Mesoscopic theory of critical fluctuations in isolated granular gases

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] [1] H. M. Jaege , S. R. Nagel, and R. P. Beh inge , Re . Mod. Phys. 68, 1259 (1996). [2] I. Goldhi sch, Annu. Re . Fluid Mech. 35, 267 (2003). [3] J. J. B ey, M. I. Ga cı ´a de So ia, P. Mayna , and M. J. Ruiz- Mon e o, Phys. Re . Le . 94, 098001 (2005). [4] S. T. B amwell, P. C. W. Holdswo h, and J.-F. Pin on, Na u e (London) 396, 552 (1998). [5] S. T. B amwell, K. Ch is ensen, J.-Y. Fo in, P. C. W. Holdswo h, H. J. Jensen, S. Lise, J. M. Lo ´pez, M. Nicodemi, J.-F. Pin on, and M. Selli o, Phys. Re . Le . 84, 3744 (2000). [6] L. Landau and E. M. Li shi z, Fluid Mechanics (Pe agamon P ess, New Yo k, 1959). [7] R. B i o and M. H. E ns , Eu ophys. Le . 43, 497 (1998). [8] P. K. Ha , J. Fluid Mech. 134, 401 (1983). [9] T. P. C. an Noije, M. H. E ns , R. B i o, and J. A. G. O za, Phys. Re . Le . 79, 411 (1997). [10] I. Goldhi sch and G. Zane i, Phys. Re . Le . 70, 1619 (1993); S. McNama a and W. R. Young, Phys. Re . E 50, R28 (1994). [11] J. J. B ey, M. J. Ruiz-Mon e o, and D. Cube o, Phys. Re . E 60, 3150 (1999). [12] J. J. B ey, M. I. Ga cı ´a de So ia, P. Mayna , and M. J. Ruiz- Mon e o, Phys. Re . E 70, 011302 (2004). [13] N. B illian o , C. Saluen ˜a, T. Schwage , and T. Po ¨schel, Phys. Re . Le . 93, 134301 (2004). -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