Validi y o he Bol zmann equa ion o desc ibe low-densi y g anula sys ems
J. Ja ie B ey and M. J. Ruiz-Mon e o
Fı
´sica Teo
´ ica, Uni e sidad de Se illa, Apa ado de Co eos 1065, E-41080 Se illa, Spain
共Recei ed 15 July 2003; published 29 Janua y 2004兲
The depa u e o a g anula gas in he ins able egion o pa ame e s om he ini ial homogeneous cooling
s a e is s udied. Resul s om molecula dynamics and om di ec Mon e Ca lo simula ion o he Bol zmann
equa ion a e compa ed. The esul s indica e ha he Bol zmann equa ion accu a ely p edic s he low-densi y
limi o he sys em. The ele an ole played by he pa alleliza ion o he eloci ies as ime p oceeds and he
dependence o his e ec on he densi y is analyzed in de ail.
DOI: 10.1103/PhysRe E.69.011305 PACS numbe 共s兲: 45.70.⫺n, 51.10.⫹y, 05.20.Dd, 47.20.⫺k
I. INTRODUCTION
The s udy o g anula sys ems has a ac ed much in e es
in he las yea s. The p ac ical impo ance o hese sys ems,
which a e p esen in many si ua ions in daily li e, oge he
wi h he ichness and complexi y o hei beha io 关1兴, has
mo i a ed many esea ches ying o desc ibe and unde s and
he physical mechanisms go e ning g anula lows. O
cou se, he a ie y o wha we call ‘‘g anula media’’ makes
i necessa y o use di e en heo e ical desc ip ions depend-
ing on he p oblem we wish o add ess. In he con ex o
apid, low-densi y, g anula lows, he applica ion o he
me hods o he kine ic heo y o molecula 共elas ic兲sys ems
has p o en o be a e y use ul ool. The s a ing poin o his
desc ip ion has been in many cases he ex ension o inelas ic
collisions o he Bol zmann equa ion, which is de i ed unde
he same hypo hesis as in elas ic sys ems. F om his kine ic
equa ion, closed hyd odynamic equa ions wi h explici ex-
p ession o he luxes and he associa ed anspo coe i-
cien s 共up o second o de in he g adien s兲 o inelas ic ha d
pa icles ha e been de i ed 关2,3兴. This hyd odynamic pic u e
has been ound o p o ide an accu a e desc ip ion o g anula
sys ems in e y di e en si ua ions, d i en and no d i en.
The Bol zmann equa ion has also been used o s udy he
eloci y dis ibu ion o a g anula gas, modeled in mos o
he cases as a sys em o inelas ic ha d pa icles. O cou se,
he shape o he dis ibu ion depends on he s a e o he sys-
em. The simples possibili y is he so-called homogeneous
cooling s a e 共HCS兲, he s a e o a homogeneous, eely
e ol ing g anula gas 关4兴. In ha case, he Bol zmann equa-
ion is shown o admi a solu ion whose ime dependence can
be scaled ou h ough i s second momen . De ia ions om
Gaussiani y in he HCS ha e been quan i ied by compu ing
he ku osis o he dis ibu ion 关5–7兴and by es ablishing he
exis ence o exponen ial eloci y ails 关6,8,9兴. The possible
solu ions o he Bol zmann equa ion in he case o a ib a ed
sys em in he absence o g a i y ha e also been in es iga ed,
and he condi ions o he exis ence o a s eady solu ion
whose space dependence is scaled ou also h ough he sec-
ond momen ha e been es ablished 关10兴.
In spi e o i s ex ended use and clea success in many
cases, he alidi y o he Bol zmann equa ion o desc ibe
g anula lows has been ques ioned since he ea ly de elop-
men s o he kine ic heo y o g anula sys ems. One o he
i s objec ions aised agains i s use was based on he en-
dency o hese sys ems o o m densi y clus e s. Since i is
only alid in he low-densi y limi , he Bol zmann equa ion
is no sui able o desc ibe he clus e e olu ion. This is no a
undamen al objec ion, and in ac i can also be aised
agains he applicabili y o he Bol zmann equa ion o mo-
lecula sys ems in inhomogeneous s a es. The alue o he
densi y limi s indeed he applicabili y o he Bol zmann de-
sc ip ion, bu bo h in he elas ic and he inelas ic case. On he
o he hand, i we s a om a homogeneous, low-densi y,
ini ial con igu a ion o a g anula gas, unde condi ions ha
clus e s will de elop e en ually, i can be expec ed ha he
Bol zmann equa ion will be alid o desc ibe he i s s ages
o he clus e o ma ion, as long as egions wi h a oo la ge
densi y do no show up. A di e en objec ion conce ns he
alidi y o he molecula chaos hypo hesis, which is on he
basis o he Bol zmann desc ip ion. As collisions in g anula
sys ems end o make he pa icle eloci ies mo e pa allel,
eloci y co ela ions may de elop om he ea ly s ages o
he e olu ion o a g anula gas, making he molecula chaos
hypo hesis in alid. This p oblem was di ec ly add essed by
So o e al. 关11,12兴, who s udied he sho - ange eloci y co -
ela ions in he HCS o a g anula luid, concluding ha hey
we e no ele an in he low-densi y case. Pagonaba aga
e al. 关13兴also s udied he alidi y o he molecula chaos
hypo hesis in a homogeneous g anula sys em, bu in his
case d i en by a andom o ce. Again, i was ound ha , o
dilu e sys ems, de ia ions om molecula chaos we e no
signi ican . To pu his wo k in a p ope con ex , i should be
aken in o accoun ha he d i ing o ce in oduced by he
au ho s induced some eloci y co ela ions, as poin ed ou by
hem.
The p e ious men ioned s udies a e es ic ed o homoge-
neous s a es o a g anula gas. They co espond o e y spe-
cial si ua ions, so he conclusions canno be ex apola ed o
mo e gene al cases. A di e en si ua ion was in es iga ed
ecen ly by Nakanishi 关14兴, who s udied he ime e olu ion
o he eloci y dis ibu ion o a eely e ol ing g anula luid
in condi ions such ha he HCS is uns able. He ound ha
he ku osis o he dis ibu ion did no emain s a iona y, bu
e ol ed owa ds he Gaussian alue om he ea ly s ages o
he e olu ion o he sys em, e en be o e he clus e ing ins a-
bili y shows up. He a gues ha his is in con adic ion wi h
he p edic ions o kine ic heo ies based on he Bol zmann
equa ion, being a clea indica ion o he g ow h o eloci y
co ela ions, which in alida es he molecula chaos hypo h-
PHYSICAL REVIEW E 69, 011305 共2004兲
1063-651X/2004/69共1兲/011305共9兲/$22.50 ©2004 The Ame ican Physical Socie y69 011305-1
esis. He also ound ha he e u n o he Gaussian beha io
became slowe he mo e dilu e he sys em, bu as changing
he densi y implies changing he ‘‘clus e ing ime,’’ i was
no clea o he au ho how he Bol zmann beha io could be
eco e ed.
The objec o his wo k is o in es iga e he possible di -
e ences in he beha io o he eloci y dis ibu ion unc ion
o an ini ially dilu e, homogeneous, g anula gas and he p e-
dic ions o he Bol zmann equa ion when he sys em is unde
condi ions such ha he HCS is uns able. We will compa e
he esul s om molecula dynamics 共MD兲simula ions a
di e en a e age densi ies wi h hose om he di ec simu-
la ion Mon e Ca lo 共DSMC兲me hod, which is a me hod o
sol e nume ically he Bol zmann equa ion 关15兴.
The plan o he pape is as ollows. In Sec. II he kine ic
heo y p edic ions o he HCS a e b ie ly discussed, while
he de ails o he simula ions and he p ope ies o be s udied
a e gi en in Sec. III. The esul s o he DSMC and MD
simula ions a e p esen ed in Secs. IV and V. Some inal com-
men s and discussion a e made in Sec. VI.
II. KINETIC THEORY FOR THE HOMOGENEOUS
COOLING STATE
Le us conside a sys em o Nsmoo h inelas ic ha d pa -
icles, sphe es (d⫽3) o disks (d⫽2), o mass mand diam-
e e
. The loss o ene gy in collisions is cha ac e ized by a
cons an coe icien o no mal es i u ion
␣
, which implies
ha he eloci ies o wo colliding pa icles i,jbe o e and
a e he collision a e ela ed by
i
⬘⫽ i⫺1⫹
␣
2共g•
ˆ兲
ˆ,
j
⬘⫽ j⫹1⫹
␣
2共g•
ˆ兲
ˆ,共1兲
whe e he p imes deno e eloci ies a e he collision, g⫽ i
⫺ jis he ela i e eloci y, and
ˆa uni ec o joining he
cen e s o pa icles iand ja con ac . Le us no ice ha 0
⭐
␣
⭐1, and ha he alue
␣
⫽1 co esponds o elas ic col-
lisions. The abo e ule implies ha , in each collision, he
componen o he ela i e eloci y in he di ec ion o
ˆis
educed in a ac o
␣
,
g⬘•
ˆ⫽⫺
␣
共g•
ˆ兲.共2兲
The Bol zmann equa ion desc ibing he e olu ion o he
eloci y dis ibu ion ( , , ) o a sys em o eely e ol ing
ha d pa icles has he o m
冉
⫹ •“
冊
共 , , 兲⫽JB关 ,
兩
共 , , 兲兴,共3兲
whe e JBis he 共inelas ic兲Bol zmann collision ope a o ,
JB关 ,
兩
共 , , 兲兴⫽
d⫺1
冕
d 1
冕
d
ˆ
共g•
ˆ兲共g•
ˆ兲
⫻共
␣
⫺2b⫺1⫺1兲 共 , , 兲 共 , 1, 兲.
共4兲
He e
is he Hea iside s ep unc ion and b⫺1an ope a o
ans o ming eloci ies and 1 o i s igh in o hei p ecol-
lisional alues. As is he case wi h elas ic pa icles, he de i-
a ion o his equa ion is based on he molecula chaos hy-
po hesis, i.e., he ac o iza ion o he wo-pa icle dis ibu ion
unc ion in he p ecollisional sphe e: (2)(x1,x2, )
⫽ (x1, ) (x2, ), whe e xi⬅
兵
i, i
其
. I spa ial and/o eloc-
i y co ela ions a e p esen be ween colliding pa icles, his
ac o iza ion does no hold and he assump ion o molecula
chaos ails.
When a sys em o inelas ic pa icles such as he one de-
sc ibed e ol es eely, i s simples possible s a e is he HCS.
I is a homogeneous s a e wi h no luxes, whose empe a u e
T( ), de ined as p opo ional o he a e age kine ic ene gy,
e ol es acco ding o Ha ’s law 关16兴,
T共 兲⫽T共0兲
冉
1⫹
0
冊
2,共5兲
whe e 0is he ime cha ac e izing he ene gy decay.
The Bol zmann equa ion admi s a solu ion ( , ) desc ib-
ing he HCS, which obeys he scaling law 关5–7兴
共 , 兲⫽nH
0共 兲d
冉
0共 兲
冊
,共6兲
whe e nHis he homogeneous densi y, and 0 he he mal
eloci y o he sys em, de ined as 0⫽
冑
2kBT/m, wi h kB
he Bol zmann cons an . The e o e, in he HCS all he ime
dependence o he eloci y dis ibu ion can be scaled ou
h ough i s second momen . The unc ion
is de e mined
om he Bol zmann equa ion. Al hough i s exac exp ession
is no known, i was ound ha i does no de ia e much om
a Gaussian 关7兴. Then, i is sensible o expand i using he
Sonine polynomials S(j), whose explici exp ession can be
ound in Re . 关17兴,
共c兲⫽e⫺c2
⫺d/2 兺
j⭓0ajS(j)共c2兲,c⫽
0.共7兲
No maliza ion and scaling imply a0⫽1 and a1⫽0. The
coe icien a2is ela ed o he ku osis o he dis ibu ion
a2⫽4
d共d⫹2兲
冋
具
c4
典
⫺d共d⫹2兲
4
册
,共8兲
and i s alue has been es ima ed om he Bol zmann equa-
ion up o linea o de in a2关5,6兴. In he abo e exp ession,
he angula b acke s deno e a e ages wi h he eloci y dis i-
bu ion o he HCS. Bo h heo e ical calcula ions and nume i-
cal simula ions o he Bol zmann equa ion 关7兴show ha , o
J. J. BREY AND M. J. RUIZ-MONTERO PHYSICAL REVIEW E 69, 011305 共2004兲
011305-2
no oo inelas ic sys ems, a2is e y small. De ia ions om
Gaussiani y a e impo an i we conside he ails o he dis-
ibu ion, which a e exponen ial. Again, his has been e i-
ied om heo e ical a gumen s based on he Bol zmann
equa ion 关8,6兴and om DSMC simula ions 关9兴.
The ime 0cha ac e izing he kine ic ene gy decay in he
HCS has also been compu ed by using he Bol zmann equa-
ion 关2兴, and i s exp ession is
0
⫺1⫽
*共
␣
兲4
(d⫺1)/2
共2⫹d兲⌫共d/2兲
冉
kBT共0兲
m
冊
1/2
nH
(d⫺1),共9兲
whe e
*(
␣
) is a unc ion ha depends only on he coe i-
cien o es i u ion
␣
, and ha , o no oo inelas ic sys ems
eads 关2,3兴
*共
␣
兲⯝2⫹d
4d共1⫺
␣
2兲.共10兲
In e ms o he a e age numbe o collisions pe pa icle,
,
Ha ’s law akes he o m
T共
兲⫽T共0兲e⫺2
*
,共11兲
i.e., he kine ic ene gy decays exponen ially wi h he numbe
o collisions.
I is a well known esul 关18,19兴 ha he homogeneous
cooling s a e o a g anula gas is uns able agains long wa e-
leng h pe u ba ions. In p ac ice, his implies ha i he sys-
em size is la ge han a c i ical alue Lc, i will spon ane-
ously de elop spa ial inhomogenei ies. The alue o he
c i ical size has also been de e mined om he Bol zmann
equa ion 关3,20兴, and one ob ains
Lc⫽Cd共2⫹d兲⌫共d/2兲
2
(d⫺3)/2
冉
*
2
*
冊
1/2
H.共12兲
He e, H⫽1/(CdnH
(d⫺1)) is he homogeneous mean ee
pa h, Cd⫽2
冑
2 o d⫽2 and Cd⫽
冑
2 o d⫽3, and
*is
a unc ion o
␣
ela ed o he iscosi y o a g anula gas,
whose exp ession can be ound in Re . 关3兴. Bo h MD 关18,19兴
and DSMC 关21兴simula ions o eely e ol ing g anula
gases show ha , in he de elopmen o inhomogenei ies, i s
eloci y o ices appea in he sys em and hen he densi y
becomes also e y inhomogeneous. O cou se, once he in-
s abili y has se in, he eloci y dis ibu ion o he sys em
will no longe be he one o he HCS. The e o e, one expec s
he scaling o ail and a2 o become ime dependen . The
ques ion is whe he he Bol zmann equa ion is alid o de-
sc ibe how a low-densi y g anula sys em depa s om he
HCS. I migh happen ha , as poin ed ou by Nakanishi 关14兴,
he de elopmen o eloci y co ela ions a e e y impo an
om he ea ly s ages o his depa u e, and hen he Bol z-
mann equa ion ails. This is he main poin we wan o cla i y
in his wo k.
III. COMPUTER SIMULATIONS
Gi en ha we wan o check he alidi y o he Bol zmann
equa ion o desc ibe g anula gases, we will compa e MD
simula ions o ini ially homogeneous, low-densi y, g anula
sys ems wi h he p edic ions o he Bol zmann equa ion.
Since he ime-dependen analy ical solu ion o he la e is
no known, we will use he DSMC me hod de eloped by
Bi d 关15兴 o cons uc nume ical solu ions o he Bol zmann
equa ion unde he desi ed condi ions. I mus be emem-
be ed ha his me hod mimics he dynamics behind he Bol -
zmann equa ion by uncoupling he ee low o pa icles and
collisions du ing a small enough ime in e al. Besides, col-
lisions a e always ea ed as i he e we e no co ela ions
be ween colliding pa icles. In p ac ice, his implies ha
space is di ided in cells o size smalle han he mean ee
pa h, and pa icles wi hin a cell collide wi h a p obabili y
p opo ional o hei ela i e eloci y. Technical de ails abou
he DSMC me hod ha e been ex ensi ely discussed in he
li e a u e 关15,22兴. The only poin we wan o s ess is ha ,
when using his me hod, he simula ed sys em is by de ini ion
in he low-densi y limi , whe e he Bol zmann equa ion is
supposed o apply, no ma e how many simula ed pa icles
he e a e in a gi en space egion.
The simula ions we will p esen in he ollowing co e-
spond o a wo-dimensional sys em o eely e ol ing ha d
disks in a squa e box o side L, la ge han he c i ical size Lc
gi en by Eq. 共12兲. Pe iodic bounda y condi ions will be con-
side ed in all he cases. In he MD simula ions, he e en
d i en algo i hm 关23兴will be used. Fo ou pu poses, i is
impo an o s udy he e ec o lowe ing he densi y on he
e olu ion o he sys em. Fo ha eason, wo di e en , low
alues, o he densi y will be s udied in he MD simula ions,
namely, nH⫽0.05
⫺2and nH⫽0.0125
⫺2. Ne e heless, i
mus be no iced ha , o a gi en
␣
, changing he densi y
implies changing he c i ical size o he sys em, as ollows
om Eq. 共12兲. As a consequence, i he sys ems ha e he
same size L, he cha ac e is ic ime go e ning he g ow h o
ins abili ies will be also changed, and he depa u e om he
HCS will be as e in he dense sys em. The e o e, i we
wan o ha e he same ‘‘dis ance o s abili y’’in sys ems wi h
di e en densi ies, hei sizes mus be changed co espond-
ingly, so L/H, and, he e o e, L/Lc, emain unchanged.
The same alue o his scaled size should be conside ed in
he DSMC simula ions, again o expec he same cha ac e -
is ic ime go e ning he depa u e om he HCS.
Mo eo e , we ha e used a ixed alue o he es i u ion
coe icien ,
␣
⫽0.9, which is no oo inelas ic, bu lies ou -
side o wha can be conside ed he quasielas ic egion. Fo
his alue o
␣
, Eq. 共12兲leads o Lc⯝23.36H. As we wan
he sys em o be well inside he uns able egion, we ha e
chosen he sys em size o be L⫽80H. Then, o nH
⫽0.05
⫺2 he numbe o pa icles in he sys em in he MD
simula ions was N⫽16000, while o nH⫽0.0125
⫺2,N
⫽64000. In he DSMC simula ions, N⫽2.048⫻106pa -
icles we e conside ed, bu i mus be eminded ha his
numbe has only a s a is ical meaning. In all he cases, we
s a ed wi h he pa icles homogeneously dis ibu ed in he
sys em and wi h a Gaussian eloci y dis ibu ion. This si u-
VALIDITY OF THE BOLTZMANN EQUATION TO . . . PHYSICAL REVIEW E 69, 011305 共2004兲
011305-3
a ion was le o e ol e wi h elas ic collisions du ing se e al
collisions pe pa icle, un il he sys em was equilib a ed.
Then, he inelas ici y was swi ched on, and his ime is aken
as he o igin, ⫽0, o ou simula ions. The ini ial elas ic
pe iod ensu es ha he s uc u e o he luid a ⫽0 is he
equilib ium one.
S udied p ope ies
As he aim o he pape is o in es iga e he depa u e o
he eloci y dis ibu ion o he sys em om he HCS o m,
his will be one o he p ope ies o be s udied in he simu-
la ions. To be mo e p ecise, we will compu e he second
具
2
典
and ou h
具
4
典
eloci y momen s o he o al eloci y dis i-
bu ion, and om hem we will compu e also a2, gi en by
Eq. 共8兲. As a as he sys em s ays in he HCS, a2will be
cons an . I is impo an o no ice ha , as in Re . 关14兴,we
will s udy he global eloci y dis ibu ion o he sys em, e en
i inhomogenei ies a e p esen , causing local a e ages o he
physical quan i ies o be qui e di e en om hei global
alues.
The g ow h o inhomogenei ies will be con olled by ol-
lowing he e olu ion o he densi y n( , ), and eloci y
u( , ), ields. To compu e hem, he sys em is di ided in o
30⫻30 squa e cells o size lc⬃2.7H, and p ope ies a e
a e aged in each cell. Again, while he sys em emains in he
HCS, n( , ) should be cons an and u( , )⫽0共aside om
s a is ical noise兲. Once he ins abili y de elops, o ices and
also densi y inhomogenei ies will show up. The ques ion e-
mains o how he depa u e o he eloci y dis ibu ion om
he HCS one is ela ed o he de elopmen o hese inhomo-
genei ies.
The possible ailu e o he molecula chaos hypo hesis
will be in es iga ed by he s udy o se e al collisional a e -
ages, i.e., a e ages o e colliding pai s o pa icles. These
a e ages con ain in o ma ion abou he wo-pa icle dis ibu-
ion o colliding pa icles, which is he unc ion whose ac-
o iza ion is assumed in he molecula chaos hypo hesis. The
i s o he collisional a e ages we will conside is he pai
dis ibu ion unc ion a con ac , g(
⫹). In an elas ic sys em,
i he e a e no co ela ions be ween colliding pa icles, as i
is in ac assumed by he Bol zmann equa ion, g(
⫹)⫽1. In
he case o a sys em o inelas ic disks, he pai dis ibu ion
unc ion a con ac can be easily compu ed by using 关24兴
共no ice ha he e is a missp in in he exp ession o g(
⫹)
p o ided in he ci ed e e ence兲
g共
⫹, 兲⫽1⫹
␣
␣
1
NnH
1
⌬ 兺
␥
苸⌬
1
兩
ˆ•g
␥
兩
共⫺
ˆ•g
␥
兲,
共13兲
whe e we a e summing o e all collisions
␥
aking place in
he in e al ⌬ , and he
unc ion implies ha we a e using
he p ecollisional alues o he quan i ies in he sum. When
he sys em is in he HCS, i has been ound 关24,11兴 ha
gHCS(
⫹) does no depend on ime, and akes he alue
gHCS共
⫹兲⫽1⫹
␣
2
␣
g0共
⫹兲,共14兲
wi h g0 he equilib ium elas ic pai dis ibu ion unc ion a
con ac , which in he wo dimensional case is accu a ely
gi en by 关25兴
g0共
⫹兲⫽1⫹
共25⫺4n
兲n
4共4⫺n
兲2.共15兲
In ou simula ions, ime will be disc e ized so he a e age
o e collisions in Eq. 共13兲will be done no o e he whole
simula ion ime, bu o e collisions occu ing in a gi en in-
e al. In his way, we can s udy he e olu ion o he colli-
sional a e ages and, in pa icula , o he pai dis ibu ion
unc ion a con ac . Needless o say, g(
⫹, ) will only be
compu ed in he MD simula ions, as in he DSMC me hod i s
alue is gi en.
Ano he es o he alidi y o he molecula chaos hy-
po hesis will be p o ided by he p obabili y dis ibu ion o
he impac pa ame e , b⫽(
/2)sin
, whe e
is he angle
o med by he impac ela i e eloci y gand he uni ec o
ˆ. In a sys em o ha d disks, and i he e a e no co ela ions,
he impac pa ame e is uni o mly dis ibu ed. In he HCS
his dis ibu ion has also been measu ed 关11,26兴by compu e
simula ions, and no de ia ions om uni o mi y ha e been
ound in he low-densi y limi . He e, we will conside he
same disc e iza ion discussed abo e o cons uc he dis ibu-
ion o impac pa ame e s in each ime in e al, and s udy i s
possible ime dependence. Again, he impac pa ame e dis-
ibu ion will be s udied only in he MD simula ions, as he
DSMC me hod assumes i s uni o mi y.
The possible eloci y co ela ions will also be s udied. A
quan i y ha has been used o s udy hem is he collisional
a e age ⌫( ) de ined as
⌫共 兲⫽1
N
␥
兺
␥
苸⌬
ci
␥
•cj
␥
兩
ci
␥
⫺cj
␥
兩
共⫺
ˆ•g
␥
兲,共16兲
wi h N
␥
he numbe o collisions in he in e al ⌬ , and i,j
he pa icles in ol ed in collision
␥
. I he e is nei he mac-
oscopic eloci y ield no eloci y co ela ions in he sys-
em, i is ⌫⫽0. The e o e, in he HCS, nonze o alues o ⌫
a e a clea signal o he p esence o eloci y co ela ions in
he sys em. When he sys em lea es he HCS, he e is an-
o he possible eason o hese nonze o alues. The buildup
o eloci y o ices implies ha u( , ) is di e en om ze o,
and hen, e en i he e a e no eloci y co ela ions, ⌫will be
di e en om ze o. Le us no ice ha ⌫( ) can be measu ed
bo h in molecula dynamics and DSMC simula ions.
Finally, he dis ibu ion o he angle
o med by he
eloci ies i, j, o colliding pa icles will also be com-
pu ed. We a e no awa e o any analy ic exp ession o his
quan i y in he elas ic case, and, he e o e, we will use he
ini ial, elas ic pa , o he simula ions o ob ain he elas ic
dis ibu ion o he
angle. The pa alleliza ion mechanism
inhe en o inelas ic collisions may cause de ia ions om
his beha io , e en in he HCS. In any case, his dis ibu ion
shows i he e is a p edominance o collisions be ween pa -
J. J. BREY AND M. J. RUIZ-MONTERO PHYSICAL REVIEW E 69, 011305 共2004兲
011305-4
icles mo ing mo e o less pa allel in he sys em. Again, his
dis ibu ion will be measu ed bo h in MD and DSMC simu-
la ions.
I is impo an o ealize he di e en physical in o ma ion
behind he dis ibu ions P(b, ) and P(
, ). The o me is
ela ed o he spa ial dis ibu ion o he inciden lux o e
colliding pa icles, while he la e con ains only in o ma ion
abou he ela i e di ec ion o he eloci ies o colliding pa -
icles. In pa icula , i is easy o see ha all he alues o
a e compa ible wi h any gi en alue o b.
IV. RESULTS
The esul s we will p esen co espond, as we ha e al-
eady s a ed, o MD and DSMC simula ions o a eely
e ol ing sys em o ha d disks wi h
␣
⫽0.9 and L⫽80H,
i.e., in condi ions such ha he HCS is well inside he un-
s able egion. In he ollowing, he mass mo he pa icles
will be used as he uni o mass, and he ini ial kine ic ene gy
pe pa icle as he uni o ene gy. The esul s we will p esen
ha e been a e aged o e se e al ajec o ies in all he cases
in o de o imp o e he s a is ics. Besides, he ime e olu ion
o he sys em will be exp essed in e ms o he numbe o
collisions pe pa icle,
. As he sys em is p epa ed in an
ini ially homogeneous si ua ion, we expec i o s ay in he
HCS o a ansien pe iod, un il he ins abili y se s in. F om
ha momen , he di e en physical p ope ies will depa
om hei HCS alues.
In Fig. 1 we ha e plo ed he e olu ion o he a e age
kine ic ene gy pe pa icle,
具
2
典
/2, in he sys em. This quan-
i y is p opo ional o he g anula empe a u e as a as he e
is no mac oscopic eloci y ield, i.e., while u⫽0. Also plo -
ed is Ha ’s law, Eq. 共11兲, desc ibing he e olu ion o his
p ope y in he HCS. In all he cases he e is an ini ial pe iod
in which he ene gy ollows Ha ’s law and, a e ha , he
a e age kine ic ene gy o he sys em decays slowe han in
he HCS. The depa u e o m Ha ’s law occu s soone in he
MD simula ions han in he DSMC one, bu i is impo an o
no ice ha , i we compa e he wo MD simula ions, i occu s
soone in he case o la ge densi y. The e o e, i seems ha ,
al hough he e a e quan i a i e di e ences be ween he be-
ha io o a ini e densi y g anula gas and he p edic ions o
he Bol zmann equa ion, hey a e smalle he smalle he
densi y, being simila he shape o he cu es. Then, i seems
sensible o expec ha he Bol zmann equa ion p edic ions
p o ide indeed he co ec pic u e in he analy ic low-densi y
limi , n→0.
The e olu ion o a2is plo ed in Fig. 2. In Fig. 2共a兲, he
ini ial e olu ion o his quan i y is shown: he e is an ini ial,
e y as , decay om he ini ial Gaussian, a2⫽0, alue o he
HCS one, ollowed by a s eady pe iod o which he eloci y
dis ibu ion o he sys em seems o emain wi h he HCS
o m. The comple e ime e olu ion o he sys em is gi en in
Fig. 2共b兲. Again, in all he cases we obse e a simila beha -
io . A e he ini ial s eady pe iod, a2g ows, eaches a maxi-
mum, and a e wa ds i decays in ime. The app oxima e du-
a ion o he s eady pe iod,
s ,is
s ⬃15 o nH
⫽0.05
⫺2,
s ⬃30 o nH⫽0.0125
⫺2, and
s ⬃50 in he
DSMC simula ion. The posi ion
*o he maximum o a2is
*⬃40 o nH⫽0.05
⫺2,
*⬃50 o nH⫽0.0125
⫺2, and
FIG. 1. E olu ion o he a e age kine ic ene gy o he simula-
ions discussed in he ex . The uni s a e chosen such ha he ini ial
alue is equal o 1. Also shown is he heo e ical p edic ion o he
HCS, i.e., Ha ’s law. Time is measu ed in a e age numbe o col-
lisions pe pa icle.
FIG. 2. Time e olu ion o he dimensionless coe icien a2 o
he simula ions discussed in he main ex . The sho ime beha io
共a兲and he comple e e olu ion 共b兲a e displayed in di e en plo s
o he sake o cla i y.
VALIDITY OF THE BOLTZMANN EQUATION TO . . . PHYSICAL REVIEW E 69, 011305 共2004兲
011305-5
*⬃70 in he DSMC simula ion. The heigh o he maxi-
mum is also la ge he la ge he densi y. A i s conclusion
ha can be ex ac ed om Fig. 2 is ha he quali a i e be-
ha io o a2 ha ollows om he Bol zmann equa ion is he
same ound in he MD simula ions. Besides, he di e ences
be ween he MD simula ions and he DSMC one a e smalle
he smalle he densi y in he o me , showing again a en-
dency o he Bol zmann beha io in he limi o e y low
densi y.
Compa ison o Figs. 1 and 2 is in e es ing in o de o
de e mine he sensi i i y o Ha ’s law o he exac shape o
he eloci y dis ibu ion unc ion as measu ed by i s ou h
momen . I is ound ha , o
⫽
s , when a2begins o
depa om he s eady alue, he empe a u e akes he alue
p edic ed by Ha ’s law in all he cases, as is expec ed. Wha
migh be su p ising is ha , a he maximum
*, he de ia-
ions om he Ha alue a e ela i ely small, he empe a-
u e being a mos 1.3 imes he Ha ’s alue. This is in
ag eemen wi h simula ions o homogeneous sys ems o in-
elas ic ha d disks, which show ha he e olu ion o he em-
pe a u e ollows qui e app oxima ely Ha ’s law, unless he
ou h momen o he eloci y dis ibu ion is e y la ge as
compa ed wi h he second one 关27兴.
Once he alidi y o he Bol zmann desc ip ion as a limi
o he beha io o a g anula gas o anishing densi y has
been shown o be consis en wi h he simula ion esul s 共a
leas wi h ega ds o he beha io o he second and ou h
momen s o he eloci y dis ibu ion兲, a na u al eme ging
ques ion is which is he mechanism aking he eloci y dis-
ibu ion ou o he HCS shape. Wi h ha pu pose, he e o-
lu ions o he densi y and eloci y ields ha e been s udied. I
mus be no iced ha , as we expec hem o be inhomoge-
neous, and hei spa ial dis ibu ion changes om one eal-
iza ion o ano he , hese ields canno be a e aged o e di -
e en uns.
Le us conside i s he e olu ion o he densi y ield. In
all he cases we obse e ha he sys em emains homoge-
neous du ing wha we ha e called he s eady pe iod
s .
Be ween
s and
*small densi y inhomogenei ies begin o
show up, bu hey a e no e y signi ican . The si ua ion
changes om
*on: he e is a e y as g ow h o densi y
inhomogenei ies, and in all he cases, elonga ed clus e s o
pa icles a e o med, simila o hose obse ed in p e ious
s udies o eely e ol ing g anula sys ems. This beha io o
he densi y ield is quan i ied in Figs. 3 and 4. In he i s o
hem, he e olu ion o he dispe sion o he densi y luc ua-
ions,
␦
⫽
冑
具
关n( , )⫺nH兴2
典
is plo ed. Le us no ice ha his
quan i y is nonze o e en in a homogeneous s a e, due o
s a is ical noise, and i s alue in he homogeneous s a e de-
pends on he numbe o pa icles pe cell, which is di e en
in each o he simula ions. Fo ha eason
␦
has been scaled
wi h i s alue in he elas ic pa o he simula ion,
␦
el .I is
obse ed in he igu es ha , in all he cases,
␦
has no in-
c eased much o e i s elas ic alue a
⫽
*, bu om hen
on he e is a e y sha p g ow h o he densi y luc ua ions as
a consequence o he o ma ion o clus e s in he sys em. A
simila beha io is exhibi ed by he maximum alue o he
densi y, nM, which is plo ed in Fig. 4, scaled wi h he ho-
mogeneous densi y. While he sys em is in a homogeneous
s a e, nMis a measu e o he s a is ical noise. Le us emem-
be ha he sys ems we e di ided in 30⫻30 cells o compu e
he hyd odynamics ields: as he numbe o pa icles is
smalle in he dense sys em, he noise in he ields will be
la ge , and ha is he eason why, in he homogeneous pa o
he e olu ion, nM/nHis la ge he la ge he densi y. I mus
be also poin ed ou ha , once he clus e ing begins, he e is a
ime in e al in which he g ow h o nMcan be i ed o an
exponen ial,
nM⬃nH⫹Cesn(
⫺
1),共17兲
whe e Cand
1a e cons an s whose alue is no ele an o
he p esen discussion, and sn⬃0.13 o n⫽0.05
⫺2,sn
⬃0.163 o n⫽0.0125
⫺2, and sn⬃0.167 in he DSMC
simula ion. Again, he disc epancies be ween he Bol zmann
beha io and he one a ini e densi y a e smalle as we lowe
he densi y, he g ow h a e being almos he same o he
lowe densi y case and he DSMC simula ion.
In a eely e ol ing g anula sys em, i has been shown
ha , when using
as he ime a iable, he g ow h a e o he
scaled ans e sal eloci y mode is
FIG. 3. E olu ion o he a e age alue o he densi y luc ua-
ions, scaled wi h hei alue in he elas ic case, o he simula ions
discussed in he main ex .
FIG. 4. E olu ion o he maximum alue o he scaled densi y
o he simula ions discussed in he main ex . The symbols a e
om he simula ions, and he do ed line is jus a guide o he eye.
J. J. BREY AND M. J. RUIZ-MONTERO PHYSICAL REVIEW E 69, 011305 共2004兲
011305-6
s⬜⫽
*⫺
*
2k2,共18兲
whe e kis a nondimensional wa e ec o de ined as k
⫽2
/(
冑
nH
l), wi h l he wa eleng h o he pe u ba ion.
In a simula ion, he maximum allowed wa eleng h, which is
he one ha g ows as e , is l⫽L, due o he bounda y con-
di ions. Fo he alues o he pa ame e s used in his wo k,
he heo e ical p edic ion is s⬜⬃8.64⫻10⫺2. I has been a -
gued 关18,28兴 ha he g ow h o densi y inhomogenei ies in a
eely e ol ing g anula sys em is a consequence o he non-
linea coupling be ween he ans e sal eloci y mode and
he o he hyd odynamic ields, in pa icula , he densi y. I
ha is indeed he case, he g ow h a e o he densi y, a leas
in i s i s s ages, should be 2s⬜. He e, 2s⬜⬃0.173, which is
e y close o he alue o sn ound in he DSMC and in he
MD lowes densi y simula ion, con i ming once again he
pic u e desc ibed abo e. This ag eemen indica es he hyd o-
dynamical cha ac e o he densi y luc ua ions in he clus e -
ing egime.
The g ow h o he scaled eloci y ield has also been ol-
lowed in ou simula ions, and we ound ha o ices begin o
de elop qui e soon in he sys em. To quan i y his, we in o-
duce
␦
u, he scaled a e age alue o u2,as
␦
u
⫽
具
n( , )u2( , )
典
. Again, due o s a is ical noise,
␦
uwill be
di e en om ze o when compu ed om a simula ion, e en
i he e a e no luxes. Fo ha eason, in Fig. 5 we ha e
plo ed
␦
uscaled wi h i s alue in he elas ic pa o he
simula ion,
␦
u
el . All he simula ions show ha he eloci y
ield g ows om he e y ea ly s ages o he e olu ion. In
ac , he e is a i s pa o he g ow h o
␦
u/
␦
u
el , which can
be qui e well app oxima ed by an exponen ial, which is com-
mon o all he simula ions. A e ha , he e is a sa u a ion
e ec ha ansla es in o de ia ions om he exponen ial
g ow h o la ge imes, occu ing soone he la ge he den-
si y. In ac , he sa u a ion occu s o imes o he o de o
*, which is he ime when he clus e ing is igge ed.
V. COLLISIONAL AVERAGES
The e olu ion o he collisional a e ages in he sys em
has been compu ed by disc e izing he ime in in e als ⌬
⫽5. P ope ies a e a e aged o e collisions aking place in
each in e al. Also, all he esul s ha e been a e aged o e
se e al uns.
In Fig. 6 we ha e plo ed he e olu ion o he pai co e-
la ion unc ion a con ac , g(
⫹), om he MD simula ions.
Also included is he heo e ical p edic ion in he HCS o he
wo alues o he densi y displayed in he igu e, calcula ed
om Eq. 共14兲. In bo h cases i is ound ha , when he eloc-
i y dis ibu ion begins o depa om he HCS one, i.e., a
⫽
s , he pai co ela ion unc ion akes s ill he HCS alue
and, in ac , de ia ions om i a e qui e small e en a
⫽
*. A e ha , he e is a e y as inc ease o g(
⫹) due o
he o ma ion o clus e s in he sys em. The e o e, he in-
c ease in a2shown in Fig. 2 is no due o he de elopmen o
posi ional co ela ions o colliding pa icles. These co ela-
ions do appea , bu a a he la e imes. We ha e also s ud-
ied he e olu ion o he impac pa ame e dis ibu ion P(b, )
and ound ha in none o he MD simula ions discussed in
his wo k i de ia ed signi ican ly om uni o mi y o he
imes shown in his pape .
Veloci y co ela ions we e in es iga ed h ough he beha -
io o ⌫de ined in Eq. 共16兲, and o he dis ibu ion unc ion
o he angle o med by he eloci ies o colliding pa icles,
. In Fig. 7, he e olu ion o ⌫ o he h ee simula ions is
shown. I mus be no iced ha , in a ini e sys em, e en i
he e a e no co ela ions, ⌫ akes a ini e alue ha depends
on he numbe o pa icles, which is di e en in he h ee
cases. So, e en in he elas ic pa o he simula ion, ⌫is
di e en in each case, being la ge in he dense sys em, as i
has less pa icles. Besides, he alue o ⌫is di e en in an
elas ic sys em and in an inelas ic one in he same condi ions.
The e o e, when he inelas ici y is swi ched on a ⫽0, he e
is a e y as inc ease o ⌫ o i s HCS alue. Then, he e is a
pe iod o e which ⌫does no change oo much, and ha
las s longe he lowe he densi y, ollowed by an almos
exponen ial inc ease o ⌫. I is ound ha he g ow h a e in
his pe iod depends on he densi y, becoming la ge as n
dec eases. Finally, he e is a slowing down in he inc ease o
⌫, and i seems ha i sa u a es in he end o a alue which
FIG. 5. E olu ion o he luc ua ions o he eloci y ield, scaled
wi h i s alue in he elas ic case, o he simula ions discussed in he
main ex .
FIG. 6. E olu ion o he pai dis ibu ion unc ion a con ac ,
g(
⫹), ob ained in he wo MD simula ions discussed in he main
ex . The ho izon al lines a e he heo e ical p edic ion o his unc-
ion in he HCS a n⫽0.0125
⫺2and n⫽0.05
⫺2, om bo om o
op.
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is la ge he la ge he densi y. Also he slowing down begins
soone he la ge he densi y. In any case, we ind again ha
by lowe ing he densi y in he MD simula ions, he beha io
o he sys em app oaches he one p edic ed by he Bol zmann
equa ion. One could be emp ed o conclude ha he inc ease
o ⌫is due o he de elopmen o co ela ions be ween e-
loci ies o colliding pa icles, his happening om he ea ly
s ages o he e olu ion o he sys em. Bu one mus be cau-
ious when in e p e ing his esul . Le us emembe ha in
ou sys em a eloci y ield is also being buil up om he
beginning o he e olu ion, as was shown in Fig. 5, and his
leads o an inc ease in he alue o ⌫ ha has no hing o do
wi h he ailu e o he ac o iza ion o he wo-pa icle dis i-
bu ion unc ion o colliding pa icles, i.e., wi h a iola ion o
he molecula chaos hypo hesis. In ac , he beha io o ⌫
displayed in Fig. 7 is qui e eminiscen o he one o he
luc ua ions o he eloci y ield. Also, in he DSMC simula-
ion, i mus be aken in o accoun ha he molecula chaos
hypo hesis is assumed in he e y basis o he algo i hm, so
he inc ease o ⌫in ha case canno be due o he p esence
o p ecollisional eloci y co ela ions. We conclude hen ha
he g ow h o ⌫displayed in Fig. 7 is due o he ins abili y o
he scaled ans e sal eloci y mode, which leads o he o -
ma ion o o ices and, as a consequence, he eloci ies o
colliding pa icles become mo e and mo e pa allel.
The exis ence o he pa alleliza ion mechanism is in es i-
ga ed also by s udying he e olu ion o he dis ibu ion o he
angle o med by he eloci ies o colliding pa icles, P(
).
In Fig. 8 his dis ibu ion is plo ed a di e en imes o he
MD simula ion wi h n⫽0.0125
⫺2. We ha e included he
dis ibu ion a
⫽0, which is cons uc ed om he elas ic
pa o he simula ion. While he sys em emains in he HCS,
he dis ibu ion o he
angle is indis inguishable om he
elas ic one, so he e a e no appa en angle co ela ions in his
pa o he e olu ion o he sys em. The si ua ion changes a
⬃
s (
s ⬃30 in his case兲, when de ia ions om he elas-
ic dis ibu ion begin o show up. The ela i e numbe o
collisions wi h la ge ela i e angles o he eloci y begin o
dec ease wi h espec o he elas ic case. As ime p oceeds,
his endency becomes s onge , and o he inal imes con-
side ed in he simula ion, mos o he collisions co espond
o pa icles ha a e mo ing almos pa allel. The dis o ion o
he angle dis ibu ion begins p ecisely a he ime when he
ku osis o he o al eloci y dis ibu ion begins o depa
om he HCS alue. I seems sensible o conclude hen ha
he beha io o he eloci y dis ibu ion o he sys em is a
consequence o he pa alleliza ion mechanism, which is in-
he en o inelas ic collisions, and which, when he sys em is
uns able, induces a collec i e beha io o he eloci ies o
pa icles, which ansla es in o a depa u e o he eloci y
dis ibu ion om he HCS one.
The quali a i e beha io o P(
) discussed o n
⫽0.0125
⫺2also holds o he la ge densi y MD simula ion
and o he DSMC one. O cou se, he de ia ion om he
elas ic dis ibu ion occu s soone he la ge he densi y, bu i
is also p esen in he Bol zmann desc ip ion. To ha e a clea
pic u e o he pa alleliza ion o he eloci ies o colliding
pa icles in he h ee cases, in Fig. 9 we ha e plo ed he
e olu ion o
具
典
in he h ee simula ions. The alue o
his quan i y in an equilib ium, elas ic luid, is
具
典
⯝0.59
(⬃107°). In he h ee cases, he sys em emains
wi h he elas ic alue o
具
典
du ing wha we ha e called he
s eady pe iod, and a e wa ds i dec eases wi h ime, indica -
FIG. 7. E olu ion o ⌫ o he simula ions discussed in he main
ex . FIG. 8. P obabili y dis ibu ion o he angles o he eloci ies
be ween colliding pa icles P(
) om he MD simula ion wi h n
⫽0.0125
⫺2.
FIG. 9. E olu ion o he a e age alue o he angle be ween he
eloci ies o colliding pa icles,
具
典
, o he simula ions discussed
in he main ex .
J. J. BREY AND M. J. RUIZ-MONTERO PHYSICAL REVIEW E 69, 011305 共2004兲
011305-8
ing ha he e is a endency o ha e mo e collisions wi h
eloci ies ha o m a small angle. I mus be also emem-
be ed ha , in he MD simula ions, we ha e obse ed ha he
impac pa ame e dis ibu ion did no show de ia ions om
uni o mi y o he imes conside ed in his wo k.
VI. DISCUSSION
The simula ion o a sys em on eely e ol ing ha d disks
in condi ions such ha he HCS is uns able shows ha he
Bol zmann desc ip ion seems o p o ide a alid pic u e o
he beha io o a low-densi y g anula gas, con a y o he
s a emen made in Re . 关14兴. This has been es ablished by
showing ha he e a e no quali a i e di e ences be ween he
esul s ob ained in low-densi y MD simula ions and hose
ha ollow by using he DSMC me hod. I has been shown
ha , when he densi y is lowe ed in he sui able way in he
MD simula ions, i.e., lea ing he cha ac e is ic ime o he
de elopmen o ins abili ies unchanged, he beha io o he
sys em ends o he Bol zmann one. Ne e heless, i is also
ue ha he de ia ions om he Bol zmann beha io a e
la ge in a sys em in hese condi ions o ins abili y han in a
s able one, o he same alues o he densi y and es i u ion
coe icien . In o he wo ds, i could be said ha he ange o
densi ies o which he Bol zmann equa ion holds depends
a he s ongly on he s a e o he sys em, and i canno be
gi en a gene al ule. In he si ua ion conside ed he e, he
eason o his seems o be ha , when he HCS in uns able,
he e is a pa alleliza ion o eloci ies o colliding pa icles
ha is mo e e icien he highe he densi y, al hough i is
also p esen in he Bol zmann desc ip ion.
The ole o he spa ial co ela ions be ween colliding pa -
icles has been in es iga ed in his wo k by s udying he pai
dis ibu ion a con ac and he p obabili y dis ibu ion o im-
pac pa ame e s. The o me only de ia es om he HCS
alue when densi y inhomogenei ies a e al eady de eloped
in he sys em, while he la e emains always uni o m. This
implies ha , in a low-densi y g anula gas, he de elopmen
o spa ial co ela ions does no play a signi ican ole in he
ea ly depa u e om he HCS.
The abo e esul s seem o indica e ha , when ying o
ex end he inelas ic Bol zmann equa ion o ini e highe den-
si y, he e ec o eloci y co ela ions be ween colliding pa -
icles mus be inco po a ed. A leas , in he physical si ua ion
conside ed in his pape , eloci y co ela ions become impo -
an qui e be o e he sys em de elops signi ican spa ial co -
ela ions, as measu ed by he pai dis ibu ion unc ion a
con ac . I is sensible o expec ha his e ec inc eases as
he inelas ici y o he sys em inc eases. I his pic u e we e
igh , kine ic equa ions o inelas ic dense gases should no
be based on Enskog-like equa ions, aking in o accoun only
spa ial co ela ions, bu new app oxima ions inco po a ing
he e ec o eloci y co ela ions in he p ecollisional sphe e
a e needed. This implies a a he s ong depa u e om he
adi ional me hods o kine ic heo y o elas ic sys ems.
ACKNOWLEDGMENTS
We acknowledge pa ial suppo om he Minis e io de
Ciencia y Tecnologı
´a共Spain兲 h ough G an No. BFM2002-
00303 共pa ially inanced by FEDER unds兲.
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