Table Of ContentGranular Matter manuscript No.
(will be inserted by the editor)
7
0
0
J. Javier Brey · M.J. Ruiz-Montero
2
n Average energy and fluctuations of a granular gas in the
a
J threshold of the clustering instability
1
1
]
h
c
e
m Received:date
-
t
a Abstract The behavior of an isolated dilute granular some simple limiting cases have been recently investi-
st gas near the threshold of its clustering instability is in- gated [3; 4].
. vestigated by means of fluctuating hydrodynamics and Here, some results for the total energy fluctuations
t
a the directsimulationMonteCarlomethod.The theoret- in a freely evolving granular gas, below the critical size
m
icalpredictionsfromthe formerareshowntobe ingood for the onset of the clustering or shear instability [5],
- agreement with the simulation results. The total energy willbe reported.The model consideredis a systemofN
d
of the the system is renormalized by the fluctuations of smoothinelastichardspheresofmassmanddiameterσ.
n
the vorticity field. Moreover, the scaled second moment Theinelasticityisassumedtobedescribedbyaconstant,
o
c oftheenergyfluctuationsexhibitsapower-lawdivergent velocityindependent, coefficientofnormalrestitutionα.
[ behavior It is well known that the simplest possible state for this
systemistheso-calledhomogeneouscoolingstate(HCS).
1 Keywords Homegeous cooling state clustering
Atthe levelofhydrodynamics,this state is describedby
v ·
instability critical behavior
9 · a constant number density nh, a vanishing flow velocity,
4 andatime dependenttemperatureTh(t),thatobeysthe
2 equation
1
1 Introduction ∂ T (t)= ζ (t)T (t), (1)
0 t h − h h
7 where ζh[nh,Th(t)] is a cooling rate that must be spec-
0 Granularmaterialsoftenexhibitflowsthataresimilarto ified from a microscopic theory. For the case of hard
/ those found in molecular fluids. In fact, phenomenolog- spheres considered here, there is no microscopic energy
t
a ical hydrodynamic-like equations are frequently used to scale associated with the collision model. Therefore, the
m describe them [1]. Nevertheless, in general, fluctuations temperature dependence of the cooling rate can be de-
- of the hydrodynamic fields are much larger in granular termined by dimensional arguments as ζ (t) T1/2(t).
d systems than in ordinary fluids [2], as a consequence of h ∝ h
TheHCSisunstableagainstlongwavelengthspatialper-
n
the number ofparticlesbeing manyordersofmagnitude
o turbations, leading to the formation of velocity vortices
smallerintheformerthaninthelatter.Thusfluctuations
c andhighdensityclustersofparticles[5].Linearstability
: are expected to play a more significant role in granular analysis of the hydrodynamic equations indicates that
v
flows than in molecular fluid flows. In addition, it must
i this instability is driven by the transversal shear mode
X be realized that a new source of noise and fluctuations
[5; 6]. Using phenomenological Navier-Stokes equations,
ispresentingranularsystems,associatedwiththe local-
r it is found that the criticalwavelengthL beyondwhich
c
a ized character of the energy dissipation. The properties
the system becomes unstable is given by
of this intrinsic noise remain largely unknown, although
2η∗ 1/2
L =2π ℓ . (2)
J.J. Brey c ζ∗ 0
Fisica Te´orica, Universidad de Sevilla, Apartado de Correos (cid:18) (cid:19)
1065. 41080 Sevilla. Spain Here, ℓ0 (nhσ2)−1 is proportional to the mean free
≡
Tel.: +34-954550936 path,
E-mail: [email protected]
ζ (t)ℓ
∗ h 0
ζ , (3)
≡ v (t)
M.J. Ruiz-Montero h
Fisica Te´orica, Universidad de Sevilla, Apartado de Correos and
1065. 41080 Sevilla. Spain η(T )
∗ h
E-mail: [email protected] η , (4)
≡ mn ℓ v (t)
h 0 h
2
where v (t) [2T (t)/m]1/2 is a thermal velocity and The total energy E(t) of the system, either isolated
h h
≡
η(T ) is the shear viscosityofthe granularfluid. For the orwithperiodicboundaryconditions,weareconsidering
h
particular case of a dilute gas described by the inelastic can be expressed as e
Boltzmann equation, which is the case to be considered
in the following, the explicit expressions of η and ζh(t) E(t) N mVi2(t)
are given in [7]. ≡ 2
i=1
Although the initial set up of the clustering instabil- X
e 3 m
ityhasbeenextensivelystudied,muchlessattentionhas = dx N (x,t)T(x,t)+ N (x,t)u2(x,t) ,
x x
2 2
been payed to the behavior of the system as the insta- Z (cid:20) (cid:21)
bility is approachedfrombelow,i.e.asthe linear system e e e e (5)
size L increases towards L . The aim of this paper is
c where V (t) is the velocity of particle i at time t, the
to analyzethe behavior of the totalenergy of ofa dilute i
tilde is used to indicate that the quantities are treated
granulargasofhardspheresinthatregion,byusingfluc-
as stochastic variables, and the x axis has been taken
tuating hydrodynamicsandthe directsimulationMonte
perpendiculartothelayersdescribedabove.Intheequa-
Carlo (DSMC) method [8]. A similar study for a system
tion,N (x,t)isthenumberofparticlesperunitoflength
of harddisks, where the simulations were carriedout by x
in the x direction, u(x,t) is the flow velocity field, and
meansofmoleculardynamics,hasalreadybeenreported
T(x,t)ethetemperature.Thelasttwoquantitiesarecoarse-
elsewhere [9; 10].
grained over the ceells. The microscopic definitions of
tehese fields are
N
2 Fluctuating hydrodynamics and average Nx(x,t)= δ[x Xi(t)], (6)
−
energy i=1
X
e
N
One of the main advantages of the DSMC method [8] N (x,t)u(x,t)= V (t)δ[x X (t)], (7)
x i i
is that it allows to explote the symmetry of the state −
i=1
to be investigated, when dividing the system into cells X
e e
to apply the algorithm. For instance, to study average
properties of the HCS, the system can be forced to stay 3 m N
N (x,t)T(x,t)= [V (t) u(x,t)]2δ[x X (t)],
homogeneous by considering a single cell, the position 2 x 2 i − − i
of the particles then becoming irrelevant. This allows Xi=1
to significantly increase the statistics of the results, de- e e e (8)
creasingthe uncertainty.Of course,by doing this allthe
thatcorrespondtotheidealizedlimitofinfinitelynarrow
spatial hydrodynamic fluctuations and correlations that
layers.Notethat,consistentlywiththediscussionabove,
may occur in the system are by definition eliminated.
only the x coordinate of the particles at time t, X (t),
This implies, in particular, that the clustering instabil- i
appearsinthedefinitionofthecoarse-grainedfields.Ex-
itycannotdevelopanditsinfluence onthepropertiesof
pansionofEq.(5)aroundtheaveragevaluesofthefields
the system can not be studied with a single cell.
in the HCS, Y , retaining up to second order terms in
α,h
Nevertheless, it is possible to consider intermediate
the deviations, δY (x,t) Y Y (x,t), yields
situationswheretheclusteringinstabilityispresent,but α ≡ α,h− α
thesystemisforcedtohavesomekindofsymmetrythat
δE(t) E(t) E (t)
allows to increase the statistical accuracy. This can be ≡ − eh e
1
achieved, in particular, by dividing the system into a = dx 3N δT(x,t)+3δN (x,t)δT(x,t)
given number of parallel layers of the same width. Each e 2e x,h x
Z n
layerisconsideredasacellwhenapplyingtheDSMCal- +mN [δu(x,t)e]2 , e e (9)
x,h
gorithm. Since the positions of particles inside the same
o
cellplaynoroleindeterminingtheircollisionprobability, with E (t) 3NT (t)/2 and N N/L .
it follows that the dynamics is somehow coarse-grained In ohrder≡to elimhienate from txh,eh ≡analysisxthe time de-
over each layer and variations of the properties inside pendence ofthe reference HCS, it is convenientto intro-
it can not be analyzed. On the other hand, variations duce dimensionless length ℓ and time s scales by
of the hydrodynamic fields between different layers, i.e.
x v (t)dt
along the coordinate perpendicular to them, can show h
ℓ= , ds= . (10)
up. Therefore, the clustering instability appears when ℓ0 ℓ0
thelengthofthesystemalongthatdirection,L ,reaches
x Also, dimensionless hydrodynamic fields are defined as
the critical value L . This is the coarse-graineddescrip-
c
tionforwhichthetheorywillbedevelopedinthefollow- δN (x,t)
x
ρ (ℓ,s) , (11)
ing. x ≡ N
x,h
e
3
δu(x,t) field grow in time. On the other hand, for Λ < Λ =
ω(ℓ,s) , (12) x c
≡ v (t) 2π/k , the long time solution of Eq. (17) is
h c
e s
θ(ℓ,s) δT(x,t). (13) ωk,⊥(s)= ds′e(s−s′)λ⊥(k)ξk(ω,⊥)(s′), (19)
≡ Th(t) Z−∞
Moreover,tehe Fourier transform is introduced through with λ⊥(k) = ζ2∗ −η∗k2. Then, by using Eq. (18) it is
obtained:
ρx,k(s)=Z dℓe−ikℓρx(ℓ,s), (14) <ωk,⊥(s)ωk′,⊥(s′)> =−2ΛN2xλη⊥∗k(k2)e(s−s′)λ⊥(k)δk,−k′I
andsimilarlyfor the otherfields. Then,Eq.(5) becomes
(20)
δE(t) and, in particular,
ǫ(t)
≡ E (t)
h Λ2η∗k2
= θ0Λe(s) + Λ12 ρx,k(s)θ−k(s)+ 23|ωk(s)|2 ,(15) <|ωk⊥(s)|2 >=−Nxλ⊥(k). (21)
x x Xk (cid:20) (cid:21) Equation (20) holds for s>s′ 1.
≫
withΛx Lx/ℓ0.Then,tocalculateǫ(t),expressionsfor A Langevin equation for the energy is obtained by
≡
theFouriercomponentsofthefluctuatinghydrodynamic linearizing the macroscopic averageequation for it,
fields are needed. It will be assumed that, at the meso-
d 9
scopiclevelusedhere,theyobeyLangevinequationsob- δE(t)= ζ (T )N dxδT(x,t). (22)
h h x,h
tained by linearizing the Navier-Stokes equations for a dt −4
Z
granular gas around the HCS. Moreover, if attention is
For aenormal fluid, the right haned side identically van-
restricted to the quasi-elastic limit, i.e. α very close to
ishessincethecoolingrateiszero.Then,anequationfor
unity, it can be expected that a good approximation be
the scaled energy fluctuations is easily found,
obtainedbyusingthesameexpressionsfortheproperties
∗
of the noise terms in the Langevin equations as those in dǫ(s) ∗ 3ζ
ζ ǫ(s)= θ (s). (23)
theLandau-Langevinequationsfornormalfluids[11;12]. ds − −2Λ 0
x
ConsidertheFouriertransformoftheflowfield,ω (s).
k The noise term mentioned above, intrinsic to the local
The vorticity field or transverse flow field in a general
energydissipationincollisions,hasbeenomitted,sinceit
situationisbydefinitionitscomponentperpendicularto
isexpectedtogivesmallcontributionsascomparedwith
the vector k, i.e. in the present case perpendicular to
those to be kept, in the quasi-elastic limit. Combination
the x axis. It will be denoted by ωk,⊥(s). The coarse- of Eqs. (15) and (23) yields
grained velocity field ω(ℓ,s) is related with the actual
velocity field w(l,s) by dǫ(s) ǫ(s) 3
∗
= ζ
V∗ ds − (cid:26) 2 − 2Λ2x
dℓ⊥w(ℓ,s)= ω(ℓ,s), (16)
Z Λx ρk,x(s)θ−k(s)+ 2 ωk,⊥ 2 , (24)
where ℓ = r/ℓ0, ℓ⊥ denotes the vector component of ℓ × k (cid:20) 3| | (cid:21))
∗ X
perpendicular to the xaxis,andV is the volume ofthe
systeminthedimensionlessscale,V∗ =V/ℓ3.Usingthis, In the threshold of the clustering instability, the fluctu-
0 ationsofthe transversalcomponents ofthe flowvelocity
it is easily obtained that the Landau-Langevinequation
areexpectedtodominateoverthedensityandthelongi-
for ωk,⊥(s) is
tudinal velocity fluctuations, so the above equation can
∂ ζ∗ be reduced to
∂s − 2 +η∗k2 ωk,⊥(s)=ξk(ω,⊥)(s). (17) ∗
(cid:18) (cid:19) dǫ(s) ζ 2
The termξk(ω,⊥)(s)isa Gaussianwhite noiseverifyingthe ds =− 2 "ǫ(s)− Λ2x k |ωk,⊥|2#. (25)
X
fluctuation dissipation relation
¿From this equation, it follows that the averagevalue of
<ξk(ω,⊥)ξk(ω′,)⊥(s′)>= ΛN2xδ(s−s′)δk,−k′η∗k2I, (18) the total en2ergy of the system is 2 η∗k2
Ibrbaeciknegtsthdeenuontiitngteanvseorragoef odvimerenthsieonno2iseanrdeatlihzeataionngsu.laItr <ǫ>st= Λ2x k h|ωk,⊥(s)|2ist =−N k λ⊥(k). (26)
X X
is worth to remark that the clustering instability mani- This is the resultexpected fromthe expressionobtained
fests itself in Eq. (17), which shows that for k < kc for a system without introducing the coarse-grain aver-
| | ≡
(ζ∗/2η∗)1/2,thefluctuationsofthescaledtransverseflow age over the parallel layers [10].
4
For δL (L L )/L 1 and positive, the main time required to reach it increases very fast as the in-
c x c
≡ − ≪
contribution to Eq. (26) is given by those modes hav- stability is approached. For instance, for α = 0.9, the
ing the lfargest possible wavelength,i.e. those with k = systembecomesstationaryafterabout100collisionsper
| |
k =2π/Λ . For them, it is particle for L /L = 0.88, while it needs of the order of
min x x c
2000 collisions per particle for L /L =0.997. Once the
∗ 2 x c
ζ 2π
∗ system is in the steady state, the different macroscopic
λ⊥(k) =λ⊥(kmin)= η
2 − (cid:18)Λx(cid:19) properties of interest have been computed by taking av-
ζ∗ L 2 erages of the microscopic values. In addition, in all the
= 1 c ζ∗δL. (27) results reported here, the data have also been averaged
2 " −(cid:18)Lx(cid:19) #≃− over 1000 independent trajectories.
It is at this point when the cofarse-grain used in the The height of the cells used in the simulations has
simulations deserves some attention. In a normal simu- been ∆x = 0.04ℓ0, significantly smaller than the mean
lation, for instance using molecular dynamics of a cubic freepathλ=0.225ℓ0,asrequiredbytheDSMCmethod.
wceiltlh, tthhee pnuermiobdeirc obfoumnoddaersywcoitnhditkio=nskshmoinuldcobmep6a,ttibwloe TNh/eLxnu=m2b5e0r0ℓo−0f1p.aTrthieclnesumpberercoelflcwelalsst1o00b,escoonNsixd,here≡d
for each of the axis of the system [10]. Nevertheless, the in each case was determined from the the above figures
coarse-graininthedirectionsperpendiculartothelayers, andthevalueofLx beingsimulated.Itisworthtostress
actually kills all the modes in those directions. Conse- that the number ofparticles does not affectthe effective
quently only two modes with k = k survive. Then, dynamicsusedintheDSMCthatremains,bydefinition,
min
±
from Eqs. (26) and (27), that of an N particle system in the low density (Boltz-
mann) limit [8].
2
ǫ =A (δL)−1, A = , (28) For a given value of α, the series of simulations were
st ǫ ǫ
h i N L
x,h c started with a system size L significantly smaller than
x
where it hafs been used that near the instability it is the critical values predicted by Eq. (2), namely around
N N L . This equation shows that a renormaliza- 0.8 times that value. From there, the size of the system
x,h c
≃
tionbyfluctuationsofthe totalenergyofthe HCStakes was increased systematically and its properties studied
place as the clustering instability is approached.In fact, ineachcase.Specialattentionwasgiventoverifytheho-
it is seen that the energy of the system becomes larger mogeneityofthehydrodynamicpropertiesofthesystem.
than the prediction based on Haff’s law. When inter- Oncetheformationofvelocityvorticeswasobserved,the
preting this theoretical prediction, it must be kept in series was stopped since this is the indication that the
mind the the approximation carried out here only holds size is larger than the critical length Lc.
while the fluctuations are small as compared with their In Figs. 1 and 2 the quantity ǫ −1 is plotted as a
h ist
average values far from the instability. This requires, in function of Λ =L /ℓ for α=0.9 and α=0.8, respec-
x x 0
particular,that the righthandside ofEq.(28) be small. tively. In both cases, an increase of the average energy
relative to the Haff’s law prediction is observed as the
size of the system is decreased approaching the critical
3 DSMC method results value. It must be noted, however, that this increase re-
mains quantitatively small,namely below 10%inall the
The simulation results to be reported in the following parameter region studied.
have been obtained by using the DSMC method, parti- If the theoretical prediction in Eq. (28) is verified,
tioning the system in parallel layers as described above. the above plot must lead to a straight line, and that is
The simulations started with a configuration in which in fact was is obtained when L increases approaching
x
the particles were homogeneously distributed along the thecriticallength.Thisconfirmstheexponent 1inEq.
−
length L of the system and their velocities obeyed a (28). From the slope and the ordinate at the origin, the
x
Maxwellian distribution. In order to increase the num- simulation values of the critical size L and the critical
c
ber of statistical averages and avoid the technical prob- amplitude A are obtained. These fitting parameters as
E
lems inherent to the continuous cooling of the HCS, the well as the theoretical predictions for them are given in
steady representation of the latter was used. This con- Table1.Thetheoreticalpredictionforthecriticallength
sists in a change in the time scale that implies a mod- has been obtained by means of Eq. (2) and the expres-
ification of the dynamics, but does not involve any in- sions for the shear viscosity and the cooling rate in the
ternal property of the system. This dynamics leads to a first Sonine approximation given in [7]. It is observed
steady state whose properties are exactly mapped into that the agreement between the theory and the simula-
those of the HCS. The details of the method have been tion results for the critical size L is really good. This is
c
extensively analyzed elsewhere [13] and will be not re- consistent with the well established result that the first
peated here. Then, the system is allowed to evolve until Sonineapproximationprovidesgoodvaluesfortheshear
thesteadystateisreached.Itisimportanttoverifythat viscosity and the cooling rate of a dilute granulargas in
the system has actually reached stationarity, since the the range of values of α considered here.
5
600 Table 1 Theoretical prediction, L(ct), and values obtained
from the simulation data for the average energy, Lc(ǫ), and
fromthesecondmomentofthefluctuations,L(cσ) forthecrit-
icallengthcharacterizingtheclusteringinstabilityofadilute
three-dimensional granular gas, as a function of the coeffi-
400 cient of restitution α. In all cases, Lc is measured in units
of ℓ0 = (nhσ2)−1. The parameters reported in the last two
−1>st bcoelubmotnhseinqvuoallvteoth2e, acrcictoicradlinagmtpolittuhdeetshAeoǫraentidcaAlσpraenddicsthioonusld.
ε
<
200 α L(ct) Lc(s,ǫ) Lsc,σ L(cǫ)NxA(ǫs) (A(σs)NxL(cσ))2
0.95 7.86 7.93 7.91 2.66 2.08
0.90 5.70 5.71 5.71 2.72 2.78
0.85 4.77 4.75 4.75 2.82 2.43
0.80 4.23 4.19 4.19 2.85 2.68
0 0.75 3.88 3.82 3.82 3.35 2.88
4.8 5 5.2 5.4 5.6 5.8
L /l
x 0
Fig. 1 Dimensionless quantity hǫi−1, defined in the main 4 Critical energy fluctuations
text, as a function of the size of thste system Lx, measured
in units of ℓ0 ≡ (nhσ2)−1, for α = 0.9. The circles are the Consider next the fluctuations around its average value
DSMCresultsandthesolidlinealinearfitneartheclustering
of the total energy of the system, againin the threshold
instability.
of the instability. Since it has been shown above that
the averagevalue of the energy differs from the Haff law
prediction, let us define
400
E(s) E(s)
′
δ ǫ(s) ǫ(s) <ǫ> = −h i, (29)
st
≡ − E (s)
H
e e
′
300 δ ω(s) ω(s) ω(s) st, (30)
≡ −h i
with
2
−1<ε>st200 ω(s)≡ Λ2x Xk |ωk,⊥(s)|2. (31)
Equation (26) shows that < ω > =< ǫ > and, there-
st st
fore, Eq. (25) is equivalent to
100
∗
d ζ
′ ′ ′
δ ǫ(s)= [δ ǫ(s) δ ω(s)]. (32)
ds − 2 −
03.7 3.8 3.9 4 4.1 4.2 Taking into account that cumulants of ωk,⊥ of order
Lx /l0 higher than two vanish since ξ⊥(ω(s) is assumed to be
Gaussian, it follows from Eqs. (18) and (19) that
Fig. 2 The same as in Fig. 1, but for a system of particles
with α=0.8. <δ′ω(s)δ′ω(s′)>= 2 e−2(s−s′)ζ∗δL(δL)−2. (33)
N2 L2
x,h c
Upon deriving this equation, it has beene ufsed that only
modeswithkinthedirectionofthexaxiscanbeexcited,
On the other hand, the comparisonbetween the the- due to the way in which the simulations are carriedout.
oretical prediction and the DSMC result for the critical The solution of Eq. (32), once the initial value has been
amplitude of the energy A is not so good. In Table 1 forgotten, is
ǫ
Ethqe.m(2e8a)siutrsehdouvaldlubeesefoqruaNlxtLocaAcǫonisstgainvte,ni.nAdecpcoenrddienngttoof δ′ǫ(s)= ζ∗ s ds1e−ζ∗(s−s1)/2δ′ω(s1). (34)
2 −∞
the inelasticity,namely 2.Instead,althoughthe orderof Z
magnitudisaccuratelypredicted,definitelylargervalues Then, by means of Eq. (33) it follows that
areobtainedshowing,inaddition,arelevantdependence
on α. Something about the possible origins of this dis- <δ′ǫ(s)δ′ǫ(s′)>= 2 e−2(s−s′)ζ∗δL(δL)−2
N2 L2
crepancyandtheimprovementofthetheorywillbesaid x,h c
at the end of the paper. = <δ′ω(s)δ′ω(s′)>,e f (35)
6
′
s s 1.TermsinvolvingpowersofδLlargerthan 2
≥ ≫ −
havebeenconsistentlyneglectedinthisexpression.Thus 4
boththetotalenergyfluctuationsandtfhefluctuationsof
the energy associated with the transversal modes decay
with acharacteristictime (2ζ∗δL)−1, indicating adiver- 3
−1
gent behavior of the relaxation time as δL . Also, the
ψ
second moment of the energyffluctuations is predicted
′ 2
to diverge. For s = s, Eq. (35) leads tofthe stationary
value
<(δ′ǫ)2 > =A2δL−2, A = √2 . (36) 1
st σ σ N L
x,h c
It is f
0
4.8 5 5.2 5.4 5.6 5.8
<(E(s) <E(s)>)2 >
σ2 − <(δ′ǫ)2 >. (37) L /l
E ≡ <E(s)> ≃ x 0
e e Fig. 3 PlotoftheDSMCresults(circles)forthedimension-
To compare with the DSMC method results, it must be
less quantity defined in Eq. (38) as a function of the size of
realized that Eqe. (36) gives the moment of the energy the system Lx, for α=0.9. The solid line is a linear fit near
fluctuations that dominate near the instability, but in a theclustering instability.
regionwhere the fluctuations are stillsmall. Then,what
hasbeendoneistomeasureσ2 asafunctionofL ,start-
E x α= 0.95, there is a good agreement. The most relevant
ing from values quite smaller than L . There, the value
c
disagreementbetweentheory andsimulations is the def-
of σ is not affected by the presence of the instability,
E
and Nσ2 is independent of L [3]. It will be denoted by inite dependence oninelasticity ofthe parametercombi-
(Nσ2) E. Then, the quantity cxonsidered has been nation given in Table 1 shown by the latter, while the
E h former predicts a constant value (namely, 2).
N L σ2 −1/2 Itcanbewonderedwhichistheaspectofthetheoret-
ψ x,h c E 1 . (38) ical approachdeveloped here that should be modified in
≡ (Nσ2) −
(cid:20) E h (cid:21) order to improve the accuracy of the predicted expres-
The measured values for this quantity are plotted as a sions for the critical amplitudes. Of course, a first im-
function of Lx in Figs.3 and4, for α=0.9 and α=0.8, portant limitation of the theory is its restriction to the
respectively. As predicted by Eq. (36), a linear behav- almost elastic limit, implied by the use of the Landau
ior is observed as the critical size is approached. Al- fluctuation-dissipationrelations.But it must be realized
though the fit is reasonably good for all the values of that up to now those expressions have not been gener-
α reported here, it becomes clearly worse as the inelas- alized for non-conservative interactions. A more modest
ticityincreases.Moreover,the ‘critical’region,identified stepinthisdirectionshouldbetoincludeinEq.(22)the
by the values of Lx for which the predicted qualitative intrinsic noise term associated with the cooling rate, as
critical behavior is actually observed, is smaller for the already indicated.
second moment of the energy than for its average. This Inrefs.[9]and[10],ascalingpropertyoftheprobabil-
canbeverifiedbycomparing,forinstance,Figs.2and4. ity distribution of the energy fluctuations in the thresh-
In contrast to what happened with the average energy, oldoftheclusteringinstabilityofatwo-dimensionalgran-
the growthexhibited by the fluctuations is quite fast.In ular fluid was identified. Moreover, the scaling function
the region plotted in Figs. 3 and 4, the second moment was very well fitted by the same expression as found in
σE increases more than one order of magnitude. several equilibrium and non-equilibrium molecular sys-
¿From the slope and ordinate in the origin of the tems [14; 15]. The possible existence of a similar scaling
fitting straight line, the simulation values of the crit- for the three-dimensional granular gas considered here
ical length Lc and amplitude Aσ are directly derived. has also been investigated, finding similar results.
The values obtained in this way are also included in Ta-
ble 1. Again, the results for the critical length L are
c Acknowledgements This research was supported by the
in excellent agreement with the theoretical predictions.
Ministerio de Educaci´on y Cienc´ıa (Spain) through Grant
Moreover,the fact that the same values follow from the No. FIS2005-01398 (partially financed by FEDER funds).
measurementsofboththeaverageenergyandthesecond
momentsprovidesatestoftheinternalconsistencyofthe
theory.With regardto the criticalamplitude A , signif-
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