PHYSICAL REVIE%' 8VOLUME 43, NUMBER 11JANUARY 1991
Con inuous cooling o aone-dimensional bonded luid: AMon e Ca lo simula ion s udy
J.J.B ey and M. J.Ruiz-Mon e o
Depa amen o de I'Isica Teo ica, Uniue sidad de Seuilla, Apa ado de Co eos 1065, Sec o Su , E-41080Seuilla, Spain
(Recei ed 23 July 1990)
Akine ic one-dimensional bonded Quid model is o mula ed by ex ending amodel o wa e due
o Bell. The beha io o he sys em unde con inuous cooling is analyzed by use o aMon e Ca lo
simula ion me hod. I shows aphenomenon simila o he labo a o y glass ansi ion. The depen-
dences o he glass ansi ion empe a u es and o he esidual ene gy on he cooling a e a e dis-
cussed.
I. INTRODUCTION
In spi e o he in ense scien i ic e o ca ied ou in e-
cen yea s, he na u e o he liquid-glass ansi ion is a
om being we11 unde s ood. Al hough i is clea ha he
glass ansi ion as obse ed in eal expe imen s is akine -
ic phenomenon associa ed wi h he depa u e o he
liquid om he equilib ium, li le is known abou he
physical p ocesses con olling he beha io o he sys em
h ough he ansi ion. 'In some heo ies and models he
labo a o y glass ansi ion is he mani es a ion o an un-
de lying he modynamic o kine ic phase ansi ion,
whe e one o mo e elaxa ion imes di e ge. Ne e he-
less, i has also been shown ha an unde lying phase
ansi ion is no a necessa y condi ion o amodel o
p esen glass ansi ion phenomena simila o hose ob-
se ed in eal glasses.
A ypical way o p epa ing aglass is by he as cooling
o aliquid. In ac , i seems ha any liquid can be led o
aglassy s a e i he cooling a e is la ge enough. Besides,
he p ope ies o he esul ing glass depend on he cooling
p ocedu e. The e o e, he s udy o he ime e olu ion o
sys ems whose empe a u e is ex e nally changed shows
up as ac i ical s ep o he unde s anding o he glass
ansi ion. This poses us wi h a o midable ask om a
heo e ical poin o iew. I is no e en clea how o o -
mula e, in an app op ia e way, akine ic equa ion inco -
po a ing he mechanism con olling he empe a u e. O
cou se, his can be done, in p inciple, by conside ing
ime-dependen bounda y condi ions in he kine ic equa-
ion o an isola ed sys em. Ne e heless, his in oduces
inhomogenei ies in o he sys em and ende s he p oblem
ex emely di%cul .
Asimple , al hough unjus i ied, possibili y is o assume
ha he equa ion desc ibing he e olu ion o asys em a
cons an empe a u e can be di ec ly ex ended by o mal-
ly eplacing he cons an alue o he empe a u e by he
gi en ime unc ion. Mos o he models in en ed o ana-
lyze he dynamics o glasses a e o mula ed in his way.
Ne e heless, e en wi h his choice, he explici analy i-
cal solu ion o he case o con inuous cooling is known
only o a ew e y simple models in some asymp o ic
limi s. '
Recen ly, se e al models ha e been p oposed ha
mimic he dynamics o eal iuids nea he glass ansi-
ion. Al hough no ully sa is ac o y heo y has been
de eloped o explain hei kine ic p ope ies ye , hei
simplici y allows e y e Ficien compu e simula ions. As
F edickson has poin ed ou , agene al cha ac e is ic o
hese models is ha hey con ain some dynamical con-
s ain s ha a e spa ially sho anged, bu lead o ahigh
coope a i i y in he dynamics a la ge dis ances.
The pu pose o his pape is o in es iga e, using
Mon e Ca lo simula ion me hods, he ime e olu ion un-
de con inuous cooling o amodel o one-dimensional
bonded iuid. This model was in oduced by Bell as a
model o liquid wa e . I is ala ice model o which
only he s a ics was speci ied. The ee Gibbs ene gy can
be easily compu ed and, om i , analy ical exp essions
o he he modynamic p ope ies a e de i ed. He e we
in oduce adynamics o he bonded iuid by amas e
equa ion wi h ansi ion p obabili ies obeying de ailed
balance.
I mus be no iced ha Bell's model has been p e ious-
ly used o s udy he p ope ies o glasses by Singh and
Ko ac. Ne e heless, hei app oach is qui e di e en
om he one we p esen he e. They assume ha , in he
quenched iuid, he e is some kind o diso de associa ed
wi h agi en dis ibu ion o bond ene gies. No a emp o
desc ibe he dynamics o he glass ansi ion is made and,
he e o e, he kine ics o he model was no de ined.
Sec ion II de ines ou model. Explici exp essions o
some o he he modynamical p ope ies a e gi en and
he kine ics o he sys em is es ablished. The Mon e Ca -
lo simula ion me hod employed is ou lined in Sec. III.
The esul s ob ained o he e olu ion o he sys em a e
p esen ed in Sec. IV. The e, i is seen ha he sys em ex-
hibi s abeha io simila o alabo a o y glass ansi ion.
The dependence o he esidual ene gy and he glass an-
si ion empe a u e on he cooling a e is analyzed. Sec-
ion Vcon ains some commen s and conclusions.
II. DESCRIPTION OF THK MODEL
We conside aone-dimensional la ice o Nequally
spaced si es. Each si e can be ei he occupied by apa i-
cle o i can be emp y. The numbe o pa icles is M.
Two nea es -neighbo pa icles ha e an in e ac ion ene -
43 1991 The Ame ican Physical Socie y
586 J.J.BREY AND M. J.RUIZ-MONTERO 43
E(N, M, Z,B)=—
(M —
Z)E—
Bc@ .(2.1)
The numbe o possible con igu a ions wi h gi en
alues o Zand 8is
g(N, M, Z,B)
(M —
1)!N(N —
M—
B—
1)!
Z! (Z —
1)!(M —
Z—
B)!(N —
M—
Z—
B)!
gy —
c. Also, wo pa icles sepa a ed by asingle emp y
la ice si e may es ablish be ween hem abond o ene gy
—
(8+co). Bo h Eand co a e aken as posi i e. These a e
he only in e ac ions in he sys em.
Following Bell, we will e e o asi e be ween wo
bonded pa icles as occupied by abond. The emp y si es
a e conside ed as holes.
Le us hink o agi en con igu a ion o he sys em.
We will assume pe iodic bounda y condi ions. The ene -
gy o he sys em can be exp essed in e ms o he numbe
o bonds 8and he numbe o pa icle-hole con ac s,
deno ed by 2Z, as
Re s. 3and 4, Eq. (2.6) does. In ac , he la e can be
di ec ly de i ed by using he ans e ma ix me hod.
By he di e en ia ion o he Gibbs ee ene gy, all he
equilib ium he modynamical p ope ies o he sys em
can be de i ed. In pa icula , he equa ion o s a e eads
aG
N=— ADJ '+J(J—
1)
ADJ '+ A+(J—
1)
(2.7)
and he a e age ene gy is
BG —AE+ADJ '(E+co)
ADJ-'+ A+(J—
1)-'
(2.8)
O he p ope ies can be de i ed by using he iso he m-
isoba ic dis ibu ion. The a e age numbe o bonds is
gi en by
(2.2) 8ink
BD ADJ
ADJ '+ A+(J—
1) (2.9)
Since his exp ession di e s om hose epo ed by Bell
and Singh and Ko ac, ade i a ion o i is p esen ed in
he Appendix. Equa ion (2.2) is use ul in es ima ing
ini e-size e ec s in acompu e simula ion o he model.
In ou case i has been checked ha hese e ec s a e
negligible o he size o he sys em we ha e conside ed.
Now he pa i ion unc ion is cons uc ed om Eqs.
(2.1) and (2.2). I is con enien o wo k in he iso he m-
isoba ic ensemble whe e he pa i ion unc ion o ou
model is gi en by
M—
Zco
6(M,p, T)=gggg(N, M, Z,B)
Z=O B=0 N=M+B+Z
Xexp[ 13(E+pNl) j, —
(2.3)
whe e Tis he empe a u e, P=(k~T) ', pis he p es-
su e, and Iis he dis ance be ween neighbo ing si es.
Subs i u ion o Eqs. (2.1) and (2.2) in o Eq. (2.3) yields,
a e ca ying ou he summa ions,
6(M,p, T)=(J—
1) 'J +'[(J—
1)
+DAJ '+ Aj
Since we a e dealing wi h aone-dimensional model
wi h sho - ange in e ac ions, he sys em does no p esen
any equilib ium phase ansi ion. In he ollowing we
will conside asys em wi h acons an leng h, i.e.,num-
be o si es X. The equilib ium p essu e a each empe a-
u e is de e mined by Eq. (2.7).
Once he he modynamics o ou sys em is known, we
speci y i s dynamics. I is assumed o be go e ned by a
mas e equa ion. Taking in o accoun ha he numbe o
si es and pa icles a e kep ixed, he allowed ansi ions
consis ei he in he c ea ion (o des uc ion) o abond,
o in he mo ion o apa icle o anex emp y si e. O
cou se, abond can only be c ea ed s a ing om a
con igu a ion in which wo pa icles a e sepa a ed by a
single hole. Also, apa icle canno jump o asi e ha is
being "occupied" by abond. Be o e ha , he bond mus
be des oyed. This implies ha a low empe a u es„
when he concen a ion o bonds is la ge, he elaxa ion
o he model equi es he sys em o jump o e high ene -
gy ba ie s.
I xand x' deno e wo con igu a ions o he sys em
connec ed by an allowed single ansi ion, he ansi ion
a e used is
X[J '+2AD(J —
1)J AE
I—
anh 2kB T(2.10)
+AJ '(J —
1)+(J—
1) 'j,
wi h A=e~', D=e~, and J=e~~.
The Gibbs ee ene gy 6is
6=—
kBT ink
and, in he limi o Mgoing o in ini y, one ge s
6=MkBT ln J
ADJ '+ A+(J—
1)
(2.4)
(2.5)
(2.6)
Al hough Eq. (2.4) does no ag ee wi h hose de i ed in
whe e QE =Q(x') E(x) and zo is a
—
cons an ha ixes
he na u al ime uni o ou sys em. The dynamics
de ined in his way sa is ies he de ailed balance condi-
ion. Besides, i is clea ha all he con igu a ions can be
eached om agi en one by as ing o ansi ions wi h
nonze o p obabili y, i.e., he Ma ko p ocess de ined by
ou equa ion is i educible. I ollows ha any ini ial dis-
ibu ion o con igu a ions will end owa ds he equilib i-
um canonical dis ibu ion.
I mus be no iced ha we ha e no in oduced in he
dynamics any cons ain o he han he pu ely ene ge ic
43 CONTINUOUS COOLING OF AONE-DIMENSIONAL BONDED. ..
ones associa ed wi h he des uc ion o bonds. In his
sense ou model di e s om he acili a ed Ising mod-
els 'and also om he iling models. '
Fo con enience we ha e in oduced an adimensional
cooling a e as
III. MONTE CARLO SIMULATION
~0
5= (E+co) (4.2)
The simula ion o he sys em has been ca ied ou by
means o aMon e Ca lo algo i hm simila o ha o mu-
la ed by Bo z, Kalos, and Lebowi z. 'Fo compu a ion-
al easons, his me hod was p e e ed o he s anda d
Me opolis Mon e Ca lo me hod, whe e he p obabili y
o ejec ion o amo emen inc eases e y as as he em-
pe a u e is lowe ed. Gi en he con igu a ion o he sys-
em a a ime , he ansi ion a es o all he possible
ansi ions a e compu ed acco ding o Eq. (2.10). The o-
al ansi ion a e om he gi en s a e
O(x) =gW(x ~x') (3.1)
—
lna
Q(x) (3.2)
whe e ais ano he andom numbe be ween 0and 1. As
indica ed be o e, he ime was measu ed in Mon e Ca lo
s eps, aking so= 1in Eq (2.10.
). Fo he in e ac ion en-
e gies we chose c.=200 and co =500.
The da a p esen ed in his pape ha e been ob ained
wi h ala ice o 200 si es and 100 pa icles, and pe iodic
bounda y condi ions ha e been used. Fo he highes
cooling a es conside ed, we ha e checked ha he esul s
a e no signi ican ly al e ed when he numbe o pa icles
is inc eased o 1000 (2000 si es). Besides, by using he
ans e ma ix me hod we ha e calcula ed he no mal-
ized equilib ium densi y co ela ion unc ion. O cou se,
i is an oscilla ing unc ion whose ampli ude mono o-
nously decays wi h dis ance. Fo kz T=400 i is smalle
han 10 o dis ances g ea e han 10 la ice spacings.
Fo kz T=200, ha is he smalles empe a u e a which
he sys em emains nea equilib ium in ou simula ion,
he co ela ion has al eady decayed o 10 o dis ances
o 30 la ice spacings. Fo each cooling a e, 100 in-
dependen cooling uns we e pe o med and he esul s
we e a e aged. In spi e o he a he small size o ou
sys em, he uns a he lowes cooling a es equi ed mo e
han 35 hin aCONVEX-220 compu e .
IV. CONTINUOUS COOLING
We ha e cooled he sys em ollowing alinea law o
he a ia ion o he empe a u e, i.e.,wi h he cons an
cooling a e
d(k~ T)
d (4.1)
is also calcula ed. A andom numbe uni o mly dis i-
bu ed be ween 0and 1is hen gene a ed o de e mine
which o he possible ansi ions akes place. This is done
by di iding he in e al (0,1) in o anumbe o in e -
als equal o he numbe o possible ansi ions. To
each o hem i co esponds apo ion o leng h
W'(x~x )/Q(x). The ime in e al associa ed wi h his
change in he con igu a ion o he sys em is
C3
C3
C3 —0.1
(D
~= a.o
00
e,.mo0
"i 00
0
0
0
0
200 300 400 500
T
FIG. 1. A e age ene gy as a unc ion o he empe a u e in
con inuous cooling expe imen s o =1, 0.1, and 0.01. Also,
he equilib ium alues ha e been plo ed (ci cles).
We s a ed wi h he equilib ium con igu a ion a P= —
ca
ha , o he sys em conside ed he e, is an open s uc u e
wi h ahole be ween e e y wo pa icles. As is he case
o he acili a ed Ising model, he ac ha he empe a-
u e is nega i e does no p esen any physical p oblem
since he model does no show pa hological beha io in
passing om nega i e o posi i e empe a u es. Besides,
he s a e a 13= —
~is only used in ou simula ion as a
con enien ini ial s a e, om which we ca y ou an in-
s an aneous quench o alow empe a u e. This empe a-
u e was chosen in such away ha signi ican depa u es
om he equilib ium beha io when cooling he sys em
a he conside ed a e show up well below i . I s alue
anged be ween k~T=500 o =1 and k~T=300 o
=0.005.
The aim o he quench was o sa e compu e ime.
Then, he sys em was equilib a ed a acons an empe a-
u e un il he ela i e di e ence be ween he a e aged en-
e gy o e 100 uns and he equilib ium alue gi en by Eq.
(2.8) was o he o de o 1%. This equi ed uns o ypi-
cally 10 Mon e Ca lo ime s eps. A e ha , he sys em
was con inuously cooled. Ten di e en alues o he con-
s an a e in he ange 2.5X10 ~ ~1ha e been in es-
iga ed. Tha in e al co esponds o 3.6X10 ~6
&1.4X10 '.
Figu e 1shows he ime e olu ion o he a e age ene -
gy as a unc ion o he empe a u e in some o he con-
inuous cooling expe imen s. Fo he sake o cla i y only
he esul s co esponding o h ee alues o he cooling
a e a e ep esen ed, namely =1,0.1, and 0.01. Ne e -
heless, he ollowing commen s apply o all he cases
s udied. I is seen ha he sys em has abeha io simila
o he labo a o y glass ansi ion obse ed in eal glasses.
Fo agi en alue o he cooling a e, he e olu ion o he
a e age ene gy ollows he equilib ium cu e a high em-
pe a u es, bu he sys em alls ou o equilib ium when
J.J.BREY AND M. J.RUIZ-MONTERO 43
e„,(5)=5 (4.3)
wi h a=0.199. Fo 0.05 & &1 he i gi es o,"=0.139.
To see whe he aloga i hm dependence on he cooling
a e is mo e accu a e, in Fig. 3we ha e plo ed he loga-
i hm o he esidual ene gy agains ln( —
ln5). Again,
wo linea egions show up qui e clea ly. Each o hem
can be i ed o abeha io o he o m
e„,(5)=(—
ln5) (4.4)
wi h y=2.184 o &0.05 and y' =1.087 o 0.05 & &1.
Al hough he i ing in Fig. 3is sligh ly be e han in
Fig. 2, ou esul s a e no a all conclusi e abou which o
he laws gi en by Eqs. (4.1) and (4.2) desc ibes bes he
0
0
FIG. 2. Double loga i hm ep esen a ion o he esidual en-
e gy as a unc ion o he cooling a e.
su Bcien ly low empe a u es a e eached, and he ene gy
depa s om i s equilib ium alue.
I we de ine he glass ansi ion empe a u e Tas ha
co esponding o he poin whe e he sys em alls ou o
equilib ium, hen Tdec eases as he cooling a e de-
c eases. Amo e p ecise de ini ion o Twill be gi en
la e on. Subsequen cooling leads he ene gy o a alue
p ac ically cons an ; ha is, no a ec ed by a u he de-
c ease o he empe a u e. The di e ence be ween he
alue o he ene gy and he ene gy o he g ound s a e is
he so-called esidual ene gy. In ou model he ene gy
pe pa icle in he g ound s a e is —
(E+co). Figu e 1also
shows ha he esidual ene gy becomes smalle as he
cooling a e is dec eased.
In an a emp o iden i y he dependence o he esidu-
al ene gy on he cooling a e, wo di e en ep esen a-
ions ha e been used. They co espond o he wo laws
usually conside ed in he bibliog aphy. In Fig. 2adouble
loga i hm ep esen a ion is shown. Al hough one could
y o i all he da a o asingle s aigh line, i seems ha
wo di e en egions can be iden i ied, wi h ac osso e
a ound =0.05 (5=7.1X10 ). Asimila beha io o
he esidual ene gy was ound by Kob and Schilling" o
adi e en one-dimensional con igu a ional glass model.
Fo &0.05, he esul s i qui e well o he powe law
2.22.4in(-i&b)
FIG. 3. Loga i hm o he esidual ene gy s ln (—
ln6).
p(T)=p, (T~)+m„(T—
TI), (4.5)
whe e mis he a ia ion o pwi h he empe a u e asso-
cia ed wi h he glass elaxa ion. The ic i e empe a u e
depends, in gene al, on he p ope y unde conside a ion.
In acon inuous cooling expe imen T& eaches as a-
iona y alue ha can be iden i ied as he labo a o y glass
ansi ion T.I s alue is easily ob ained by ex ending
he glass beha io un il eaching he equilib ium cu e.
The empe a u e o he in e sec ion poin is he glass
ansi ion empe a u e. In ou case, he sys em becomes
ozen a low empe a u es. The e o e, Tis gi en by he
equilib ium empe a u e o he sys em when i has a
alue o he mac oscopic p ope y equal o i s esidual
alue in he glass.
F om he abo e discussion i is clea ha di e en
p ope ies can lead o di e en alues o T o agi en
alue o he cooling a e. We ha e e alua ed wo glass
ansi ion empe a u es o each o ou cooling expe i-
men s by using he esidual alues o he ene gy and he
numbe o bonds, espec i ely. The esul s a e plo ed in
Fig. 4. Is is seen ha he e is aclea co ela ion be ween
he alues o bo h ansi ion empe a u es. Besides, hei
di e ence seems o go o ze o as he cooling a e de-
c eases. I mus be no iced ha , acco ding o Eq. (2.1),
he ene gy is no de e mined by he numbe o bonds and
ice e sa.
Figu e 4shows ha , om ap ac ical poin o iew, Tg
is qui e accu a ely gi en by alinea unc ion o he loga-
i hm o he cooling a e. Asimila beha io has been
ound om phenomenological heo ies and also obse ed
in some eal glasses.
dependence o he esidual ene gy on he cooling a e.
Aconcep ha has p o ed o be e y use ul in phe-
nomenological heo ies o glassy elaxa ion is ha o
ic i e empe a u e. The ic i e empe a u e T& o a
glass is he empe a u e a which he equilib ium liquid
has app oxima ely he same s uc u e as he glass. Usu-
ally, i is de ined wi h e e ence o agi en p ope y po
he sys em by
43 CONTINUOUS COOLING OF AONE-DIMENSIONAL BONDED. ..589
C3
O0
0
ha e ied o iden i y in ou model he ene gy o apossi-
ble con igu a ional de ec and he co esponding ba ie
heigh leading o he alues o aand a' epo ed below
Eq. (4.3). Ne e heless, we ha e no ound ag eemen ,
e en app oxima e, o any o he possible simples de-
ec s. The ela ion be ween he low- empe a u e elaxa-
ion p ope ies o ou model and an ensemble o wo-le el
sys ems is an impo an poin ha dese es u he s udy.
ACKNOWLEDGMENTS
—
12
We acknowledge pa ial suppo om he Di eccion
Gene al de In es igacion Cien i ica yTecnica (Spain)
h ough G an No. PB86-0205.
FIG. 4. Glass ansi ion empe a u e ob ained om he ene -
gy (ci cles) and om he numbe o bonds (as e isks).
V. DISCUSSION
The one-dimensional bonded luid model discussed in
his pape exhibi s aphenomenon analogous o he labo-
a o y glass ansi ion obse ed in eal glasses. The mod-
el is e godic and i s he modynamic p ope ies a e
known. I does no p esen any kind o he modynamic
o kine ic ideal glass ansi ion. In his sense, i is simila
o he acili a ed Ising models and di e s om he iling
models. '
He e we ha e es ic ed ou sel es o con inuous cool-
ing compu e expe imen s. The linea and nonlinea e-
laxa ion p ope ies a cons an empe a u e will be dis-
cussed elsewhe e, bo h analy ically and by means o a
Mon e Ca lo simula ion. Ne e heless, i can be an ici-
pa ed ha , a low empe a u es, he elaxa ion o he
model is domina ed by ade ec di usion pic u e. A such
empe a u es, he concen a ion o bonds is e y high,
and he concen a ion o holes and pa icle-pa icle con-
ac s is e y low. In o de o dec ease he ene gy, he
holes mus mo e un il hey ind apa icle-pa icle con-
ac , and hen c ea e anew bond. The p ocess equi es
he des uc ion o all he bonds sepa a ing he ini ial posi-
ions o he hole and he pa icle-pa icle con ac . O
cou se, he bonds can be c ea ed again a e he hole (o
he pa icle) has jumped. We a i e hen o apic u e o
de ec s di using in ape iodic po en ial. The same kind
o pic u e was ound o he one-spin acili a ed Ising
model.
The esidual ene gy o he model can be i ed qui e
well o bo h powe laws and loga i hm laws in ce ain
anges o he cooling a es. No clea conclusion has been
es ablished abou which o he wo laws leads o abe e
i o he da a. Bo h o hem ha e been jus i ied heo e i-
cally by conside ing an ensemble o wo-le el sys-
ems. "' The loga i hm law co esponds o asmoo h
dis ibu ion o low-exci a ion ene gies, while he powe
law is ob ained when he exci a ion ene gy is he same o
all he wo-le el sys ems. In his la e case he exponen
is gi en by he a io be ween he exci a ion ene gy and
he po en ial ba ie .
In an a emp o elucida e be ween he wo laws, we
APPENDIX
In his appendix we p esen ade i a ion o Eq. (2.2).
We s a om Mpa icles wi hou holes o bonds be-
ween hem and le us inse Zholes and 8bonds wi h
he condi ion ha e e y wo o hem mus be sepa a ed
by a leas one pa icle. I is con enien o analyze he six
possible ollowing cases sepa a ely.
(a) Con igu a ions ha ing one pa icle o he le and
ano he pa icle o he igh . I s numbe is easily seen o
be
(M —
1)!
Z!8!(M —
Z8—
1)!— (Al)
(b) Con igu a ions ha ing one hole o he le and, o
cou se, ending by one pa icle. The e a e
(M —
1)!
(Z —
1)!8!(M —
Z—
8)! (A2)
o hem.
(c) Con igu a ions ha ing one hole o he igh and, o
cou se, beginning by one pa icle. This numbe is also
gi en by Eq. (A2).
(d) Con igu a ions ha ing one bond o he le and, o
cou se, one pa icle o he igh :
(M —
1)!
Z!(8 —
1)!(M —
Z—
8)! (A3)
(N —
M8—
1)!—
(Z —
1)!(N —
M8—
Z)!— (A4)
(b) Now holes can also be in oduced o he igh o he
la ice wi hou changing he alue o Z. The numbe o
di e en ways o in oducing he new holes is
(e) Con igu a ions ha ing one bond o he igh and one
pa icle o he le a e gi en by Eq. (A3).
The nex s ep is o in oduce in each o he abo e
con igu a ions N—
(M+8+Z) holes in all he possible
ways, bu wi hou c ea ing any new pa icle-hole con-
ac s. This implies ha he new holes can only be inse -
ed nex o one o he Zholes al eady p esen . In he p o-
cess, he pe iodic bounda y condi ions mus be aken in o
accoun . Le us conside each o he abo e g oup o
con igu a ions.
(a) I is seen ha he numbe o possibili ies o e e y
con igu a ion in his g oup is

590 J.J.BREYAND M. J.RUIZ-MONTERO 43
(N —
M—
8)!
Z!(N —
M—
8—
Z)! (N —
M—
8—
l)!
(Z —
1)!(N —
M—
8—
Z)! (A6)
In his way we gene a e all he con igu a ions ha ing a
leas ahole o he le .
(c) Only hose inal con igu a ions wi h one pa icle o
he le and one o mo e holes o he igh ha e o be con-
side ed he e. Then, o each o he s a ing
con igu a ions gi en by Eq. (A2) we ha e
possibili ies.
(d) and (e) Equa ion (A4) also holds.
The inal s ep is o collec all he abo e esul s, namely
mul iply Eq. (Al) by Eq. (A4), Eq. (A2) by Eq. (AS), and
so on, adding all he esul ing p oduc s. This 1eads e y
easily o Eq. (2.2).
'C. A. Angell and M. Golds ein, Dynamical Aspec s o S uc u
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G. W. Sche e , Relaxa ion in Glasses and Composl' es (Wiley,
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T. A. Webe , G. H. F edickson, and F. H. S illinge , Phys.
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