G een-Kubo ep esen a ion o he iscosi y o g anula gases
J. Ja ie B ey
Á ea de Física Teó ica, Uni e sidad de Se illa. Apa ado de Co eos 1065, E-41080, Se illa, Spain
Abs ac . The G een-Kubo ep esen a ion o he shea iscosi y o a dilu e gas o inelas ic ha d disks de i ed om he
Bol zmann equa ion is e alua ed by means o he di ec simula ion Mon e Ca lo me hod. The ela ionship be ween he one-
pa icle dynamics o he o iginal exp ession and he N-pa icle dynamics needed o he simula ion is analyzed. The p esence
o eloci y co ela ions in he la e is iden i ied and hei possible implica ions a e discussed.
INTRODUCTION
The p o o ypical model o g anula gases is an ensemble o smoo h inelas ic ha d sphe es o disks wi h a cons an
coe icien o no mal es i u ion
α
. Fo his model, he me hods o kine ic heo y ha e been applied and, in pa icula ,
he Bol zmann equa ion has been ex ended [1, 2]. Then, by using a modi ica ion o he Chapmann-Enskog p ocedu e,
hyd odynamicequa ions ha e been de i ed [3, 4]. As he e e ence s a e, he iso opic and homogeneouscooling s a e
(HCS) o a g anula sys em is used. Due o he ene gy dissipa ion, he HCS is no s a iona y, bu i s ene gy dec eases
mono onically. Hyd odynamics has been employed wi h success o desc ibe he beha io o g anula lows.
As o molecula sys ems, he Chapmann-Enskogme hod leads o complica ed di e en ial equa ions ha ha e o be
sol ed in some app oxima ions.I has been shown ha he a o emen ionedequa ions a e equi alen o a ep esen a ion
o he anspo coe icien s ha has a simila s uc u e o he G een-Kubo ela ions o elas ic sys ems, al hough wi h
signi ican di e ences [5]. Equi alen exp essions ha e been de i ed by iden i ying he hyd odynamic pa o he
spec um o he linea ized Bol zmann equa ion and assuming i domina es o long long imes and wa eleng hs [6, 7].
He e, an e alua ion o he G een-Kubo ela ion o he shea iscosi y by means o N-pa icle simula ions, namely
he di ec simula ion Mon e Ca lo (DSMC) me hod [8], is p esen ed. This equi es going om he one-pa icle
e ec i e dynamics, as de ined by a linea ized Bol zmann ope a o , o a supposed equi alen N-pa icle dynamics.
The ela ionship be ween bo h desc ip ions will u n ou o be impo an o he analysis o he esul s. The exis ence
an exac mapping o he HCS on o a s eady s a e will be exploi ed as he basis o he simula ion me hod [9, 10].
THE GREEN-KUBO EXPRESSION FOR THE SHEAR VISCOSITY
By using he Chapman-Enskog me hod o eigen unc ions expansions [5, 6], he shea iscosi y
η
o a dilu e g anula
gas o Nsmoo h inelas ic ha d disks (d=2) o sphe es (d=3) o mass mand diame e
σ
can be exp essed in he o m
η
(T) = nm` 0(T)Z∞
0dse−s
ζ
0/2Zdc1
χ
HCS(c1)∆xy(c1,s)Φ2,xy(c1).(1)
He e nis he densi y, T he empe a u e, 0(T) = (2kBT/m)1/2wi h kB he Bol zmann cons an , `= (n
σ
d−1)−1, and
∆xy(c) = cxcy,Φ2,xy(c) = −cx
∂
ln
χ
HCS(c)
∂
cy.(2)
The unc ion
χ
HCS(c1)is de ined om he solu ion o he Bol zmann equa ion desc ibing he HCS, HCS( 1, ), as [1]
HCS( 1, ) = n −d
0[T( )]
χ
HCS(c1),c1= 1
0[T( )] ,(3)
815
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G een-Kubo ep esen a ion o he iscosi y o g anula gases
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S anda d Fo m 298 (Re . 8-98)
P esc ibed by ANSI S d Z39-18
and i is an iso opic unc ion o he ec o c1. Finally, he ime dependence o ∆xy is gi en by
∆xy(c1,s) = es¯
Λc(c1)∆xy(c1),(4)
¯
Λc(c1) = Zdc2
χ
HCS(c2)Zdb
σ
Θ(c12 ·b
σ
)c12 ·b
σ
[b
σ
(c1,c2)−1](1+P12)+
ζ
0
2c1·
∂
∂
c1,(5)
whe e c12 =c1−c2,Θis he Hea iside s ep unc ion, b
σ
is he uni ec o poin ing om he cen e o pa icle 2 o he
cen e o pa icle 1 a con ac , he ope a o P12 in e changes he subindexes 1 and 2 o i s igh , and b
σ
(c1,c2)is an
ope a o eplacing all he eloci ies c1and c2appea ing o i s igh by he pos collisional alues c0
1and c0
2gi en by
c0
1≡b
σ
c1=c1−1+
α
2(b
σ
·c12)b
σ
,c0
2≡b
σ
c2=c1+1+
α
2(b
σ
·c12)b
σ
.(6)
The las e m on he igh hand side o Eq. (5) con ains he ime-independen educed cooling a e o he HCS,
ζ
0,
de ining he e olu ion o he empe a u e in ha s a e,
∂
T( ) = −
ζ
HCS(T)T,
ζ
0=`
ζ
HCS
0(T).(7)
Equa ion (1) di e s om he s anda d G een-Kubo exp ession o dilu e molecula sys ems in se e al aspec s:
•The a e age is aken o e he eloci y dis ibu ion o he HCS and no o e he Maxwellian cha ac e izing he
equilib ium s a e. Bo h di e o
α
<1.
•The ime co ela ion unc ion couples he ans e sal momen um lux ∆xy and a “modi ied lux” Φ2,xy.
•The dynamics includes an accele a ing s eaming be ween collisions.
•The ime in eg al con ains, in addi ion o he co ela ion unc ion, an exponen ially dec easing in ime ac o .
To ende Eq. (1) sui able o e alua ion by means o N-pa icle simula ions, some issues mus be add essed.
Fi s , i p esen s he echnical di icul y ha Λcin ol es he cooling a e
ζ
0 ha , he e o e, mus be known a p io i.
Consequen ly, i is use ul o make a change o scale [9, 10],
τ
=
ζ
0
2
ω
0s,w=2
ω
0`
ζ
0c(8)
wi h
ω
0being an a bi a y ime-independen dimensionless equency. Then, Eq. (1) ans o ms in o
η
(T) = nm 0(T)
e 0,s NZ∞
0d e−
ω
0 h∆xy( , )Φ2,xy ( /e 0,s )is .(9)
He e, we ha e enamed he ime and eloci y a iables, e 0,s ≡(2kBe
Ts /m)1/2whe e
e
Ts =2m
kB
ω
0`
ζ
02
,(10)
and he angula b acke s deno e a e age de ined by
ha( , )b( )is ≡Zd Zd e
s ( )a( , )b( ),e
s ( ) = ne −d
0,s
χ
HCS 
e 0,s .(11)
Mo eo e , he ime dependence is now gi en by
a( , ) = e ¯
Λs ( )a( )(12)
wi h
¯
Λs ( 1) =
σ
d−1Zd 2e
s ( 2)Zdb
σ
Θ( 12 ·b
σ
) 12 ·b
σ
[b
σ
( 1, 2)−1](1+P12)+
ω
0 1·
∂
∂
1.(13)
816
N-PARTICLE REPRESENTATION
In Eq. (9), he dynamics is de ined by means o he linea ope a o e
Λs . Be o e compu a ing i by N-pa icle simula ion
me hods, i mus be ew i en in e ms o a well de ined pa icle dynamics. The s uc u e o e
Λs sugges s o in oduce
an accele a ion s eaming be ween collisions,
∂
∂
Ri( ) = Vi( ),
∂
∂
Vi( ) =
ω
0Vi( ),(14)
while he e ec o a collision be ween pa icles iand jis o ins an aneously modi y hei eloci ies acco dingly wi h
he same ules as in he o iginal dynamics and gi en in Eqs. (6), i.e.
Vi→V0
i≡b
σ
Vi=Vi−1+
α
2(b
σ
·Vij)b
σ
,Vj→V0j≡b
σ
Vj=Vj+1+
α
2(b
σ
·Vij)b
σ
.(15)
Fo his dynamics, he Liou ille equa ion can be ob ained. Then, by means o any o he s anda d p ocedu es, he
Bol zmann equa ion can be de i ed in he low densi y limi . I has he o m

∂
∂
+
ω
0
∂
∂
· + ·
∂
∂
 ( , , ) = J[ , ],(16)
whe e J[ , ]is he same Bol zmann collision ope a o as in he o iginal dynamics. I is now easily e i ied ha e
s ( )
is a s eady solu ion o his equa ion [10]. Le us assume ha he Liou ille equa ion has a s a iona y solu ion
ρ
s (Γ),
whe e Γdeno es a poin in phase space. A su icien condi ion o his is ha he dis ibu ion unc ion o he HCS in
he ac ual, o iginal dynamics has he scaling p ope y
ρ
HCS(Γ, ) = ( 0[T( )])−dN
ρ
∗
HCS ({Rij,Vi/ 0[T( )]}).(17)
Conside ime co ela ion unc ions o he o m
CAB,s ( ) = hA( )BiN,s −hAiN,s hBiN,s ,(18)
whe e
hAiN,s =ZdΓ
ρ
s (Γ)A(Γ),hBiN,s =ZdΓ
ρ
s (Γ)B(Γ),(19)
hA( )BiN,s =ZdΓ
ρ
s (Γ)A(Γ, )B(Γ),(20)
wi h
A(Γ) =
N
∑
i=1a(Vi),B(Γ) =
N
∑
i=1b(Vi).(21)
The dynamical a iable A(Γ, )≡A[Γ( )] is gene a ed om A(Γ) h ough he dynamics de ined abo e.The low densi y
limi o CAB,s can be ob ained by using he same p ocedu es as o molecula sys ems [10]. The hypo hesis needed
a e he same as hose equi ed o de i e he Bol zmann equa ion [11], plus he addi ional assump ion ha eloci y
co ela ions a e negligible in he HCS a low densi ies. Then, i is ound ha CAB,s is gi en by Eq. (11). This p o ides
a di ec way o e alua ing he la e exp ession, and he e o e he shea iscosi y, by means o N-pa icle simula ions,
since he dynamics de ined by Eqs. (14) and (15) is easily implemen ed. O cou se, he equali y
CAB,s =ha( , )b( )is (22)
only holds a low enough densi ies and as long as eloci y co ela ions e ec s a e negligible in he HCS. The DSMC
me hod is a e y use ul ool o gene a e he low densi y dynamics o a sys em [8, 12]. This me hod is designed as a
eal N-pa icle dynamics simula ion o a low densi y gas, consequen ly p o iding i s comple e dynamical desc ip ion.
817
012345
−6
−4
−2
0
(ln Χ)’
FIGURE 1. Plo o (lnX)0≡
∂
ln
χ
HCS/
∂
cas a unc ion o c= /e 0,s o
α
=0.6. The ci cles a e he nume ical de i a i e o
he simula ion esul s and he solid line he i ed unc ion. The eloci y is measu ed in he uni s de ined in he ex .
SIMULATION RESULTS
We ha e pe o medDSMC simula ions o a sys em o N=104inelas ic ha d disks. Since we a e dealing wi h posi ion
independen quan i ies in a homogeneous s a e, i is enough o conside one cell in con igu a ionspace. This allows o
inc ease hes a is ics and alsoa oids ha he sys em spon aneouslyde elopsspa ial inhomogenei ies.This is impo an
since he HCS is uns able wi h espec o spa ial long wa eleng h pe u ba ions [13, 14].
We in oduce a dimensionless educed shea iscosi y
η
∗by scaling he iscosi y wi h i s elas ic limi in he i s
Sonine app oxima ion
η
0,
η
∗=
η
(T)
η
0(T),
η
0=1
2
σ
mkBT
π
1/2
.(23)
Then, Eq. (9) yields
η
∗=2√2
π
`e 0,s Z∞
0d J
η
( )e−
ω
0 ,(24)
wi h
J
η
( ) = 1
Nh∆xy( , )Φ2,xy( /e 0,s )is .(25)
Le us emind ha wha is ac ually compu ed in he N-pa icle desc ip ion is
J0
η
( ) = 1
N
N
∑
i
N
∑
jh∆xy( i, )Φ2,xy( j/e 0,s )iN,s .(26)
The simula ions show ha he sys em eaches, a e a ansien pe iod, a s a iona y s a e wi h a empe a u e e
Ts . Then,
we measu ed he eloci y unc ion
∂
ln
χ
HCS(c)/
∂
c,c= /e 0,s , o de e mine he modi ied lux Φ2,xy(c), de ined in Eq.
(2). Fo his, he ange o was pa i ioned in o small non-o e lapping bins and he equency dis ibu ion was buil .
A e wa ds, he nume ical de i a i e was compu ed. An example o he esul s, o
α
=0.6, is gi en in Fig. 1. He e
and in he ollowing, he uni o mass is m, he uni o leng h is `= (n
σ
)−1, and he uni o ime is `[2kBe
T(0)/m]−1/2,
whe e e
T(0)is he ini ial empe a u e. Mo eo e , we se kB=1, implying ha in ou uni s i is e
T(0) = 1/2. Fo la ge
eloci ies,
∂
ln
χ
HCS(c)/
∂
c ends o a cons an , e lec ing he exponen ial decay o
χ
HCS(c)[15].
Once he unc ion Φ2,xy has been de e mined o each o he alues o
α
conside ed, he co ela ion unc ion J0
η
( )
has been measu ed. I is seen o decay exponen ially in ime, a leas o e mo e han wo decades, o all he alues o
he es i u ion coe icien [16]. Then, J0
η
( )was i ed o an exponen ial, J0
η
( ) = J0
η
(0)e−
λη
, and he i ed pa ame e s
818

0.1 0.3 0.5 0.7 0.9
α
1.0
1.2
1.4
1.6
η∗
FIGURE 2. The educed coe icien o shea iscosi y
η
∗ o a sys em o inelas ic disks as a unc ion o he coe icien o no mal
es i u ion
α
. The symbols a e om he DSMC me hod and he solid line is he i s Sonine app oxima ion.
J0
η
(0)and
λη
we e de e mined. In e ms o hem, Eq. (24) eads
η
∗=2√2
π
J0
η
(0)
λη
+
ω
0.(27)
The alues o
η
∗ob ained in his way a e shown in Fig. 2. A good ag eemen is obse ed wi h he esul s ob ained
in he i s Sonine app oxima ion [17], al hough a sligh ly di e en beha io is also clea ly iden i ied. I is in e es ing
o conside he ini ial alue o J0
η
( )and decompose i in he o m
J0
η
(0) = J0(1)
η
(0)+J0(2)
η
(0),(28)
whe e
J0(1)
η
(0) = 1
N
N
∑
ih∆xy( i)Φ2,xy( i/e 0,s )iN,s ,J0(2)
η
(0) = 1
N
N
∑
i
N
∑
j6=ih∆xy( i)Φ2,xy( j/e 0,s )iN,s .(29)
The i s componen , J0(1)
η
(0)is iden ically he same as J
η
(0), whe e J
η
( )was de ined in Eq. (25). I is independen
o he eloci y co ela ions p esen in he sys em. In ac , i s exac alue can be easily ob ained, and o he choice
o
ω
0used in he simula ions and he uni s we a e employing, i is J0(1)
η
(0) = J
η
(0) = 1/2 o all
α
[16]. The o he
pa , J(2)
η
(0), con ains he e ec o eloci y co ela ions and was assumed o be negligible. The simula ion esul s o
J0
η
(0)a e plo ed in Fig. 3. Fo
α
<0.6, con ibu ions om J0(2)
η
become signi ican , indica ing he p esence o eloci y
co ela ions in he sys em. This esul aises some undamen al ques ions. A e he obse ed co ela ions an a i ac o
he DSMC me hod? We belie e hey a e no , since eloci y co ela ions in he HCS also ha e been ecen ly obse ed
and measu ed in molecula dynamics simula ions [18]. Why he e is such a good ag eemen be ween he i s Sonine
app oxima ion p edic ions and he simula ion esul s o he shea iscosi y, e en om s ong dissipa ion? A possible
answe is ha he Sonine expansion is in oduced, and unca ed, bo h in he ini ial condi ions and in he dynamics o
he luxes, leading o some kind o compensa ion. Fo small alues o
α
, which G een-Kubo exp ession is he igh
one, ha wi h J
η
( )o ha wi h J0
η
( )?, o a e bo h w ong? And, inally, can i be concluded om hese esul s ha
he Bol zmann equa ion should be somehow e ised o s ong inelas ici y? All hese poin s clea ly dese e u he
analysis.
Al hough in his pape we ha e es ic ed ou sel es o he shea iscosi y, he G een-Kubo exp essions o he
o he wo anspo coe icien s, associa ed wi h he hea lux, can be compu ed in a simila way. Fo no oo s ong
dissipa ion, he simula ion esul s ag ee qui e well wi h he i s Sonine app oxima ion p edic ions. Ne e heless, he
disc epancies g ow a he as as
α
dec eases below
α
≈0.7. This migh be, a leas pa ially, due o he p esence
819
0 0.2 0.4 0.6 0.8 1
α
0.45
0.5
0.55
0.6
FIGURE 3. DSMC esul s o he co ela ion unc ion J0
η
(0)( illed ci cles) and i s diagonal pa J0(1)
η
(0)(emp y ci cles).
o eloci y co ela ions in he DSMC esul s. A de ailed discussion o hese anspo coe icien s will be published
elsewhe e [16].
ACKNOWLEDGMENTS
We acknowledge inancial suppo om he Minis e io de Ciencia y Tecnología (Spain) h ough G an No. BFM2002-
00303 (pa ially inanced by FEDER unds)
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