C i ical beha io and conse a ion in di ec ed sandpiles
Romualdo Pas o -Sa o as and Alessand o Vespignani
The Abdus Salam In e na ional Cen e o Theo e ical Physics, P.O. Box 586, 34100 T ies e, I aly
共Recei ed 20 June 2000兲
We pe o m la ge-scale simula ions o di ec ed sandpile models wi h bo h de e minis ic and s ochas ic
oppling ules. Ou esul s show he exis ence o wo dis inc uni e sali y classes. We also p o ide nume ical
simula ions o di ec ed models in he p esence o bulk dissipa ion. The nume ical esul s indica e ha he way
in which dissipa ion is implemen ed is i ele an o he de e mina ion o he c i ical beha io . The analysis o
he sel -a ine p ope ies o a alanches shows he exis ence o a subse o supe uni e sal exponen s, whose
alue is independen o he uni e sali y class. This ea u e is accoun ed o by means o a phenomenological
desc ip ion o he ene gy balance condi ion in hese models.
PACS numbe 共s兲: 05.65.⫹b, 05.70.Ln
I. INTRODUCTION
Sandpile cellula au oma a a e he mos amous example
o sel -o ganized c i ical 共SOC兲beha io 关1–3兴. Unde an
ex e nal d i e consis ing o a slow addi ion o sand 共ene gy兲
g ains and he ac ion o dissipa ion h ough he loss o en-
e gy on he la ice bounda ies, hese models each a s a ion-
a y s eady s a e. In he limi o in ini esimal d i ing and dis-
sipa ion 共 his las achie ed in he he modynamic limi 兲, he
s a iona y s a e o sandpile models exhibi s di e ging e-
sponse unc ions associa ed o a cha ac e is ic a alanche dy-
namics. This is he hallma k o a c i ical beha io ha has
a ac ed an eno mous amoun o in e es as a plausible ex-
plana ion o he a alanche-like c i ical beha io empi ically
obse ed in many na u al sys ems 关3兴.
Sandpile models ha e been a he cen e o an in ense
esea ch ac i i y made o bo h analy ical s udies and nume i-
cal simula ions. Despi e he simple de ini ion o hese au-
oma a, i u ns ou ha hei ull analy ical unde s anding is
a e y p oblema ic ask 关4兴. As a u he complica ion, also
he nume ical inspec ion o hese models esul s o be pa -
icula ly di icul . Fo example, he p ecise iden i ica ion o
uni e sali y classes has esis ed o many yea s e en he
mos ca e ul nume ical analysis, and only ecen esul s ha e
pa ially se led his p oblem 关5–8兴. On he o he hand, hese
e ined analyses ha e poin ed ou ha se e al sandpile mod-
els do no ollow he simple ini e size scaling 共FSS兲 o m
usually adop ed in he desc ip ion o c i ical beha io 关9兴.
Fo ins ance, he mo e sophis ica ed mul iscaling app oach
关10–12兴seems o be equi ed o a ull desc ip ion o he
scaling p ope ies o he o iginal Bak, Tang, and Wiesen eld
共BTW兲model 关1,2兴.
Many sandpile ea u es ha e been unde lined as he pos-
sible o igin o hese scaling anomalies. The de e minis ic dy-
namical ules o he BTW model induce none godic e ec s
关8兴, ha a e ce ainly missing in s ochas ic models, such as
he Manna model 关13,4兴, which shows a pe ec FSS beha -
io , e en o mode a e sys em sizes. A u he complica ion
o sandpile au oma a s ems om he peculia ole o he
bounda y dissipa ion, ha makes he la ice size scaling en-
angled wi h he sys em dynamics. In such cases, he he mo-
dynamic limi is essen ial o he dissipa i e dynamics o
la ge a alanches. A clea unde s anding o he in e play be-
ween dissipa ion and size scaling has no ye been achie ed
and i has been ecen ly he subjec o se e al s udies 关11,14兴.
In his pape we add ess some o he a o emen ioned
p oblems in he case o di ec ed sandpile models 关10,15–18兴.
In his case Dha and Ramaswamy ob ained an exac solu ion
o he Abelian de e minis ic di ec ed sandpile 共DDS兲关15兴,
ha can be used as a ouchs one o check he nume ical
simula ion analysis. Di ec ed sandpiles hus become an in e -
es ing es ield o s udy how he c i ical beha io is a ec ed
by he in oduc ion o s ochas ic elemen s and dissipa ion.
We pe o m la ge scale nume ical simula ions o wo di-
ec ed sandpile au oma a: he de e minis ic di ec ed sandpile
model 关15兴and he s ochas ic di ec ed sandpile model 关18兴.
We s udy bo h models in he case o bounda y and bulk
dissipa ion 关19–22兴. We ind, in ag eemen wi h he esul s
in Re . 关18兴, ha he models de ine wo di e en uni e sali y
classes. In addi ion we show ha he uni e sali y class o he
models does no depend on he way in which dissipa ion is
implemen ed. Finally we analyze he p ope ies o aniso-
opic models in which he dynamics is no ully di ec ed
关10,23兴. In his case we obse e ha on la ge scales he c i i-
cal beha io is he same o ha o ully di ec ed models.
Resul s o he s ochas ic models a e compa ed wi h a ecen
heo e ical app oach by Paczuski and Bassle 关24兴, ha p o-
ides alues o he c i ical exponen s in pe ec ag eemen
wi h nume ical simula ions. These esul s a e also eco e ed
in Re . 关25兴.
The nume ical analysis also poin s ou ha some c i ical
exponen alues, such as he co ela ion leng h exponen s o
he a ini y exponen 共 o be de ined la e on兲, a e independen
o he pa icula uni e sali y classes and common o all mod-
els conside ed. In o de o explain his nume ical e idence,
we p o ide a phenomenological cha ac e iza ion o di ec ed
sandpiles based on he basic symme ies in oduced by he
conse ed dynamics o hese au oma a. Following balance o
ene gy a gumen s inspi ed in Re s. 关26–28兴, we de i e a
se ies o esul s and p edic ions on he alue o c i ical ex-
ponen s which a e a s aigh o wa d consequence o conse -
a ion. These gene al esul s can be conside ed as supe uni-
e sal, because hey cha ac e ize he c i ical beha io o all
di ec ed sandpiles wi h local dynamical ules, independen ly
on he speci ic uni e sali y class. The esul s p esen ed he e
p o ide a gene al pic u e o di ec ed models and he ole o
PHYSICAL REVIEW E NOVEMBER 2000VOLUME 62, NUMBER 5
PRE 62
1063-651X/2000/62共5兲/6195共11兲/$15.00 6195 ©2000 The Ame ican Physical Socie y
bounda y and bulk dissipa ion in he p ocess o sel -
o ganiza ion.
The pape is a anged as ollows. In Sec. II we in oduce
and de ine he a ious di ec ed models conside ed. Sec ions
III and IV p esen and discuss, om he s andpoin o uni-
e sali y, he nume ical esul s o di ec ed models wi h
bounda y and bulk dissipa ion. In Sec. V we in oduce an-
iso opic models, and p esen he nume ical esul s ob ained,
in compa ison wi h hose o di ec ed models. Sec ion VI is
de o ed o an analy ical app oach based on he conse a ion
o ene gy. Finally, in Sec. VII we d aw ou conclusions and
pe spec i es.
II. DIRECTED MODELS
Sandpile models a e usually de ined on a d-dimensional
hype cubic la ice o size L. To each node o he la ice is
assigned an in ege a iable zi, called ‘‘ene gy.’’ Ene gy is
added o he sys em uni o mly a andomly chosen si es (zi
→zi⫹1). When a si e becomes ac i e, ha is, when i s en-
e gy becomes la ge han o equal o a ce ain h eshold zc,
i opples. A oppling si e loses an ene gy zc, ha is dis ib-
u ed among i s neighbo s acco ding o a ce ain se o ules.
The neighbo s ha ecei e ene gy can become ac i e and
opple on hei u n, hus gene a ing an a alanche. The slow
d i ing condi ion is e ec i ely imposed by s opping he an-
dom ene gy addi ion du ing he a alanche sp eading. This
means ha he d i ing ime scale is in ini ely la ge wi h e-
spec o he oppling cha ac e is ic ime scale.
The models we conside in his sec ion a e di ec ed, in he
sense ha he ene gy is always anspo ed along a p e e ed
ixed di ec ion. We deno e his p e e ed di ec ion by he
coo dina e x
储
, whose posi i e di ec ion is usually de ined as
‘‘downwa ds.’’ The ans e se di ec ion 共subspace o dimen-
sion d⫺1 pe pendicula o x
储
) will be deno ed by x
ជ
⬜.
The oppling ules o he models de ine wo main classes.
共i兲De e minis ic di ec ed sandpile 共DDS兲:Inddimen-
sions, he h eshold is se o zc⫽2d⫺1. When a si e in a
gi en hype plane x
储
opples, i sends de e minis ically one
g ain o ene gy o each one o i s nea es and nex -nea es
neighbo s on he hype plane x
储
⫹1关see Fig. 1共a兲兴. Ou de i-
ni ion is somewha di e en om he o iginal model o Dha
and Ramaswamy 关15兴, in bo h he d i ing and he o ien a ion
o he la ice. Bo h models, howe e , a e expec ed o sha e
he same uni e sali y class, being de e minis ic and di ec ed.
Nume ical simula ions con i m indeed his poin 关18兴.
共ii兲S ochas ic di ec ed sandpile 共SDS兲: In his case, he
h eshold is zc⫽2, independen ly o he dimensionali y o
he la ice. When a si e in he hype plane x
储
opples, i sends
wo g ains o ene gy o wo si es, andomly chosen among i s
2d⫺1 nea es and nex -nea es neighbo s on he hype plane
x
储
⫹1. The oppling ules o his model can be de ined ex-
clusi e i he wo ene gy g ains a e always dis ibu ed on
di e en si es, Fig. 1共b兲. On he o he hand, he model can be
de ined nonexclusi e i he dynamics allows he ans e o
wo ene gy g ains on o he same si e, Fig. 1共c兲. We he e o e
epo simula ions on he exclusi e s ochas ic di ec ed sand-
pile 共ESDS兲and on he nonexclusi e s ochas ic di ec ed
sandpile 共NESDS兲. In spi e o he s ochas ic na u e o hese
models, we mus bea in mind ha hey a e ne e heless
Abelian 关4兴. The discussion he e o e ocuses on he di e -
ence be ween s ochas ic and de e minis ic models.
Once he oppling ules ha e been de e mined, he models
a e inally de ined by speci ying he dissipa ion mechanism.
Fo sys ems wi h bounda y dissipa ion, we impose pe iodic
bounda y condi ions in he ans e se di ec ions x
ជ
⬜and open
a he hype plane x
储
⫽L. In his way, he models a e locally
conse ed; ene gy can only lea e he sys em a he bo om o
he la ice. In models wi h bulk dissipa ion, we impose pe i-
odic bounda y condi ions in bo h he x
储
and x
ជ
⬜di ec ions.
Dissipa ion is implemen ed by allowing a oppling si e o
lose an ene gy zcwi hou ans e ing i wi h p obabili y p
关26,21兴. This means ha , on a e age, an ene gy
⑀
⫽zcpis
dissipa ed in each oppling. In he limi
⑀
→0, he sys em
shows c i ical beha io 关26兴.
In he s a iona y s a e we can de ine he p obabili y ha
he addi ion o a single ene gy g ain is ollowed by an a a-
lanche o oppling e en s. A alanches a e hen cha ac e ized
by he o al numbe o opplings sand he ime du a ion .In
he limi o in ini esimal d i ing 共slow d i ing condi ion兲 he
sys em shows scaling beha io and he p obabili y dis ibu-
ions o hese quan i ies ollow he ini e-size scaling 共FSS兲
o ms
P共s兲⫽s⫺
␶
sG共s/sc兲,共1兲
P共 兲⫽ ⫺
␶
F共 / c兲,共2兲
whe e scand ca e he cha ac e is ic size and ime, espec-
i ely. The exponen s
␶
sand
␶
cha ac e ize he c i ical be-
ha io and de ine he uni e sali y classes o which he mod-
els belong. In he c i ical egion he cha ac e is ic ime and
size a e de e mined only by he sys em size Lo he dissipa-
ion
⑀
, in he case o bounda y and bulk dissipa ion, espec-
i ely. In di ec ed models, he a ini y exponen
␨
is o pa -
icula impo ance; i ela es he a alanche cha ac e is ic
leng hs in he pe pendicula di ec ion,
␰
⬜, and in he pa allel
di ec ion,
␰
储
, h ough he ela ion
␰
⬜⬃
␰
储
␨
. This exponen
cha ac e izes he deg ee o aniso opy due o he p e e en ial
di ec ion p esen in he anspo o he ene gy. In o he
wo ds, i exp esses he sel -a ine p ope ies in he scaling o
a alanches. A gene al esul conce ns he a e age a alanche
size
具
s
典
, ha also scales linea ly wi h L关10,15,23兴; his esul
can be exac ly ob ained by inspec ing he conse a ion sym-
me y o he model as we shall see in Sec. VI.
Fo he DDS, he exac analy ical solu ion in d⫽2 yields
he exponen s
␶
s⫽4/3 and
␶
⫽D⫽3/2 关15兴. The uppe c i i-
cal dimension is ound o be dc⫽3, and i is also possible o
FIG. 1. Toppling ules in d⫽2 o di ec ed sandpiles. Filled
ci cles ep esen ac i e 共 oppling兲si es; emp y ci cles a e s able
si es. In he de e minis ic model 共a兲an ac i e si e sends one g ain o
each o i s h ee neighbo s on he nex downwa ds ow. In he
s ochas ic models, exclusi e 共b兲and nonexclusi e 共c兲, one g ain is
sen o wo andomly chosen downwa ds neighbo s.
6196 PRE 62
ROMUALDO PASTOR-SATORRAS AND ALESSANDRO VESPIGNANI
ind exac ly he loga i hmic co ec ions o scaling 关15,29兴.
The in oduc ion o s ochas ic ing edien s in he oppling dy-
namics o di ec ed sandpiles has been s udied ecen ly in a
model ha andomly s o es ene gy on each oppling 关16兴.
This model is s ic ly ela ed o di ec ed pe cola ion and de-
ines a uni e sali y class ‘‘pe se.’’ I would be e y in e es -
ing o in es iga e he possible connec ion be ween his model
and he s ochas ic di ec ed one p esen ed in his wo k.
III. NUMERICAL SIMULATIONS WITH BOUNDARY
DISSIPATION
In his sec ion we epo esul s om compu e simula-
ions o de e minis ic and s ochas ic di ec ed sandpiles, pe -
o med wi h bounda y dissipa ion. The sys em sizes consid-
e ed ange om L⫽100 o L⫽6400. The s a is ical
dis ibu ion unc ions ha e been compu ed a e aging o e
107nonze o a alanches.
In he case o bounda y dissipa ion, he la ice size Lis he
only cha ac e is ic leng h p esen in he sys em. App oaching
he he modynamic limi (L→⬁), he a alanche cha ac e is-
ic size and ime in Eqs. 共1兲and 共2兲di e ge as sc⬃LDand
c⬃Lz, espec i ely. The exponen Dde ines he ac al di-
mension o he a alanche clus e and zis he usual dynamic
c i ical exponen . The di ec ed na u e o he model in o-
duces a d as ic simpli ica ion, since i imposes z⫽1. In o de
o compu e he di e en exponen s cha ac e izing he dy-
namics o he a alanches, we ha e pe o med he momen
analysis o he dis ibu ions, in analogy o he me hod de el-
oped by De Menech e al. 关11,12兴. We de ine he q h mo-
men o he a alanche size dis ibu ion on a la ice o size L
as
具
sq
典
L⫽
兰
dssqP(s). I he FSS hypo hesis 共1兲is alid in
he asymp o ic limi o la ge s, hen he q h momen has he
ollowing dependence on sys em size:
具
sq
典
L⫽LD(q⫹1⫺
␶
s)
冕
dy y(q⫺
␶
s)G共y兲⬃L
␴
s(q).共3兲
The exponen
␴
s(q)⫽D(q⫹1⫺
␶
s) is compu ed as he slope
o he log-log plo o
具
sq
典
Las a unc ion o L. Fo la ge
enough alues o q关i.e., away om he egion whe e he
in eg al in Eq. 共3兲is domina ed by i s lowe cu o 兴, one can
compu e he ac al dimension Das he slope o
␴
s(q)asa
unc ion o q:D⫽
⳵␴
s(q)/
⳵
q. On he o he hand, since he
i s momen mus scale linea ly wi h L, we ha e
␴
s(1)⫽1.
Once Dis known we can es ima e
␶
susing he ela ion
␴
s(1)⫽D(2⫺
␶
s)⫽1.
Along he same lines we can ob ain he momen s o he
a alanche ime dis ibu ion. In his case,
具
q
典
L⬃L
␴
(q), wi h
⳵␴
(q)/
⳵
q⫽z. Analogous conside a ions o small qapply
also o he ime momen analysis. He e, an es ima e o he
asymp o ic con e gence o he nume ical esul s is p o ided
by he cons ain z⫽1, ha mus hold o la ge enough sizes.
Then, he
␶
exponen can be ound using he scaling ela ion
(2⫺
␶
)⫽
␴
(1).
Once he exponen s ha e been es ima ed nume ically, we
can check he accu acy o he momen analysis’ p edic ions
using he FSS hypo hesis. I he FSS hypo hesis o Eqs.
共1兲,共2兲is co ec , hen he plo s o he dis ibu ions, unde he
escaling s→s/LDand P(s)→P(s)LD
␶
sand co espond-
ingly → /Lzand P( )→P( )Lz
␶
, should collapse on o he
same uni e sal unc ion, o di e en alues o L.
In Table I we epo he exponen s ound o he DDS,
ESDS, and NESDS models in d⫽2. Figu e 2 shows he
momen s
␴
s(q) and
␴
(q). Figu es 3 and 4 plo he FSS da a
collapse o sizes and imes, espec i ely. The exponen s ob-
ained o he DDS a e in pe ec ag eemen wi h he ex-
pec ed analy ical esul s. This ac suppo s he idea ha he
sys em sizes used in he p esen wo k allow o eco e he
co ec asymp o ic beha io . Resul s o he ESDS and
NESDS a e iden ical wi hin he e o ba s, poin ing ou ha
hese wo models a e in he same uni e sali y class. On he
o he hand, he ob ained exponen s p o e beyond any doub s
ha de e minis ic and s ochas ic di ec ed sandpile models do
no belong o he same uni e sali y class.
We ha e also di ec ly compu ed he cha ac e is ic leng hs
in he pa allel and ans e sal di ec ions
␰
储
and
␰
⬜as a unc-
FIG. 2. Plo o 共a兲
␴
s(q) and 共b兲
␴
(q) o he d⫽2 models
DDS, ESDS, and NESDS wi h bounda y dissipa ion.
TABLE I. C i ical exponen s o di ec ed sandpiles wi h bound-
a y dissipa ion in d⫽2. DR: Dha and Ramaswamy’s exac solu-
ion; DDS, de e minis ic di ec ed model; ESDS and NESDS, s o-
chas ic di ec ed models. Figu es in pa en heses deno e s a is ical
unce ain ies.
Model
␶
sD
␶
z
␨
DR 4/3 3/2 3/2 1 1/2
DDS 1.34(1) 1.51(1) 1.51(1) 1.00(1) 0.50(1)
ESDS 1.43(1) 1.74(1) 1.71(3) 0.99(1) 0.51(1)
NESDS 1.43(1) 1.75(1) 1.74(4) 0.99(1) 0.51(1)
PRE 62 6197CRITICAL BEHAVIOR AND CONSERVATION IN . . .
ion o he sys em size. The aniso opy o he sys em is e-
lec ed in he di e en de ini ions o bo h cha ac e is ic
leng hs. In his sense, we de ine hem wi h he same spi i as
in di ec ed pe cola ion 关30兴.
Conside a gi en a alanche, labeled
␣
, ha has s a ed a
he si e (x
储
(0) ,x
ជ
⬜
(0)), and has a ec ed he se o di e en si es
兵
(x
储
(i),x
ជ
⬜
(i))
其
, o i⫽0•••a⫺1共i.e., i has co e ed an a ea
a). Le us de ine he quan i ies
R
储
共
␣
兲⫽1
a兺
i⫽1
a⫺1
兩
x
储
(0)⫺x
储
(i)
兩
共4兲
and
R⬜
2共
␣
兲⫽1
a兺
i⫽1
a⫺1
共x
ជ
⬜
(0)⫺x
ជ
⬜
(i)兲2.共5兲
Fu he mo e, le us de ine R
储
(a) and R⬜
2(a) as he a e ages
o he p e ious quan i ies, o e all a alanches o he same
ixed a ea a. Le P(a) be he p obabili y o obse ing an
a alanche o a ea a. We de ine he co ela ion leng hs by
␰
储
⫽兺aR
储
共a兲aP共a兲
兺aaP共a兲,
␰
⬜
2⫽
兺aR⬜
2共a兲aP共a兲
兺aaP共a兲.共6兲
The di e en de ini ions 共4兲and 共5兲a e ob iously due o he
di e en na u e o he a alanche sp eading in he di ec ions
x
储
and x⬜. In he o me case, he sp eading is iso opic, and
hus he second momen o he ela i e dis ance dis ibu ion
is needed o de ine a meaning ul co ela ion leng h. In he
la e case, on he o he hand, he sp eading is always in he
di ec ion o g owing x
储
, and he e o e he i s momen is
su icien .
The sys em being c i ical, bo h co ela ion leng hs should
scale wi h he sys em size, de ining he exponen s
␯
储
and
␯
⬜
by
␰
储
⬃L
␯
储
,
␰
⬜⬃L
␯
⬜.共7兲
The a ini y exponen , de ined by
␰
⬜⬃
␰
储
␨
共8兲
is hus gi en by
␨
⫽
␯
⬜/
␯
储
.
We ha e calcula ed he co ela ions leng hs in he models
DDS, ESDS, and NESDS, gi en by he de ini ion 共6兲. The
esul s, plo ed in Fig. 5, gi e he ollowing dependence o
he co ela ion leng hs wi h sys em size o all models:
␰
储
⬃L,
␰
⬜⬃L1/2.共9兲
These ela ions de ine he exponen s
␯
储
⫽1 and
␯
⬜⫽1/2, and
an a ini y exponen
␨
⫽1/2. I is in e es ing o no e ha his
exponen is independen o he uni e sali y class o he
model, de ining a so o supe uni e sal p ope y o di ec ed
models.
As poin ed ou in Re . 关18兴, he s ochas ic dynamics o
SDS models in oduces mul iple oppling e en s on he same
si e, which a e by de ini ion absen in he de e minis ic case.
This gi es ise o a e y di e en a alanche s uc u e, e en-
FIG. 3. Da a collapse analysis o he in eg a ed a alanche size
dis ibu ion o he d⫽2 s ochas ic models wi h bounda y dissipa-
ion 共a兲ESDS and 共b兲NESDS. Sys em sizes a e L⫽400, 800, 1600,
3200, and 6400.
FIG. 4. Da a collapse analysis o he in eg a ed a alanche ime
dis ibu ion o he d⫽2 s ochas ic models wi h bounda y dissipa-
ion 共a兲ESDS and 共b兲NESDS. Sys em sizes a e L⫽400, 800, 1600,
3200, and 6400.
6198 PRE 62
ROMUALDO PASTOR-SATORRAS AND ALESSANDRO VESPIGNANI
ually e lec ed in he di e en asymp o ic c i ical beha io .
I is wo h ema king ha he uni e sali y class o SDS ap-
pea s obus o modi ica ions o he s ochas ic mic oscopic
dynamics as poin ed ou in Re . 关31兴, whe e i is shown ha
modi ica ions o SDS models wi h s ochas ic oppling h esh-
old s ill belong o he same uni e sali y class. Recen ly, Pac-
zuski and Bassle 关24兴, ha e p oposed a heo e ical app oach
ha allows he calcula ion o c i ical exponen s in di ec ed
models wi h mul iple opplings. The analysis goes h ough
he mapping o he a alanche e olu ion in o he dynamics o
an in e ace mo ing in a andom medium, as also p oposed
in Re s. 关32,33兴. This heo e ical esul gi es he exponen s
␶
s⫽10/7 and
␶
⫽D⫽7/4, in pe ec ag eemen wi h he al-
ues ob ained by nume ical simula ions, Table I. The same
exponen alues a e also ound in he app oach o Re . 关25兴.
IV. NUMERICAL SIMULATIONS WITH BULK
DISSIPATION
In his sec ion we epo esul s om compu e simula-
ions o de e minis ic and s ochas ic sandpiles, pe o med
wi h bulk dissipa ion. In his case, dissipa ion is imple-
men ed as desc ibed in Sec. II. Tha is, in a sys em wi h
pe iodic bounda y condi ions, each oppling si e has a p ob-
abili y
⑀
/zco losing an ene gy zc, and a p obabili y 1
⫺
⑀
/zco ans e ing i o i s neighbo s. The dissipa ion
a es ange om
⑀
⫽0.0016 o 0.0512, and he 共 ixed兲sys em
size conside ed is L⫽6400. S a is ical dis ibu ion unc ions
ha e been compu ed a e aging o e 107nonze o a alanches.
In he p esence o bulk dissipa ion he cha ac e is ic sizes
a e de e mined by he dissipa ion a e
⑀
, which de ines he
only cha ac e is ic leng h in he sys em. App oaching he
limi
⑀
→0, he a alanche cha ac e is ic size and ime di-
e ge as sc⬃
⑀
⫺⌬sand c⬃
⑀
⫺⌬ , espec i ely. I is also e y
easy o ela e he mean a alanche size o he dissipa ion a e
⑀
. On a e age, each added g ain mus be dissipa ed in he
e olu ion o he a alanche, esul ing in
⑀
具
s
典
⫽1. This eadily
yields
具
s
典
⫽
⑀
⫺1. In his case i is ex emely impo an ha
he cha ac e is ic leng h o he a alanche
␰
储
is always smalle
han he size o he la ice used. This allows us o s udy only
ini e size e ec s in oduced by he dissipa ion p obabili y,
wi hou spu ious e ec s due o he ini e la ice size.
The momen analysis can be s aigh o wa dly gene al-
ized o sys ems wi h bulk dissipa ion. In his case he ole o
he sys em size Las scaling pa ame e is played by he dis-
sipa ion
⑀
. I he FSS hypo hesis holds, he q h momen o ,
say he size dis ibu ion, has an explici dependence on he
dissipa ion a e ha eads
具
sq
典
⑀
⬃
⑀
⫺⌬s(q⫹1⫺
␶
s)⫽
⑀
⫺
␳
s(q).共10兲
The new momen
␳
s(q)⫽⌬s(q⫹1⫺
␶
s) can be es ima ed by
linea eg ession in a log-log plo o
具
sq
典
⑀
as a unc ion o
⑀
⫺1. Once his momen is compu ed, he exponen ⌬sis
gi en by ⌬s⫽
⳵␳
s(q)/
⳵
q. The ela ion
具
s
典
⫽
⑀
⫺1imposes
␳
s(1)⫽1, and om he e, once known ⌬s, we compu e
␶
s
using he ela ion
␳
s(1)⫽⌬s(2⫺
␶
s). Analogous conside -
a ions allow us o compu e he exponen s o he ime dis i-
bu ion ⌬ and
␶
. Finally, o check he exponen s wi h he
da a collapse echnique, one mus plo he escaled unc ions
P(s)
⑀
⫺⌬s
␶
sas a unc ion o s/
⑀
⫺⌬sand P( )
⑀
⫺⌬
␶
as a unc-
ion o /
⑀
⫺⌬ , espec i ely.
FIG. 5. Co ela ion leng hs
␰
储
and
␰
⬜as a unc ion o L o he
models wi h bounda y dissipa ion DDS (䊊), ESDS (䉭), and
NESDS (〫). The dashed lines a e guides o he eye wi h slope
1.00 and 0.50.
FIG. 6. Plo o 共a兲
␳
s(q) and 共b兲
␳
(q) o he d⫽2 models
DDS, ESDS, and NESDS wi h bulk dissipa ion.
TABLE II. C i ical exponen s o di ec ed sandpiles wi h bulk
dissipa ion in d⫽2. DR: Dha and Ramaswamy’s exac solu ion;
DDS, de e minis ic di ec ed model; ESDS and NESDS, s ochas ic
di ec ed models. Figu es in pa en heses deno e s a is ical unce ain-
ies.
Model
␶
s⌬s
␶
⌬
␨
DR 4/3 3/2 3/2 1 1/2
DDS 1.32(1) 1.50(1) 1.52(1) 1.00(1) 0.51(1)
ESDS 1.42(1) 1.72(2) 1.70(4) 0.98(2) 0.51(1)
NESDS 1.43(1) 1.75(2) 1.70(5) 0.99(2) 0.50(1)
PRE 62 6199CRITICAL BEHAVIOR AND CONSERVATION IN . . .

In Table II we epo he exponen s compu ed in d⫽2 o
he di ec ed models DDS, ESDS, and NESDS wi h bulk dis-
sipa ion. The co esponding momen s
␳
s(q) and
␳
(q) a e
shown in Figs. 6, while Figs. 7 and 8 plo he da a collapse
o sizes and imes, espec i ely.
To conclude ou analysis o di ec ed sandpiles wi h bulk
dissipa ion, we ha e p oceeded o compu e he co ela ion
leng h o he models. In his case, he scaling o he co ela-
ion leng hs wi h anishing dissipa ion de ine he scaling ex-
ponen s
␰
储
⬃
⑀
⫺
␯
储
⬘,
␰
⬜⬃
⑀
⫺
␯
⬜
⬘,共11兲
and an a ini y exponen
␨
⫽
␯
⬜
⬘/
␯
储
⬘. Using an analogous de i-
ni ion as in he case o bounda y dissipa ion, we compu e he
exponen s
␯
储
⬘⫽1,
␯
⬜
⬘⫽1/2, and
␨
⫽1/2, as shown in Fig. 9.
Tha is, he co ela ion leng h exponen s a e iden ical o
bo h bounda y and bulk dissipa ion. These esul s again im-
ply an a ini y exponen
␨
⫽1/2 in all he models s udied so
a .These esul s con i m ha he c i ical beha io o models
wi h bounda y o bulk dissipa ion is iden ical. In ac , all
c i ical exponen s
␶
s,
␶
,z, and
␨
a e equal in bo h cases
关34兴. This u he con i ms he comple e equi alence o bo h
poin s o iew wi h espec o sandpiles and shows ha , a
leas in he di ec ed case, he open bounda y condi ions usu-
ally implemen ed in simula ions do no a ec he scaling
beha io in a peculia way. O cou se, he open bounda y
condi ions b eaks he ansla ional in a iance o he sys em,
bu in he he modynamic limi his e ec is negligible o
he asymp o ic c i ical beha io . Finally, hese esul s ali-
da e heo e ical app oaches in which i is assumed a homo-
geneous dissipa ion ha is much easie o ea analy ically.
As a las obse a ion i is wo h ema king ha also in his
case, a se ies o exponen s such as
␨
and
␯
⬜
⬘assume alues
independen ly o he uni e sali y class o he model unde
s udy. This so o supe uni e sali y can be explained in
e ms o ene gy conse a ion as we shall see in Sec. VI.
V. NUMERICAL SIMULATIONS OF ANISOTROPIC
MODELS
An impo an ques ion o s udy in di ec ed sandpile mod-
els is he e ec on he scaling p ope ies o any amoun o
FIG. 7. Da a collapse analysis o he in eg a ed a alanche size
dis ibu ion o he d⫽2 s ochas ic models wi h bulk dissipa ion 共a兲
ESDS and 共b兲NESDS. Dissipa ions a e
⑀
⫽0.0256, 0.0128, 0.0064,
0.0032, and 0.0016.
FIG. 8. Da a collapse analysis o he in eg a ed a alanche ime
dis ibu ion o he d⫽2 s ochas ic models wi h bulk dissipa ion 共a兲
ESDS and 共b兲NESDS. Dissipa ions a e
⑀
⫽0.0256, 0.0128, 0.0064,
0.0032, and 0.0016.
FIG. 9. Co ela ion leng hs
␰
储
and
␰
⬜as a unc ion o
⑀
o he
models wi h bulk dissipa ion DDS (䊊), ESDS (䉭), and NESDS
(〫). The dashed lines a e guides o he eye wi h slope 1.00 and
0.50.
6200 PRE 62
ROMUALDO PASTOR-SATORRAS AND ALESSANDRO VESPIGNANI
di usion along he p e e ed di ec ion o anspo x
储
. One
would expec ha he b oken symme y in oduced by he
p e e en ial di ec ion should p e ail on la ge scales, so ha
he dynamical scaling in di ec ed and simply aniso opic
sandpiles become indis inguishable in he he modynamic
limi . This ac hin s owa ds he possibili y o a unique uni-
e sali y class o bo h di ec ed and aniso opic sandpiles.
This uni e sali y class is de e mined uniquely by he lack o
symme y along he x
储
di ec ion, and he p esence o absence
o s ochas ic elemen s in he de ini ion o he models.
In o de o es his conjec u e, we ha e pe o med nu-
me ical simula ions o an aniso opic s ochas ic sandpile
model, de ined acco ding o he ollowing ules: on a hype -
cubic la ice o size L, we conside a model wi h h eshold
zc⫽2. When a si e opples, i sends wo g ains o ene gy o
wo si es, andomly selec ed among he 2d⫺1 nea es and
nex -nea es neighbo s on he hype plane x
储
⫹1, and he
nea es neighbo on he hype plane x
储
⫺1, see Fig. 10. The
ules in his model a e de ined non-exclusi e, in such a way
ha he same si e can ecei e he wo sand g ain expelled by
an ac i e si e. The model is clea ly aniso opic, because he
p obabili y o ans e ene gy in he downwa ds di ec ion is
h ee imes la ge han in he upwa ds di ec ion. I would
hus co espond o a nonexclusi e s ochas ic aniso opic
sandpile 共NESAS兲. We conside only he case o bounda y
dissipa ion, pe o ming simula ions o sizes anging om
L⫽100 up o 6400, and a e aging o e 107nonze o a a-
lanches.
In Fig. 11 we plo he co ela ion leng h
␰
储
and
␰
⬜, mea-
su ed acco ding o he ules gi en in Eqs. 共6兲. We con i m
he expec a ion ha aniso opic models ha e he same scal-
ing p ope ies, as ega ds he scaling o he co ela ion
leng hs, as di ec ed models wi h he same de e minis ic o
s ochas ic ing edien s. We ha e also measu ed he exponen s
␶
s,
␶
,D, and z o his model, using he momen analysis
echnique. The alues ound a e
␶
s⫽1.43(1), D⫽1.75(1),
␶
⫽1.72(2), z⫽0.98(2). These esul s, compa ed wi h
Tables I and II, show ha his aniso opic models belongs o
he same uni e sali y class o he ESDS and NESDS di ec ed
models, con i ming he i ele ance o he di usion along he
p e e ed di ec ion x
储
.
VI. THE ROLE OF CONSERVATION
IN SANDPILE MODELS
We ha e seen in he p eceding sec ions ha a subse o
c i ical exponen s cha ac e izing he c i ical beha io o di-
ec ed and aniso opic models ha e an in e es ing supe uni-
e sal p ope y; i.e., hey a e independen o he uni e sali y
class o he models. In o de o unde s and his ea u e we
pe o m a heo e ical analysis based on he conse a ion o
ene gy, ha is he basic symme y in s anda d sandpile au-
oma a. We shall see in he ollowing ha he supe uni e sal
cha ac e o some c i ical exponen s is dic a ed by simple
ene gy conse a ion conside a ions. The use o his app oach
also allows us o es ablish a ela ion be ween bounda y and
bulk dissipa ion models by in oducing an e ec i e dissipa-
ion ha depends on he sys em size.
The a alanche dynamics in sandpile models is implici ly
due o he imposed in ini e ime scale sepa a ion be ween
d i ing and dissipa ion 关26,27,35兴. In o de o de ise a
heo y ha can ake in o accoun he symme y in oduced by
he ene gy conse a ion, one mus i s egula ize he ules o
he models in such a way ha a single ime scale is uling he
dynamics. One way o do so is o in oduce a nonze o d i -
ing a e, de ined as he p obabili y pe uni ime ho a si e o
ecei e a g ain o ene gy 关26,35兴. This d i ing a e plays he
ole o an ex e nal ield and leads o he SOC beha io in he
limi h→0⫹. On he o he hand, gi en ha he oppling ules
a e conse ed, ene gy can lea e he sys em only a he
bounda ies. Bounda y dissipa ion is a na u al choice in com-
pu e simula ions. Howe e , i in oduces undesi able com-
plica ions due o i s singula cha ac e in a local heo y. I is
he e o e con enien o use an homogeneous e ec i e dissi-
pa ion
⑀
, de ined as he a e age ene gy los in each oppling
e en . As obse ed in p e ious sec ions, one can de ine mod-
els wi h pe iodic bounda y condi ions and buil -in bulk dis-
sipa ion. When cons uc ing he local heo y o models wi h
open bounda y condi ions, he bulk dissipa ion
⑀
amoun s o
an e ec i e pa ame e ha is o be ela ed o he sys em size
L.Wi h all hese ing edien s, we a e eady o o mula e con-
se a ion o ene gy as a con inuous equa ion. In sandpiles,
we de ine he o de pa ame e
␳
aas he densi y o ac i e
si es 共i.e., whose heigh z⭓zc). The only dynamics in he
model is ob iously due o he ield
␳
a(x
ជ
, ), which is coupled
o he local ene gy densi y E(x
ជ
, )共i.e., he local densi y o
FIG. 10. Toppling ules in d⫽2 o an aniso opic sandpile.
Filled ci cles ep esen ac i e 共 oppling兲si es; emp y ci cles a e
s able si es. An ac i e si e sends one g ain o wo andomly chosen
si es selec ed among he h ee downwa ds neighbo s and he up-
wa d nea es neighbo .
FIG. 11. Co ela ion leng hs
␰
储
and
␰
⬜as a unc ion o L o he
model wi h bounda y dissipa ion NESAS. The dashed lines a e
guides o he eye wi h slope 1.00 and 0.50.
PRE 62 6201CRITICAL BEHAVIOR AND CONSERVATION IN . . .
sand g ains兲, which enhances o supp esses he gene a ion o
new ac i e si es. A Lange in desc ip ion o sandpile au-
oma a is possible by conside ing he dynamics o he local
o de -pa ame e ield
␳
a(x
ជ
, ) in a coa se-g ained pic u e,
bea ing in mind ha he ene gy densi y E(x
ជ
, )isaconse ed
ield. In Re s. 关27,28兴, in analogy wi h abso bing-s a e phase
ansi ions 关36,37兴, a pai o coupled dynamical equa ions o
he ields
␳
a(x
ជ
, ) and E(x
ជ
, ) we e p oposed. In he ollow-
ing we elucida e he consequences o ene gy conse a ion
and we ocus only on he la e equa ion. The in e es ed
eade can ind he ull se o equa ions in Re . 关28兴.In he
nex subsec ions we shall conside sepa a ely di ec ed and
aniso opic models.
A. Di ec ed sandpiles
We seek a con inuous equa ion o he coa se-g ained lo-
cal densi y o ene gy E(x
ជ
, ). In he limi o ze o d i ing and
dissipa ion, ene gy is conse ed. The e o e, he e olu ion
equa ion ul illed by he local ield Eis
⳵
E共x
ជ
, 兲
⳵
⫽⫺ⵜ
ជ
•J
ជ
E⫺
⑀
␳
a共x
ជ
, 兲⫹h共x
ជ
, 兲⫹
␩
E共x
ជ
, 兲.共12兲
The i s e m simply ep esen s he di usion o ene gy; he
second e m accoun s o he dissipa ion ha is associa ed
wi h e e y oppling e en ; he hi d e m ep esen s he ex-
e nal d i ing. Finally, he las e m is a sou ce o s ochas ic
noise, ha accoun s o he andomness in he low o en-
e gy. The noise e m can be gene a ed by he oppling ules
in a s ochas ic model, o by he ini ial condi ions plus he
andom d i ing in a de e minis ic model. We will equi e he
noise o ha e ze o a e age
具
␩
E共x
ជ
, 兲
典
⫽0. 共13兲
The noise co ela o
具
␩
E(x
ជ
, )
␩
E(x
ជ
⬘, ⬘
典
is o undamen al
impo ance o he de e mina ion o uni e sali y classes and
he c i ical beha io o he o de pa ame e . Howe e , o ou
p esen pu poses we do no need p ecise knowledge o i s
analy ical o m 共 o a de ailed discussion, see Re s. 关27,28兴兲.
The cu en can be cons uc ed by appealing o he sym-
me ies o he model. The anspo o ene gy is due o op-
plings. These a e iso opic along he ans e sal di ec ion
x
ជ
⬜, he e o e he cu en along his di ec ion will be p opo -
ional o he g adien o he densi y o ac i e si es. In he
p e e ed di ec ion, on he o he hand, all he ene gy is ans-
e ed downwa ds; he e o e, he cu en in his di ec ion
mus be p opo ional o he densi y o ac i e si es. The inal
o m o he cu en is hen
J
ជ
E共x
ជ
, 兲⫽⫺D⬜ⵜ
ជ
⬜
␳
a共x
ជ
, 兲⫹2␭
␳
a共x
ជ
, 兲e
ជ
储
.共14兲
Plugging his exp ession in o he equa ion o he ene gy, we
ha e he inal esul
⳵
E共x
ជ
, 兲
⳵
⫽D⬜ⵜ⬜
2
␳
a共x
ជ
, 兲⫺2␭
⳵
储
␳
a共x
ជ
, 兲
⫺
⑀
␳
a共x
ជ
, 兲⫹h共x
ជ
, 兲⫹
␩
E共x
ជ
, 兲,共15兲
whe e he symbol
⳵
储
s ands o he pa ial de i a i e
⳵
/
⳵
x
储
.
This is he gene al conse a ion equa ion o any di ec ed
sandpile model. I is wo h ema king a his poin ha he
ene gy ield is a s a ic ield, in he sense ha ene gy di uses
only i ac i e si es a e p esen in he sys em. This is in u-
i i ely unde s ood in sandpile models, whe e ene gy 共sand兲
g ains di use only om oppling si es.
To analyze he consequences o Eq. 共15兲, i p o es use ul
o de ine he suscep ibili y
␹
(x
ជ
, )关28兴:
␹
共x
ជ
⫺x
ជ
⬘, ⫺ ⬘兲⫽
冓
␦
␳
a共x
ជ
, 兲
␦
h共x
ជ
⬘, ⬘兲
冔
␩
,共16兲
whe e he symbol
具典
␩
deno es an a e age o e he noise
dis ibu ion. By de ini ion, he suscep ibili y measu es he
a e age inc ease in he numbe o ac i e si es due o an im-
pulsi e pe u ba ion, ha is, o he addi ion o a single en-
e gy g ain. Since we measu e he size o he a alanches by
he o al numbe o opplings, he a e age a alanche size is
gi en by
具
s
典
⫽
冕
ddxd
␹
共x
ជ
, 兲.共17兲
Taking he unc ional de i a i e o Eq. 共15兲and a e aging
o e ime and noise, we ob ain, in he limi →⬁, in which
he sandpile is in a s a iona y s a e wi h cons an a e age
ene gy, he ollowing equa ion o he s a ic suscep ibili y:
D⬜ⵜ⬜
2
␹
共x
ជ
兲⫺2␭
⳵
储
␹
共x
ជ
兲⫺
⑀
␹
共x
ជ
兲⫽⫺
␦
(d)共x
ជ
兲.共18兲
This equa ion can be easily sol ed in Fou ie space. De ining
he ans o ma ion
␹
共x
储
,x
ជ
⬜兲⫽1
共2
␲
兲d
冕
dd⫺1kdq
␹
共q,k
ជ
兲eik
ជ
•x
ជ
⬜eiqx
储
共19兲
and subs i u ing in o Eq. 共18兲, we ob ain he solu ion
␹
共q,k
ជ
兲⫽1
D⬜k2⫹2i␭q⫹
⑀
,共20兲
which yields he suscep ibili y in eal space
␹
共x
储
,x
ជ
⬜兲⫽1
共2
␲
兲d
冕
dd⫺1keik
ជ
•x
ជ
⬜
冕
⫺⬁
⬁dq eiqx
储
D⬜k2⫹2i␭q⫹
⑀
.
共21兲
This in eg al yields he esul , se ing D⬜⫽1:
␹
共x
储
,x
ជ
⬜兲⫽1
2␭
冉
␭
2
␲
冊
(d⫺1)/2
x
储
(1⫺d)/2e⫺x
储
⑀
/2␭e⫺␭x⬜
2/2x
储
.
共22兲
Equa ion 共22兲can be con enien ly ew i en in o he scaling
o m
␹
共x
储
,x
ជ
⬜兲⫽x
储
(1⫺d)/2⌫
冉
x
储
␰
储
,x⬜
␰
⬜
冊
,共23兲
6202 PRE 62ROMUALDO PASTOR-SATORRAS AND ALESSANDRO VESPIGNANI
whe e ⌫is a cu o unc ion ha dec eases exponen ially in
bo h i s a gumen s. Compa ing his las exp ession wi h Eq.
共22兲, we can iden i y he pa allel and ans e sal co ela ion
leng hs
␰
储
⬃
⑀
⫺1,
␰
⬜⬃
⑀
⫺1/2.共24兲
In mo e gene al e ms, i we de ine he exponen s
␯
储
⬘and
␯
⬜
⬘
by Eqs. 共11兲, hen we ha e o di ec ed sandpiles
␯
储
⬘⫽1 and
␯
⬜
⬘⫽1/2. F om hese las exp essions, we can ead o a i s
exac esul o di ec ed sandpiles: he a alanches p oduced
in hose models a e elonga ed, wi h cha ac e is ic leng h in
he pa allel and ans e sal di ec ions ela ed by an a ini y
exponen
␨
⫽1/2. I is e y impo an o s ess ha hese
esul s a e independen o he pa icula model conside ed
and o he dimensionali y do he sys em, dic a ed only by
he ene gy balance in he s a iona y s a e.
We can use he esul 共24兲 o ela e he e ec i e bulk
dissipa ion wi h he sys em size in a model wi h open bound-
a y condi ions. To sus ain a s eady s a e wi h cons an a e -
age ene gy, a alanches mus each he bo om bounda y in
o de o be able o dissipa e. This means ha he cha ac e -
is ic leng h o he a alanches in he pa allel di ec ion mus be
p opo ional o he sys em size
␰
储
⬃L. We ha e he e o e ha
in bounda y dissipa ion models we can de ine an e ec i e
dissipa ion a e
⑀
ha is ela ed wi h he sys em size by
⑀
⬃L⫺1.共25兲
F om his ela ions we easily ind ha ⌬s⫽Dand ⌬ ⫽z.
These iden i ies a e eco e ed in nume ical simula ions 共see
Tables I and II兲. Finally, om Eq. 共20兲, we can eco e he
well-known esul linking he sys em size and he a e age
a alanche size,
具
s
典
⫽
␹
(q⫽0,k
ជ
⫽0)⬃
⑀
⫺1⬃L关10,15,23兴.
B. Aniso opic sandpiles
Ha ing comple ed he analysis o di ec ed sandpiles, we
u n ou a en ion o he mo e complex case o aniso opic
sandpiles. In his kind o model, he anspo o ene gy is
no s ic ly di ec ed in he pa allel di ec ion, bu is simply
s onge in he di ec ion ⫹x
储
han in he opposi e di ec ion
⫺x
储
. The p esence o backwa ds low allows he possibili y
o di usion in he p e e ed di ec ion, and hus he equa ion
o he conse a ion o ene gy becomes in his case
⳵
E共x
ជ
, 兲
⳵
⫽D⬜ⵜ⬜
2
␳
a共x
ជ
, 兲⫹D
储
⳵
储
2
␳
a共x
ជ
, 兲⫺2␭
⳵
储
␳
a共x
ជ
, 兲
⫺
⑀
␳
a共x
ជ
, 兲⫹h共x
ជ
, 兲⫹
␩
E共x
ជ
, 兲.共26兲
F om Eq. 共26兲, we can ob ain he co esponding equa ion
o he suscep ibili y. The solu ion in Fou ie space is eadily
ound o be
␹
共q,k
ជ
兲⫽1
D⬜k2⫹D
储
q2⫹2i␭q⫹
⑀
.共27兲
Upon in eg a ion o e k
ជ
and q, one ob ains he exp ession in
eal space
␹
共x
储
,x
ជ
⬜兲⫽1
共2
␲
兲d
冕
dd⫺1keik
ជ
•x
ជ
⬜
⫻
冕
⫺⬁
⬁dq eiqx
储
D⬜k2⫹D
储
q2⫹2i␭q⫹
⑀
.共28兲
This las in eg al can be pe o med analy ically in d⫽1 and
2. Fo d⬎2, e en hough we do no ha e a closed exp es-
sion, we can ob ain he leading scaling beha io . To simpli y
he calcula ions, we se , wi hou lack o gene ali y, D⬜⫽D
储
⫽1. The in eg a ion in qis done by he me hod o he esi-
dues. The in eg a ion o he k
ជ
angula pa 关38兴yields
␹
共x
储
,x
ជ
⬜兲⫽1
2
冉
␥
2
␲
冊
␯
⫹1
x⬜
⫺
␯
冕
0
⬁dzz
␯
⫹1
⫻J
␯
共
␥
x⬜z兲e⫺x
储
(
␥
冑
1⫹z2⫺␭)
共1⫹z2兲1/2 .共29兲
He e, J
␯
(z) is he i s kind Bessel unc ion o o de
␯
, and
we ha e de ined he cons an s
␯
⫽(d⫺3)/2 and
␥
⫽(␭2
⫹
⑀
)1/2. We a e in e es ed in he beha io o his in eg al o
la ge dis ances, ha is, in he limi x
储
Ⰷx⬜Ⰷ1. In his limi ,
he weigh o he in eg al is gi en by he egion o small z,
since he exponen ial supp esses la ge alues. We can hen
app oxima e he in eg al in he in e al 0⬍z⬍1 and pe o m
a Taylo expansion o he squa e oo in he exponen ial and
he denomina o . In he denomina o , we eadily ha e (1
⫹z2)1/2⯝1. The e m in he exponen ial, howe e , con ains
a cons an e m, and mus be he e o e expanded up o sec-
ond o de :
⫺x
储
共
␥
冑
1⫹z2⫺␭兲⯝⫺x
储
共
␥
关1⫹z2/2兴⫺␭兲
⫽⫺x
储
共
␥
⫺␭兲⫺x
储
␥
z2/2. 共30兲
In he limi
⑀
→0, we ha e
␥
⯝␭, and he cons an
␥
⫺␭can
be expanded o gi e
␥
⫺␭⫽共␭2⫹
⑀
兲1/2⫺␭⯝␭
冉
1⫹
⑀
2␭2
冊
⫺␭⫽
⑀
2␭.共31兲
Subs i u ing hese app oxima ions in o Eq. 共29兲, we a e led o
he exp ession
␹
共x
储
,x
ជ
⬜兲⯝1
2␭
1
共2
␲
兲
␯
⫹1x⬜
(1⫺d)e⫺x
储
⑀
/2␭
⫻
冕
0
⬁dy y
␯
⫹1J
␯
共y兲e⫺(x
储
/2␭x⬜
2)y2,共32兲
whe e we ha e pe o med he change o a iables y⫽
␥
x⬜z
and ex ended again he uppe limi o he in eg al o in ini y
共which is allowed gi en i s exponen ial con e gence兲. The
in eg al in Eq. 共32兲yields 关38兴
␹
共x
储
,x
ជ
⬜兲⯝1
2␭
冉
␭
2
␲
冊
␯
⫹1
x
储
(1⫺d)/2e⫺x
储
⑀
/2␭e⫺␭x⬜
2/2x
储
,
共33兲
PRE 62 6203CRITICAL BEHAVIOR AND CONSERVATION IN . . .