e2
a
22e2
a
/2cosq
eiq2
be
2
a
/22e2
be
2
a
111e2
a
22e2
a
/2cosq,
~2.17!
and
h
~0!5
p
/2. ~2.18!
I has he p ope y
h
(2q)52
h
(q)1
p
. The eigen ec o s
j
n(q) e i y he closu e ela ion
pn
~0!1
E
0
p
dq
j
n~q!
j
m~q!
pm
~0!5
d
nm .~2.19!
By using he abo e equa ion, any ini ial condi ion can be
exp essed as a sum o e he eigen ec o s,
55 6345GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . .
Dn~ 50![pn~ 50!2pn
~0!5
E
0
p
dqg~q!
j
n~q!,
~2.20!
wi h
g~q!5(
m50
`Dm~ 50!
j
m~q!
pm
~0!.~2.21!
Now, i is i ial o w i e he ime e olu ion o he de ia ion
om equilib ium Dn( ),
Dn~ ![pn~ !2pn
~0!5
E
0
p
dqg~q!
j
n~q!e2 l~q!.
~2.22!
This p o ides he gene al solu ion o he mas e equa ion, o
he ime independen empe a u e case. Since l(q) is s ic ly
posi i e o all q,0<q<
p
,D
n( ) goes o ze o in he in ini e
ime limi , as expec ed.
As al eady discussed, a ze o empe a u e he model e-
duces o a symme ic andom walk wi h an abso bing bound-
a y a n50. The e o e, he p obabili y dis ibu ion ends o a
s a iona y s a e wi h pn5
d
n,0 . The decay o his s a e is e y
slow and aging e ec s occu , e en i he sys em was ini ially
in equilib ium a low empe a u es. Since i is easily seen
ha a T50 ou model becomes equi alen o ‘‘model B’’
s udied in de ail in Re . 17, we will no discuss he aging
e ec s he e.
III. RELAXATION PROPERTIES
In his sec ion we a e going o s udy he elaxa ion p op-
e ies o he model a a gi en cons an empe a u e. A en ion
will be ocused on ~a! he ime au oco ela ion unc ion o
ene gy in equilib ium and ~b! he linea elaxa ion o ene gy
a e a empe a u e pe u ba ion. I mus be s essed ha bo h
quan i ies does no coincide, because he ensemble desc ip-
ion o he model does no co espond o he canonical one.
A. Ene gy ime au oco ela ion unc ion
The ime au oco ela ion unc ion o he ene gy in equi-
lib ium is gi en by
^
E~0!E~ !
&
05(
n50
`
(
m50
`
e
n
e
mp1
u
1~n,
u
m,0!pm
~0!,~3.1!
whe e p1
u
1(n,
u
m,0) is he condi ional p obabili y o inding
he sys em in s a e na ime gi en i was ini ially in s a e
m. Le us in oduce he esponse unc ion
~ !5
^
E~0!E~ !
&
02
^
E
&
0
2
^
E2
&
02
^
E
&
0
2,~3.2!
ha e i ies
~0!51, lim
→`
~ !50. ~3.3!
The condi ional p obabili y p1
u
1(n,
u
m,0) is he solu ion
o he mas e equa ion ~2.10!wi h he ini ial condi ion
p1
u
1~n,0
u
m,0!5
d
nm .~3.4!
By making use o Eq. ~2.19!i is easy o see ha
p1
u
1~n,
u
m,0!5pn
~0!1
E
0
p
dq
j
m~q!
pm
~0!
j
n~q!e2 l~q!,
~3.5!
since pn
(0) co esponds o he null eigen alue, and
j
n(q) o
l(q). The e o e, i is
^
E~0!E~ !
&
05(
n,m50
`
e
n
e
mpn
~0!pm
~0!
1(
n,m50
`
e
n
e
m
E
0
p
dq
j
n~q!
j
m~q!e2 l~q!
5
^
E
&
0
21
E
0
p
dqa2~q!e2 l~q!,~3.6!
whe e we ha e in oduced he unc ion
a~q!5(
n50
`
e
n
j
n~q!.~3.7!
Subs i u ion o Eq. ~3.6!in o Eq. ~3.2!yields
~ !5
*
0
p
dqa2~q!e2 l~q!
*
0
p
dqa2~q!.~3.8!
I ollows ha
( ) decays mono onically om i s ini ial
alue,
(0)51, o ze o. This could ha e been o eseen,
since i is a gene al p ope y o equilib ium au oco ela ion
unc ions in models whose dynamics is desc ibed by means
o mas e equa ions wi h he ansi ion a es e i ying he
de ailed balance condi ion.
The p oblem has been educed o calcula e he unc ion
a(q), de ined by Eq. ~3.7!, ha can be w i en as
a~q!5(
n51
`
e
j
n~q!52
e
j
0~q!,~3.9!
because
(
n50
`
j
n~q!50, ~3.10!
due o he o hogonali y o he eigen ec o s
j
(q) wi h e-
spec o he equilib ium dis ibu ion. F om Eqs. ~3.9!and
~2.16a!we ob ain
a~q!}cos
h
~q!.~3.11!
The p opo ionali y cons an in he abo e ela ion is i el-
e an o he calcula ion o he esponse unc ion, gi en by
Eq. ~3.8!. The unc ion
h
(q) de ined in Eq. ~2.17!is a he
in ol ed, bu simple exp essions a e de i ed bo h in he lim-
i s o sho and long imes. Fo sho imes, !1, i is
~ !;e2 lM,~3.12!
whe e
lM[
S
215
*
0
p
dql~q!a2~q!
*
0
p
dqa2~q!5e
a
21. ~3.13!
6346 55A. PRADOS, J. J. BREY, AND B. SA
´NCHEZ-REY
Thus, he elaxa ion in he sho ime egime is exponen ial,
as i is he usual case in sys ems desc ibed by mas e
equa ions.18 In he limi o long imes, a Laplace’s analysis
o Eq. ~3.8!gi es
~ !;e
a
21
2
p
1/2e9
a
/4~11e2
a
/22e2
a
!2
e2 ~12e2
a
/2!2
~12e2
a
/2!4 3/2 .
~3.14!
Aside om slow algeb aic co ec ions, he elaxa ion is again
exponen ial, bu wi h a cha ac e is ic ime
L5~12e2
a
/2!2,~3.15!
which is di e en om he one o he sho ime egime. This
is also he mos common case in models desc ibed by mas e
equa ions. This ac , oge he wi h he mono onic decay o
he equilib ium au oco ela ion unc ion, leads o a nonexpo-
nen ial elaxa ion egime a in e media e imes.18 This e-
gime is expec ed o be mo e ele an as he ime scales sepa-
a ion becomes la ge . This is he case when e
a
→1. Then,
bo h cha ac e is ic imes di e ge, bu
L@
S.~3.16!
Taking in o accoun Eq. ~2.4! o
a
, i ollows ha
e
a
→1 is equi alen o
b
→`o T→0. In his limi , bo h
Eqs. ~3.12!and ~3.14!become much simple . Fo sho imes
i is
~ !;e2
a
,~3.17!
whe eas in he long ime egion
~ !;1
p
1/2
e2
a
2 /4
~
a
2 /4!3/2 .~3.18!
This la e equa ion shows ha elaxa ion akes place o e a
ime scale
s5
a
2
4,~3.19!
which is much longe han he de ined by he ini ial expo-
nen ial. Thus, sepa a ion o ime scales comes up, and non-
exponen ial elaxa ion is o be expec ed in an in e media e
ime window.
The pic u e we ha e ob ained is simila o he one ound
in he low- empe a u e elaxa ion o Glaube ’s Ising
model.19–21 The e o e, we make use o he same echniques
o de i e he beha io o he co ela ion unc ion in he in-
e media e ime egime in he low- empe a u e limi . To be-
gin wi h, we ob ain an exp ession which is alid in he ime
scale de ined by Eq. ~3.19!. We in oduce a new a iable u
h ough
q5
a
u/2. ~3.20!
Then, a simple analysis gi es
~ ![
¯
~s!54
p
E
0
`du u2
~11u2!2e2s~11u2!,~3.21!
whe e e ms o o de
a
ha e been neglec ed. Fo e y long
imes, s@1, Eq. ~3.18!is o cou se eco e ed. Howe e , in
he egion s!1 we do no ge he low- empe a u e e sion o
he sho ime beha io , as gi en by Eq. ~3.17!, since i co -
esponds o he much sho e ime scale de ined by
a
21.
O e he ime scale s, ha beha io collapses on o he poin
s50. Ac ually, s!1 co esponds o an in e media e ime
window whe e is la ge bu s5
a
2 /4 is small (
a
!1). I is
easy o see ha
ln
¯
~s!;24
p
1/2 s1/2,s!1. ~3.22!
The e o e, om Eqs. ~3.19!and ~3.21!we ge
ln
~ !;2
S
4
a
2
p
D
1/2
,~3.23!
which is a s e ched exponen ial o Kohl ausch-William-
Wa s ~KWW! unc ion,
ln
~ !52
S
D
g
,~3.24!
wi h
g
51/2, ~3.25a!
5
p
4
a
2;
p
4e
be
.~3.25b!
Thus, a low empe a u es he elaxa ion ime
obeys he
A henius law, wi h an ‘‘ac i a ion’’ ene gy
e
. One may ask
himsel which is he physical o igin o his beha io , since
he sys em does no ha e o su moun any ene gy ba ie o
each he g ound s a e. In ou model, as in he one p oposed
by Ri o ,9 he e is an en opy ba ie . A low empe a u es
a
→0 and a symme ic andom walk is pe o med by he
sys em among all he exci ed s a es. The cha ac e is ic elax-
a ion ime will be domina ed by he di usion p ocess om
he mean posi ion in he exci ed egion o he s a e n50. The
mean posi ion in he exci ed s a es n
¯
exc is gi en by
n
¯
exc5(n51
`npn
~0!
(n51
`pn
~0!5
^
n
&
0
12p0
~0!5~12e2
a
!21,~3.26!
whe e we ha e made use o Eq. ~2.5a!. This quan i y mus
no be con used wi h he a e age numbe o pa icles in ex-
ci ed s a es. In he limi o low empe a u es
n
¯
exc;
a
21@1. ~3.27!
Then, an es ima ion o he ime needed o di use un il
n50 will be
di 5O~n
¯
exc
2!5O~
a
22!.~3.28!
The abo e equa ion can be conside ed as a quali a i e expla-
na ion o he elaxa ion ime
dependence on he empe a-
u e shown by Eq. ~3.25b!, since i is easonable o expec
ha
5O(
di ), he mean ime aken by he sys em o ge o
he ‘‘bo leneck’’ in he con igu a ion space.
A simpli ied pic u e o he e olu ion o he equilib ium
ime au oco ela ion unc ion
( ) o he ene gy can be
gi en in e ms o he h ee ime egimes we ha e ound,
55 6347GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . .

~ !5
H
e2
a
!1,
e2~4
a
2 /
p
!1/2 1! !4
a
22,
p
21/2~
a
2 /4!23/2e2
a
2 /4 @4
a
22.
~3.29!
A simila beha io has been p e iously ob ained o elax-
a ion in di e en models.18,19,22 I mus be no iced ha he
scheme desc ibed by Eq. ~3.29!is consis en wi h bo h em-
pi ical and nume ical esul s o glassy sys ems, whe e non-
exponen ial elaxa ion and KWW beha io is usually ound
o e an in e media e ime window.23
I is possible o es ima e oughly he ange o alidi y o
he KWW unc ion. One can de e mine he ime in e sec ions
iand o he KWW unc ion wi h he sho and long ime
exponen ials, espec i ely. I is ound ha i54/
p
and
a
2 .6.28. In he ime in e al ( i, ) he KWW unc ion is
expec ed o hold, and he elaxa ion unc ion e i ies
exp(24
a
/
p
)>
( )>0.06. Al hough his is a e y c ude es i-
ma ion, we conclude ha mos o he ele an pa o he
elaxa ion o
( ) a low empe a u es is gi en by he
s e ched exponen ial in Eq. ~3.29!, because
a
!1.
In Fig. 1 we ha e plo ed
( ) o
be
510, which co e-
sponds o
a
56.731023. The solid line is he KWW unc-
ion gi en by Eq. ~3.23!. As discussed in he pa ag aph
abo e, i is alid o e an in e media e ime window co e-
sponding o he ele an pa o he elaxa ion. Fo e y long
imes, elaxa ion is exponen ial, and he KWW unc ion is
no a good app oxima ion. Fo e y sho imes, elaxa ion is
also exponen ial, bu he di e ence wi h he KWW unc ion
is negligible o e he scale o he igu e.
Finally, i mus be ema ked once mo e ha he KWW
decay ound a in e media e imes ollows om he exis ence
o wo exponen ial egimes alid a e y sho and e y long
imes wi h a clea sepa a ion o hei espec i e ime scales.
A de ailed discussion can be ound in Re . 18 o any sys em
whose dynamics is desc ibed by a mas e equa ion. The main
poin is whe he mos o he ele an pa o he elaxa ion
can be desc ibed by a KWW unc ion as a consequence o a
clea ime scale sepa a ion. This happens in ou model be-
cause he elaxa ion spec um becomes e y b oad a low
empe a u es. This is no a gene al p ope y o all mas e
equa ions, and KWW elaxa ion may no show up o a gi en
choice o he ansi ion a es, i he elaxa ion spec um as-
socia ed o hem emains na ow a low empe a u es.
B. Linea elaxa ion o he ene gy
The ene gy elaxa ion a e a empe a u e pe u ba ion is
cha ac e ized by he esponse unc ion
c
~ !5
^
E~ !
&
2
^
E
&
0
^
E~0!
&
2
^
E
&
0,~3.30!
whe e
^
E~ !
&
5(
n50
`
e
npn~ !5
e
@12p0~ !#.~3.31!
Using he de ini ion o Dnin Eq. ~2.22!, we ha e
c
~ !5D0~ !
D0~0!.~3.32!
Subs i u ion o he exac solu ion o he mas e equa ion o
cons an empe a u e ob ained in Sec. II, Eqs. ~2.21!and
~2.22!, leads o
c
~ !5
*
0
p
dqg~q!cos
h
~q!e2 l~q!
*
0
p
dqg~q!cos
h
~q!.~3.33!
We ha e o calcula e g(q), om he ini ial condi ions
Dn(0). We will conside ha he sys em was in equilib ium
a a empe a u e
b
1D
b
. Then, he empe a u e was ins an-
aneously changed o
b
a 50. In he linea esponse ap-
p oxima ion,
Dn~0!5pn
~0!~
b
1D
b
!2pn
~0!~
b
!5dpn
~0!
d
b
D
b
,~3.34!
and he unc ion g(q) in Eq. ~2.21! eads
FIG. 2. Ene gy elaxa ion in he low- empe a u e egion, o a
empe a u e alue co esponding o
e
/kBT510. The diamonds a e
he nume ical e alua ion o Eq. ~3.33!, while he solid line co e-
sponds o he KWW unc ion o Eq. ~3.43!. In his loga i hmic
scale, we ha e es ic ed ou sel es o posi i e alues o he e-
sponse unc ion.
FIG. 1. Plo o he equilib ium au oco ela ion unc ion o en-
e gy, o a empe a u e alue co esponding o
e
/kBT510. The dia-
monds a e he nume ical e alua ion o Eq. ~3.8!, and he solid line
is he s e ched exponen ial o Eq. ~3.29!.
6348 55
A. PRADOS, J. J. BREY, AND B. SA
´NCHEZ-REY
g~q!5D
b
(
n50
`
j
n~q!d
d
b
lnpn
~0!.~3.35!
The exp ession o he equilib ium dis ibu ion, Eq. ~2.5!,
is equi alen o
lnpn
~0!52
be
~12
d
n0!2
a
~n11!,;n>0, ~3.36!
and subs i u ion o Eqs. ~2.16!and ~3.36!in o Eq. ~3.35!,
oge he wi h he ela ion
d
a
d
b
52
e
2
e2
be
/2
11e2
be
/2 52
e
2e2
a
2
be
/2,~3.37!
leads, a e some algeb a, o
g~q!5
S
2
p
D
1/2
e
D
b
F
e~
be
2
a
!/2cos
h
~q!11
2e2
be
22
a
cos@q1
h
~q!#22e2
a
/2cos
h
~q!1e2
a
cos@
h
~q!2q#
~11e2
a
22e2
a
/2cosq!2
G
.~3.38!
The abo e exp ession o g(q) is a he in ol ed o a bi-
a y empe a u e. In he low- empe a u e limi ,
a
→0, in o-
ducing again he ime scale sde ined by Eq. ~3.19!and he
a iable uo Eq. ~3.20!, one ge s
c
~ ![
c
¯
~s!58
p
E
0
`duu2~u221!
~11u2!3e2s~11u2!.~3.39!
The elaxa ion o he ene gy akes place o e a ime scale o
o de
a
22, as i was he case o he equilib ium ene gy au-
oco ela ion. In he s ime scale, he ini ial exponen ial e-
laxa ion does no show up, because he sho ime beha io
o
c
( ) is gi en by
c
~ !.e22 sinh
a
,~3.40!
and a low empe a u es i s cha ac e is ic ime scale
(2
a
)21collapses on o he poin s50. Fo e y long imes,
s@1, a Laplace analysis o Eq. ~3.39!yields
c
¯
~s!;22
p
1/2
e2s
s3/2 .~3.41!
The p e ious equa ion ells us ha ene gy elaxa ion is no
mono onic. In ac , i is p o ed in Appendix B ha
E
0
`d
c
~ !50, ~3.42!
implying ha
c
( ) is nega i e in a ime egion. Howe e , in
he in e media e ime window s!1 a s e ched exponen ial
decay is again ob ained, hough he gene al a gumen de el-
oped in Re . 18 canno be di ec ly applied. Fo s!1, i is
easy o show om Eq. ~3.39! ha
ln
c
~ !;2
S
16
a
2
p
D
1/2
.~3.43!
The e o e, a simpli ied pic u e o he ene gy elaxa ion a
low empe a u es is ob ained, which is simila o he one
ound be o e o he ene gy au oco ela ion. In e ms o he
h ee ele an ime egimes ha ha e a isen in ou discus-
sion,
c
~ !5
H
e22
a
!1,
e2~16
a
2 /
p
!1/2 1! !4
a
22
22
p
21/2~
a
2 /4!23/2e2
a
2 /4 @4
a
22.
~3.44!
A e y long imes, he elaxa ion unc ion
c
( ) c osses he
axis and decays o ze o om nega i e alues. This is qui e
a small e ec , since a nume ical es ima ion o he minimum
o
c
( ) gi es
c
min.20.05.
The e o e, he KWW unc ion in Eq. ~3.44!also gi es a
ele an in o ma ion abou he ene gy elaxa ion a low em-
pe a u es. In pa icula , i s elaxa ion ime
E5
p
16
a
22;
p
16e
be
,~3.45!
can be used o cha ac e ize he elaxa ion o ene gy a e a
homogenous pe u ba ion in empe a u e. I mus be e-
ma ked ha
Ealso ollows an A henius law a low em-
pe a u es. A quali a i e explana ion o his beha io , in e ms
o he di usi e mo ion o he sys em, can be gi en along he
same way as in he p e ious sec ion. Ob iously, he s e ched
exponen ial app oxima ion is no able o explain he c ossing
o he axis ha akes place a e y long imes, bu i accu-
a ely i s mos o he ele an pa o ene gy elaxa ion,
namely up o
c
.0.1.
In Fig. 2 he ene gy elaxa ion unc ion ob ained nume i-
cally is compa ed wi h he KWW unc ion in Eq. ~3.44!. The
alue o he pa ame e is he same as in Fig. 1, i.e.,
be
510 (
a
56.731023). In he a iables used in Fig. 2,
exponen ial elaxa ion co esponds o a s aigh line o uni y
slope, while KWW elaxa ion is ep esen ed by a s aigh
line o slope equal o he pa ame e
g
in Eq. ~3.24!. The
loga i hm scale used ampli ies he disc epancies, especially
o sho imes, whe e he di e ence be ween he KWW
unc ion and he ini ial exponen ial is in ac negligible.
IV. THERMAL CYCLES
He e we a e in e es ed in s udying he beha io o he
model when i is con inuously cooled down om high o low
empe a u es, and a e wa ds ehea ed. This is usually called
a he mal cycle. Upon desc ibing i , he sys em may de ia e
om equilib ium while being cooled, leading o he kine ic
phenomenon known as he labo a o y glass ansi ion. In he
hea ing p ocess, equilib ium is app oached again a high
55 6349GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . .
empe a u es, bu he sys em ollows a di e en cu e om
he cooling one, and hys e esis shows up. The kine ic beha -
io jus discussed is shown by a wide class o ma e ials,2,4
and also by some simple models.9,11–13,21,24 Ne e heless,
analy ical esul s a e sca ce,13,21 al hough qui e a gene al ex-
plana ion o he hys e esis phenomenon has been gi en.12 I
can be unde s ood as he mono onic app oach o a ‘‘no mal’’
cu e, di e en om he equilib ium one, cha ac e izing
hea ing p ocesses. As he p oo in Re . 12 was made o he
canonical ensemble, a gene aliza ion o he case conside ed
he e is p esen ed in Sec. IVB.
The emainde o his sec ion is o ganized as ollows.
Fi s , we s udy cooling p ocesses, and he exis ence o he
labo a o y glass ansi ion. Secondly, hea ing p ocesses a e
conside ed, paying special a en ion o he appea ance o
hys e esis, and ela ing i o he end o he sys em o ap-
p oach he no mal cu e.
Le us poin ou ha we ha e no been able o sol e ex-
ac ly he mas e equa ion o he case o ime-dependen em-
pe a u e, ha implies ha he ansi ion a es a e also ime
dependen . The p ocedu e de eloped in Re . 13 is alid when
he eigen ec o s o he mas e equa ion do no depend on
empe a u e. This is no he case he e, since he eigen ec o s
o he mas e equa ion, gi en by Eq. ~2.16!, a e empe a u e
dependen h ough he unc ion
h
(q) in Eq. ~2.17!. The e-
o e, we ha e pe o med a Mon e Ca lo simula ion o he
mas e equa ion, using a gene aliza ion o he Bo z-Kalos-
Lebowi z algo i hm25 o mas e equa ions wi h ime-
dependen ansi ion a es.26 Ne e heless, some analy ical
es ima ions can be done, and hey will be compa ed wi h
nume ical esul s.
A. Cooling p ocesses and labo a o y glass ansi ion
Now we a e going o s udy he con inuous cooling o he
sys em o low empe a u es. In o de o analyze he de ia ion
om equilib ium alues o he p ope ies o he sys em, we
will ollow a easoning simila o ha used in Re . 13. We
s a om he elaxa ion spec um o he mas e equa ion, Eq.
~2.15!, and no ice ha he modes ldepend on empe a u e
h ough
a
, and he e o e hey a e ime dependen in a gi en
cooling p og am T( ). In his spec um, he elaxa ion a es
o he sys em a y wi h hei label q, om he minimum
alue, co esponding o q50,
l1511e2
a
22e2
a
/25~12e2
a
/2!2,~4.1!
o he maximum one, o q5
p
,
l2511e2
a
12e2
a
/25~11e2
a
/2!2.~4.2!
Gi en a cooling law, o each o he elaxa ion modes we
can associa e a cha ac e is ic ime scale
s~q!5
E
0d 8l~q;T8!,~4.3!
whe e 0is he ex apola ed ime o which he empe a u e
would anish acco ding o he cooling p og am, and
T8[T( 8). The ime s(q) is oughly p opo ional o he e -
ec i e numbe o ansi ions le o he mode l(q;T) be o e
eaching T50. Fo imes longe han he one (q) making
s(q)51, one can conside ha he mode will no expe imen
any mo e ansi ions. Thus, o empe a u es lowe han he
one co esponding o (q) he con ibu ion o he mode will
no e ol e in ime and can be conside ed as ‘‘ ozen.’’ In his
way, we can de e mine a eezing empe a u e T(q) o each
alue o q. Equi alen ly, one can in oduce he no ion o a
‘‘dema ca ion’’ mode qD(T), such ha modes wi h
q<qD(T) a e ozen, while modes wi h q.qD(T) a e s ill
elaxing a he gi en empe a u e.13,27
The labo a o y glass ansi ion begins a he empe a u e
T1[T( 1) gi en by he ela ion
E
1
0d 8l1~T8!51~4.4!
o
qD~T1!50, ~4.5!
i.e., only he slowes elaxa ion a e is ozen, and he de ia-
ion om equilib ium s a s o . On he o he hand, he sys-
em will be comple ely ozen a a empe a u e T2[T( 2) o
which he as es elaxa ion mode does no e ol e any mo e,
namely,
E
2
0d 8l2~T8!51~4.6!
o
qD~T2!5
p
.~4.7!
A global image o he eezing phenomenon can be ob ained
by means o he ime scale
s5
E
0d 81
~T8!,~4.8!
whe e
(T) is he ime cha ac e izing he elaxa ion o he
p ope y Pwe a e in e es ed in a e a empe a u e pe u ba-
ion. Fo ins ance, in ou model
would be he KWW elax-
a ion ime
Ein Eq. ~3.45!, i we wan o desc ibe he ene gy
e olu ion du ing he cooling p ocess. An es ima ion o he
‘‘global’’ eezing empe a u e T o he p ope y Pis ob-
ained by making s51, i.e.,
15
E
0d 1
~T!,~4.9!
and hen T 5T( ). Since he labo a o y glass ansi ion is
e y na ow in empe a u e, a leas when he sys em is
slowly cooled, an app oxima ion o he ozen alue o he
p ope y Punde conside a ion would be P0(T ), i.e., he
equilib ium alue a i s eezing empe a u e T .
I is impo an o no e ha he empe a u es T1,T2, and
T depend bo h on he cooling a e cand he cooling law
(T) de ining he cooling p og am,
dT
d 52 c ~T!.~4.10!
This is also he case in o he simple models whose dynamics
is desc ibed in e ms o mas e equa ions. Fo some choices
o he cooling law (T), he sys em emains in equilib ium a
all empe a u es.6,13 We a e no going o discuss his p oblem
6350 55A. PRADOS, J. J. BREY, AND B. SA
´NCHEZ-REY
he e, bu ocus ou a en ion on he beha io o he sys em
when i is being linea ly cooled,
dT
d 52 c,~4.11!
i.e., (T)51, which is he mos usual cooling p og am in
eal expe imen s4and also in heo e ical s udies o model
sys ems.9,11,24,28
The elaxa ion modes l1and l2, Eqs. ~4.1!and ~4.2!, and
he ime cha ac e izing he ene gy elaxa ion
Ea e w i en
as unc ions o
a
, de ined by Eq. ~2.4!. Then, i is use ul o
ans o m he ime in eg al in he de ini ion o he sscales
in o an in eg al o e
a
wi h he aid o
d
a
d 52
c~12e2
a
!@ln~e
a
21!#2,~4.12!
whe e Eq. ~4.11!has been aken in o accoun , and
c52kB c
e
~4.13!
is an adimensional cooling a e, gi ing he ime scale o e
which
a
e ol es.
As discussed abo e, he beginning o he labo a o y glass
ansi ion is es ima ed o ake place a a ime 1such ha
T( 1)5T1, being T1 he empe a u e in Eq. ~4.5!, i.e., he
one a which he slowes elaxa ion mode eezes. By using
Eqs. ~4.1!and ~4.12!, we can w i e
151
c
E
0
a
1d
a
~12e2
a
/2!2
~12e2
a
!@ln~e
a
21!#2,~4.14!
whe e
a
1[
a
( 1). In he limi o slow cooling,
c!1, and i
ollows ha
a
1!1. Fo his case, Eq. ~4.14!simpli ies o
151
4
c
E
0
a
1d
aa
~ln
a
!2.~4.15!
To sol e his ela ion o
a
1, we make he change o a iable
a
5
a
1x,
E
0
a
1d
aa
~ln
a
!25
a
1
2
~ln
a
1!2
E
0
1dx x
@11~lnx/ln
a
1!#2
;
a
1
2
2~ln
a
1!2.~4.16!
The las in eg al can be done by di iding he in e al (0,1)
in o he wo subin e als (0,
u
ln
a
1
u
21) and (
u
ln
a
1
u
21,1). In he
i s in e al, he in eg and is bounded by uni y, and he in-
eg al is negligible. In he second in e al, i is
u
lnx
u
!
u
ln
a
1
u
,
gi ing ise o he esul in Eq. ~4.16!. Subs i u ion in o Eq.
~4.15!yields
151
8
c
a
1
2
~ln
a
1!2.~4.17!
By making use o he slow cooling condi ion,
a
1!1, we
ha e
2ln
a
1;ln~8
c!.~4.18!
Now, we ake in o accoun ha
a
1;exp(2
b
1
e
/2), o ge
T1;
e
kB
1
u
ln~8
c!
u
.~4.19!
In o de o calcula e he ic i e empe a u e T , we s a
om Eq. ~4.9!, wi h he elaxa ion ime o ene gy
Egi en
by Eq. ~3.45!,
15
E
0
a
d
a
d
d
a
E
21~
a
!516
p
c
E
0
a
d
aa
2
~e
a
21!@ln~e
a
21!#2.
~4.20!
As be o e,
a
is he alue o
a
co esponding o T . Fo
slow cooling, i is
a
!1, since
a
,
a
1. Then, he abo e
equa ion educes o
1516
p
c
E
0
a
d
aa
~ln
a
!2.~4.21!
In his way, we ha e a i ed a an exp ession simila o Eq.
~4.15! o
a
1. The e o e,
E
0
a
d
aa
~ln
a
!2;
a
2
2~ln
a
!2~4.22!
and
158
p
c
a
2
~ln
a
!2.~4.23!
Again, a easoning along he line o he one abo e Eq. ~4.19!
gi es us
T ;
e
kB
1
u
ln~
p
c/8!
u
.~4.24!
A simila dependence on he cooling a e is ob ained in eal
expe imen s,4and has also been ound in Glaube ’s Ising
model.13 Taking in o accoun he commen below Eq. ~4.9!
one can es ima e he esidual alue o he ene gy, i.e.,
e [lim
T→0
@
^
E
&
~T!2
^
E
&
0~T!#5
^
E
&
0~T !.~4.25!
In he limi o slow cooling,
c!1, Eq. ~4.25!leads o a
po en ial dependence on he cooling a e o he esidual en-
e gy, e }
c
1/2 . A simila beha io o he esidual p ope ies
has been ob ained in some models o glasses.5,13,6
In Fig. 3 he e olu ion o he mean ene gy o he cooling
p og am in Eq. ~4.11!, wi h an adimensional cooling a e
c50.02, is plo ed. The depa u e om equilib ium oughly
begins a he empe a u e ob ained om Eq. ~4.19!,
kBT1/
e
.0.55. The es ima ion o he eezing empe a u e,
ob ained by using Eq. ~4.24!is kBT /
e
.0.21, in good ag ee-
men wi h he nume ical esul . The ozen alue o he en-
e gy gi en by he Mon e Ca lo simula ion is
^
E
&
/
e
50.076,
while he alue ob ained om T is
^
E
&
0(T )/
e
50.081.
Again, he app oxima ed heo y p o ides a easonable es i-
ma ion o he ac ual alue.
55 6351GLASSY BEHAVIOR IN A SIMPLE MODEL WITH . . .