Mesoscopic Theo y o C i ical Fluc ua ions in Isola ed G anula Gases
J. Ja ie B ey,*A. Domı
´nguez, M. I. Ga cı
´a de So ia, and P. Mayna
Fı
´sica Teo
´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080 Se illa, Spain
(Recei ed 24 No embe 2005; published 21 Ap il 2006)
Fluc ua ing hyd odynamics is used o desc ibe he o al ene gy luc ua ions o a eely e ol ing gas o
inelas ic ha d sphe es nea he h eshold o he clus e ing ins abili y. They a e shown o be go e ned only
by o ici y luc ua ions ha also lead o a eno maliza ion o he a e age o al ene gy. The heo y p edic s
a powe -law di e gen beha io o he scaled second momen o he luc ua ions, and a scaling p ope y o
hei p obabili y dis ibu ion, bo h in ag eemen wi h simula ions esul s. A mo e quan i a i e compa ison
be ween heo y and simula ion o he c i ical ampli udes and he o m o he scaling unc ion is also
ca ied ou .
DOI: 10.1103/PhysRe Le .96.158002 PACS numbe s: 45.70.n, 05.20.Dd, 51.10.+y
G anula gases a e assemblies o mac oscopic pa icles
e ol ing independen ly be ween inelas ic collisions [1].
The me hods o nonequilib ium s a is ical mechanics, ki-
ne ic heo y, and hyd odynamics ha e been success ully
ex ended o desc ibe he obse ed mac oscopic beha io
and also, al hough in a much mo e limi ed o m, he
luc ua ions a ound i [2]. The lack o ene gy conse a ion
makes hese sys ems beha e qui e di e en ly om mo-
lecula luids. A simple widely used model o hem con-
sis s o smoo h inelas ic ha d sphe es (IHS’s), wi h mo-
men um conse ing dynamics. Inelas ici y is cha ac e ized
by means o a cons an coe icien o no mal es i u ion .
Recen ly, molecula dynamics (MD) simula ion esul s
ha e been epo ed o he o al ene gy luc ua ions o a
wo-dimensional eely e ol ing IHS gas, nea he h esh-
old o he clus e ing ins abili y [3]. The dimensionless
second momen was ound o exhibi a powe -law di e -
gen beha io wi h he dis ance o he ins abili y. Also, he
scaled cooling a e was ound o end o ze o acco ding o a
powe law, al hough in a weak way. Besides, he dis ibu-
ion unc ion o he ene gy luc ua ions, when p ope ly
scaled, u ned ou o be independen o he pa ame e s
de ining he sys em. This was associa ed wi h a scaling
p ope y o he dis ibu ion. Qui e ema kably, he scaling
unc ion was e y well i ed by he same exp ession as
se e al equilib ium and nonequilib ium molecula sys ems
[4,5]. The main goal o his Le e is o p o ide an expla-
na ion o he abo e esul s on he basis o luc ua ing
hyd odynamics [6].
Conside an isola ed sys em o NIHS’s o mass mand
diame e . The o al (kine ic) ene gy ~
Eo he sys em can
be exp essed in he o m [7]
2~
E Zd d~
n ; ~
T ; m~
n ; ~
u2 ; ;(1)
whe e dis he dimension o he sys em, ~
n ;  he numbe
densi y ield, ~
T ;  he empe a u e ield, and ~
u ;  he
low ield. The ildes indica e ha all he quan i ies a e
unde s ood as luc ua ing a iables. In he ollowing, sys-
ems in he homogeneous cooling s a e (HCS) will be
conside ed. A a mac oscopic le el, his s a e is cha ac e -
ized by a cons an uni o m densi y nH, a anishing low
ield uH0, and a uni o m ime-dependen empe a u e
obeying he law [8] @ TH HTHTH , whe e H/
TH 1=2is he cooling a e. Mo eo e , we will es ic
ou sel es o he egion in which he ampli udes o he
luc ua ions o he ields a ound hei HCS alues emain
small on he a e age [see below Eq. (12)]. Then e aining
up o quad a ic o de in he de ia ions, Eq. (1) yields
~
E ~
E EH 
1
2Zd dnH~
T ; d~
n ; ~
T ; 
mnHj~
u ; j2:(2)
He e, EH dNTH =2,~
n ; ~
n ; nH,
~
u ; ~
u ; , and ~
T ; ~
T ; TH .I is
now con enien o in oduce dimensionless posi ion, l,
and ime, s, scales by l =l0and ds  H d =l0, e-
spec i ely, whe e H2TH =m1=2is he he mal e-
loci y and l0nHd11is p opo ional o he mean
ee pa h. Mo eo e , dimensionless ields a e de ined by
l;s~
n ; =nH,!l;s~
u ; = H , and
l;s~
T ; =TH . Then, Eq. (2) akes he o m
s0s
V1
V2X
kksks2
dj!ksj2;(3)
whe e s~
Es=EHs,VLdis he olume o he
sys em in he new uni s, and he Fou ie ans o ms o he
ields ha e been in oduced. I is assumed ha a e a ime
o he o de o he mean ee ime, he sys em eaches a
egime in which all i s ene gy is s o ed in he hyd ody-
namic modes. In his egime, he hyd odynamic ields a e
expec ed o be desc ibed a a mesoscopic le el by luc ua -
ing hyd odynamic equa ions. He e, hey will be assumed o
be linea Lange in equa ions ob ained by linea izing he
Na ie -S okes equa ions o a g anula gas a ound he
HCS. Mo eo e , i is pos ula ed ha he noise e ms a e
de ined by he same p ope ies as o molecula , elas ic
PRL 96, 158002 (2006) PHYSICAL REVIEW LETTERS week ending
21 APRIL 2006
0031-9007=06=96(15)=158002(4)$23.00 158002-1 ©2006 The Ame ican Physical Socie y
gases [9]. This is no expec ed o be ue, excep in he
nea ly elas ic limi , i.e., when is e y close o uni y.
Consequen ly, he heo y will be es ic ed in he ollowing
o his limi . Thus, he ans e sal low ield o o ici y
ield, !k?, obeys he ollowing equa ion in he scaled
a iables [6,9]:
@s=2k2!k?sk?s:(4)
In his exp ession, HTH l0= H and 
HTH =mnHl0 H ,Hbeing he shea iscosi y.
The noise e m k?sis Gaussian, wi h
hk?sk0?s0i  V2
Nss0k;k0k2l;(5)
lbeing he uni enso in he subspace pe pendicula o k,
and he angula b acke s deno ing a e age o e he noise
ealiza ions. A main ad an age o using he scaled a ia-
bles is ha he coe icien s in Eq. (4) and he s eng h o he
noise a e ime independen , con a y o wha happens in he
o iginal a iables. The equa ion shows ha !k?g ows in
ime o hose alues o ksuch ha ?k=2
k2>0. Al hough his does no imply by i sel ha he
HCS is linea ly uns able, due o he ime-dependen scaling
o he eloci y in oduced abo e, simula ion esul s and
nonlinea analy ical analysis o he Na ie -S okes equa-
ions ha e shown ha his g ow h is he o igin o he
clus e ing ins abili y [10,11]. The minimum alue o k
o a sys em o linea ex en L, measu ed in he lscale,
is kmin 2=L. Then, o gi en alues o he o he pa-
ame e s, he sys em becomes uns able i L>L
c, wi h
Lc22=1=2.Fo L<L
c, he HCS is s able and
he long ime solu ion o Eq. (4) is
!k?sZs
1
ds0ess0?kk?s0:(6)
F om his exp ession, i is easily ob ained
h!k?s!k0?s0i   V2k2
2N?kess0?kk;k0l;(7)
o ss01. This shows ha as Lapp oaches Lc om
below, he ampli udes o he luc ua ions o he ans e sal
componen s o he eloci y inc ease e y as due o con-
ibu ions om alues o kclose o kc. Fo he same eason
he decay o hese luc ua ions becomes e y slow. This is
no he case o he luc ua ions o he o he hyd odynamic
ields, whose Lange in equa ions a e decoupled om
Eq. (4) [6]. The e o e, i seems possible o conside a ange
o alues o
L LcL=Lcwhe e he luc ua ions o
!k;?, al hough s ill small, domina e o e he luc ua ions
o densi y and empe a u e. Bu , al hough his is ue o
componen s wi h k>0, some ca e is needed when analyz-
ing Eq. (3), since i in ol es 0s. The Lange in equa ion
o sis ob ained om he linea iza ion a ound he HCS
o he mac oscopic a e age equa ion o he o al ene gy,
@ E d
2Zd n ; Hn; TT ; :(8)
The esul is
@sss 3
2V0s:(9)
He e, he dependence o he cooling a e on he empe a-
u e has been aken in o accoun . Mo eo e , he noise e m
discussed in Re . [12], associa ed wi h he localized cha -
ac e o he ene gy dissipa ion, has been omi ed. Al hough
i can be expec ed o be negligible a om he ins abili y in
he quasielas ic limi , his may no be he case nea he
ins abili y. Equa ion (9) shows a coupling be ween he
luc ua ions o he olume a e aged empe a u e and hose
o he o al ene gy. Use o Eq. (3) in o Eq. (9) and neglec -
ing con ibu ions om he densi y and longi udinal eloc-
i y luc ua ions gi es
@ss
2s !s;
!s 6
V2dX
kj!k?sj2;
(10)
alid in he egion
L 1. The long ime limi o he
a e age alue o sis, he e o e,
his lim
s!1h!si  3d1
Nd X
k
k2
?k:(11)
Since we a e conside ing
L 1, he sum o e kin he
abo e exp ession is domina ed by he 2dmodes wi h he
la ges wa eleng h, o which ?kmin’
L. Using
his in o Eq. (11), i ollows ha he e is a eno maliza ion
by luc ua ions o he a e age o al ene gy o he HCS,
E h~
E i, gi en by
E EH 13d1
nHLd
c
L1:(12)
Consis ency o he heo y we a e de eloping equi es ha
nHLd
c
L11, a condi ion in ol ing he inelas ici y
and he dis ance o he ins abili y. Simila ly, he e is also
a eno maliza ion o he empe a u e o he HCS, T 
h~
T is , ha can be e alua ed di ec ly om he long ime
limi o he a e age o Eq. (9),
T TH 1h0is
VTH 12d1
nHLd
c
L1:
(13)
Al e na i ely, an e ec i e empe a u e Te  can be de-
ined as Te  2E =Nd. O cou se, he o m o he
eno malized law o he empe a u e depends on he de i-
ni ion used o he la e . In Re ., [3], wha was ac ually
measu ed was 
e e Te l0= HTe , wi h e de ined by
@ Te e Te Te . Then, i is ound
PRL 96, 158002 (2006) PHYSICAL REVIEW LETTERS week ending
21 APRIL 2006
158002-2

e "13d1
nHLd
c
L1#1=2
:(14)
This esul p edic s ha nea he clus e ing ins abili y
h eshold, 2
e 2=2A
L1wi h A3d
1=nHLd
c, ha is jus he beha io obse ed in Re . [3].
De ine
Es~
EsEs=Esshis 
EHs=Esand !s !sh!is . No e ha we a e
conside ing de ia ions om he eno malized a e age al-
ues, i.e., including he luc ua ions e ec s, and no om he
mac oscopic ba e alues. A s anda d calcula ion using
Eq. (7) and exploi ing he Gaussian cha ac e o he noise,
gi es ha a he ins abili y h eshold and o ss01i is
h!s!s0is 9d1
n2
HL2d
cd
L2ess0=sc;(15)
whe e sc2
L1is a di e gen ‘‘c i ical’’ elaxa ion
ime. Now Eq. (10) can be easily sol ed wi h he esul
h
Es
Es0is h!s!s0is ;(16)
alid o ss01. Thus below he ins abili y, he scaled
o al ene gy luc ua ions decay wi h he same a e as he
luc ua ions o he kine ic ene gy associa ed wi h he ans-
e sal modes o he eloci y. Fo ss0, Eq. (16) yields
2
Eh
E2is A2

L2;(17)
wi h A2
9d1=n2
HL2d
cd. The e o e, close o he in-
s abili y poin , he ela i e dispe sion o he o al ene gy
luc ua ions Ep esen s a di e gen beha io wi h a c i i-
cal exponen 1, and an ampli ude Adepending on nH
and ( h ough he alue o he c i ical leng h Lc). Again,
his is he same beha io as epo ed in Re . [3] om MD
simula ions.
To ca y ou a mo e de ailed check o he heo y p e-
sen ed he e, we ha e pe o med MD simula ions o wo-
dimensional sys ems wi h di e en alues o and nH(see
Table I). In all cases, he dependence on
L o bo h he
cooling a e and he dispe sion o he o al ene gy, i.e., he
exponen s in he powe laws (14) and (17), was in ag ee-
men wi h he heo e ical p edic ions. This was illus a ed
in Figs. 1 and 2 o Re . [3] and no mo e de ails will be
gi en he e. The compa ison be ween he p edic ed c i ical
ampli udes and he MD esul s gi en in Table I can be
conside ed as sa is ac o y, in he sense ha he heo y
co ec ly p edic s he o de o magni ude o he ampli udes,
especially aking in o accoun he smallness o he quan i-
ies being measu ed.
Nex , le us p oceed o in es iga e he o m o he
p obabili y dis ibu ion o he ene gy luc ua ions.
Pa icula iza ion o Eq. (4) o he modes wi h he smalles
possible alue o kin he limi
L 1gi es
@s
L!k?sk?s;(18)
whe e i is unde s ood ha jkjkmin. De ine a new ime
scale d 
Lds, and a new ans e sal eloci y ield by
!
k?!k?=Ld
c1=2
E. Equa ion (18) becomes
@1!
k?
k?;(19)
wi h
h
k?
k0?0i  d1=2
6d11=2k;k00l:(20)
Equa ion (19) implies ha he p obabili y dis ibu ion o
!
k?wi h jkjkmin nea he clus e ing ins abili y depends
only on he dimension do he sys em. In ac , since he
noise e m 
k?is Gaussian, i is i ial o w i e he long
ime o m o his dis ibu ion using Eq. (7) wi h ss0,
Ps !
k?22
!d1=2e!2
k?=22
!;(21)
wi h 2
!d1=2=12d11=2. In he ime scale , and
keeping only he dominan modes, Eq. (10) eads
L@y1
2y6
dX
jkjkminj!
?kj2;(22)
whe e y=Eand he sum is es ic ed o ec o s kwi h
jkjkmin. F om he compa ison o Eqs. (19) and (22) i is
seen ha , on he scale and in he h eshold o he
ins abili y, ydecays much as e han he dominan com-
ponen s o !
k?. Consequen ly, o la ge  he solu ion o
Eq. (22) is y6
dPjkjkmin j!
?kj2, whe e he p obabili y
dis ibu ion o he modes !
?kis gi en by Eq. (21). Since
he la e does no depend on he pa ame e s o he sys em
o he han he dimensionali y, he same p ope y ollows
o he p obabili y dis ibu ion o bo h yand he a iable
E
Edd11=26
dX
jkjkminj!
k?j2:(23)
TABLE I. Compa ison be ween he p edic ed and MD alues
o he c i ical ampli udes o he cooling a e Aand he o al
ene gy dispe sion A. All he alues o he ampli udes ha e been
mul iplied by 103.
nH2A
heo y
AMD
A heo y
AMD

0.02 0.9 0.88 1.11 0.62 0.6
0.02 0.8 1.62 3.63 1.15 1.5
0.1 0.98 1.06 0.50 0.75 0.47
0.1 0.95 2.4 2.38 1.7 1.45
0.2 0.98 1.97 2.34 1.4 1.34
0.2 0.95 4.59 7.78 3.24 3.6
PRL 96, 158002 (2006) PHYSICAL REVIEW LETTERS week ending
21 APRIL 2006
158002-3
This is equi alen o saying ha he p obabili y dis ibu ion
o
E e i ies he scaling ela ion
P
E 1
E

E
E;(24)
whe e is a scaling unc ion. This is jus he p ope y
assumed in Re . [3] and e i ied by MD simula ions. Since
he p obabili y dis ibu ion unc ion o !
k?is known, i is
possible o nume ically gene a e he p obabili y dis ibu-
ion unc ion P
E. The esul is shown in Fig. 1. Also
plo ed is he unc ion

EKexexa;xbs
E;a=2;
(25)
wi h K2:14,b0:938, and s0:374, ha i s ex-
emely well he MD esul s o EP
E[3]. I is impo -
an o ema k ha luc ua ions in a la ge numbe o
equilib ium and nonequilib ium sys ems exhibi ing sel -
o ganized c i icali y as well as con ined u bulen lows
p esen he same kind o beha io [4,5]. Al hough he
ag eemen be ween bo h plo ed cu es is no so bad o
posi i e alues o
E, s ong disc epancies a e obse ed
o nega i e alues. A majo sou ce o hem is easily
iden i ied om Eq. (23), ha o d2implies
E=E
2
p, while smalle alues a e ound in he MD simula-
ions. Since Eq. (23) is a consequence o Eq. (9), i seems
plausible ha in o de o elabo a e a mo e accu a e heo y
he in insic noise associa ed wi h he cooling a e mus be
aken in o accoun .
In summa y, we ha e de eloped a mesoscopic heo y o
he luc ua ions o he o al ene gy o an isola ed g anula
gas nea he h eshold o he clus e ing ins abili y. The
heo y desc ibes accu a ely he quali a i e beha io ob-
ained in MD simula ions, namely, he di e gen beha io
o he dimensionless second momen and he dec ease o
he appa en cooling a e. Also, i is consis en wi h he
obse ed scaling p ope y o he p obabili y dis ibu ion
unc ion o he luc ua ions. On he o he hand, i seems
clea ha a mo e e ined o mula ion is needed in o de o
ge a mo e sa is ac o y quan i a i e ag eemen , especially
o he dis ibu ion unc ion. Also, i should be in e es ing
o check whe he a simila beha io occu s in mo e eal-
is ic models o g anula gases in which depends on he
ela i e eloci y and, al hough p esen , he clus e ing in-
s abili y seems o be a ansien phenomenon [13].
This esea ch was suppo ed by he Minis e io de
Educacio
´n y Ciencia (Spain) h ough G an No.
FIS2005-01398 (pa ially inanced by FEDER unds).
*Elec onic add ess: [email p o ec ed]
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-6
-4
-2
0
-2 0 2 4
ln [σE P(δ E)]
δ E
~
~
/ σE
FIG. 1. P obabili y densi y unc ion o he ela i e o al ene gy
luc ua ions EP
E o a sys em o inelas ic ha d disks. The
b oken line is he heo e ical p edic ion de i ed in his Le e and
he solid line Eq. (7).
PRL 96, 158002 (2006) PHYSICAL REVIEW LETTERS week ending
21 APRIL 2006
158002-4