Brownian motion in a granular gas

PHYSICAL REVIEW E VOLUME 60, NUMBER 6 DECEMBER 1999 Brownin motion in grnulr gs J. Jvier Brey, M. J. Ruiz-Montero, nd R. Grcí-Rojo Físic Teóric, Universidd de Sevill, E-41080 Sevill, Spin Jmes W. Dufty Deprtment of Physics, University of Florid, Ginesville, Florid 32611 Received 2 June 1999 The dynmics of hevy prticle in gs of much lighter prticles is studied vi the Boltzmnn-Lorentz eqution with inelstic collisions mong ll prticles. A forml expnsion in the rtio of gs to tgged prticle mss trnsforms the Boltzmnn-Lorentz eqution into Fokker Plnck eqution. The predictions of the ltter re tested here using direct Monte Crlo simultion of the Boltzmnn-Lorentz eqution. Excellent greement is obtined for the pproch to homogeneous cooling stte, the temperture of tht stte, pproch to diffusion, nd the dependence of the diffusion constnt on dissiption prmeters. Some results from moleculr-dynmics simultions re lso presented nd shown to gree with the theoreticl predictions. S1063-651X9908012-5 PACS numbers: 81.05.Rm, 05.20.Dd, 05.40. I. INTRODUCTION The description of low-density, rpid grnulr flow by Boltzmnn kinetic theory hs received much ttention in recent yers 1,2. A primry objective hs been the derivtion of hydrodynmic equtions nd expressions for the trnsport coefficients ppering in them s functions of the coefficient of restitution. The most ccurte pproch uses generliztion of the Chpmn-Enskog method known from the corresponding nlysis of the Boltzmnn eqution for elstic collisions 3,4. This method ssumes the existence of norml solution whose spce nd time dependence occurs only through the hydrodynmic fields. Implicit in this ssumption is the rpid relxtion of nonhydrodynmic excittions so tht the hydrodynmic description domintes on longer time scle. Such seprtion of time scles hs been questioned in the cse of grnulr flow, prticulrly for lrge degrees of inelsticity 5. The bsis for this concern seems to be the dditionl time scle set by the cooling of the homogeneous reference stte, such tht the hydrodynmic time scles re not simply determined by the degree of sptil inhomogeneity. However, detiled study of self-diffusion 6 nd kinetic models bsed on the Boltzmnn eqution 7 suggests this concern is not justified. For these cses it is shown tht the microscopic excittions lwys decy on times short compred to ll hydrodynmic times, including tht defined by the cooling rte. To further reinforce the cse for hydrodynmic description, n exct nlysis of the Boltzmnn-Lorentz kinetic eqution for tgged prticle in freely evolving gs hs been performed in the limit of symptoticlly lrge reltive mss for the tgged prticle 7. In this limit, the Boltzmnn- Lorentz eqution reduces to Fokker-Plnck eqution. An exct nlysis of the spectrum for this eqution confirms the seprtion of time scles for ll degrees of tgged prticle inelsticity. It remins to confirm the vlidity of the forml symptotic nlysis leding from the Boltzmnn-Lorentz eqution to the Fokker-Plnck eqution. One objective here is to provide this confirmtion on the bsis of direct Monte Crlo simultion of the Boltzmnn-Lorentz eqution. In the next section the Boltzmnn-Lorentz nd its Fokker- Plnck limit re reclled. A chnge of spce nd time vribles provides n exct mp of the Fokker-Plnck eqution for inelstic collisions to tht for elstic collisions. Consequently, ll the known results from the ltter cse for velocity relxtion nd the pproch to diffusive stge trnslte exctly to the cse of inelstic collisions. Severl exmples re considered explicitly. The qulittive differences occur only through the chnges in spce nd time scles. Some of the most interesting differences re s follows. i The renormlized time scle is relted logrithmiclly to rel time. Accordingly, velocity relxtion nd pproch to hydrodynmics is lgebric rther thn exponentil. ii The long time limit of the tgged prticle distribution for the homogeneous stte is Gussin, lthough the bth prticle distribution is non-gussin. iii The time-dependent temperture of the tgged prticle Gussin differs from the temperture of the surrounding bth, lthough the cooling rtes of both become the sme t long times. iv The men-squre displcement pproches ln t for lrge times with coefficient tht determines the diffusion coefficient. v The diffusion eqution t long times hs the usul form, lthough its solution is qulittively different from tht for the elstic cse due to the time dependence of the temperture in the diffusion coefficient. In Sec. III the direct simultion Monte Crlo method to obtin numericl solutions of the Boltzmnn eqution is described briefly. Comprisons re mde between the simultions of the Boltzmnn-Lorentz eqution nd the predictions of the Fokker-Plnck eqution for tgged prticle whose mss is 100 times tht of the surrounding gs prticles. The Mxwellin distribution is confirmed, s is the pproch to different tempertures for the bth nd tgged prticle t the sme cooling rtes. The men-squre displcement is mesured nd the diffusion coefficient extrcted from its long time behvior to confirm the detiled dependence on inelsticity s predicted by the Fokker-Plnck eqution. The results provide strong support for the ccurcy of the Fokker-Plnck 1063-651X/99/606/71748/$15.00 PRE 60 7174 1999 The Americn Physicl Society

PRE 60 BROWNIAN MOTION IN A GRANULAR GAS 7175 eqution s representtion of the Boltzmnn-Lorentz eqution for lrge reltive mss. Of course, ll the bove discussion relies on the vlidity of the inelstic Boltzmnn-Lorentz eqution to describe mssive tgged prticle in freely evolving gs. While this vlidity is well estblished for elstic collisions, it is sometimes questioned for inelstic collisions. To provide support for this cse, we hve performed moleculr-dynmics simultions for this system in two dimensions. The results presented in Sec. IV confirm the nlysis bsed on the kinetic eqution, indicting clerly the vlidity of kinetic equtions to study rpid grnulr flows. This section lso includes summry nd discussion of the min results in the pper. II. FOKKER-PLANCK EQUATION AND ITS SOLUTION A tgged prticle immersed in low-density gs is considered. The gs is formed by hrd spheres (d3) or disks (d2) of mss m g nd dimeter g, nd the tgged prticle is lso hrd sphere or disk, but with mss m nd dimeter. All prticles re smooth nd collide inelsticlly. Collisions re chrcterized by velocity-independent coefficients of norml restitution. For the collisions between the gs prticles it will be denoted by g, while tht for collisions between the tgged prticle nd gs prticles will be represented by. The probbility density F(r,v,t) describing the dynmics of the tgged prticle obeys the Boltzmnn-Lorentz eqution t v FJr,v,tF, f. The collision opertor J is given by 7,8 Jr,v,tF, f 0 d1 dv 1 dˆ g ˆ g ˆ 2 Fr,v,tfr,v 1,t Fr,v,tfr,v 1,t, 2.1 2.2 where f (r,v,t) is the corresponding distribution for the surrounding gs prticles, is the Heviside step function, ˆ is unit vector pointing from the center of the gs prticle 1 to the center of the tgged prticle t contct, nd 0 ( g )/2. The precollisionl or restituting velocities v nd v 1 re given by vv 1 1 g ˆ ˆ, v 1 v 1 1 1 g ˆ ˆ, 2.3 with gvv 1 the reltive velocity nd m g /m the rtio of gs to tgged prticle mss. The surrounding gs is tken to be in its homogeneous cooling stte HCS s determined from the solution to the nonliner Boltzmnn eqution. Its distribution function hs the scling form 1 f H v,tn g v d v g t, 2.4 v g t where n g is the constnt number density of the gs, v g (t) 2k B T g (t)/m g 1/2 is the therml velocity of the gs prticles t time t with k B being the Boltzmnn constnt, nd T g (t) is the temperture of the gs which cools ccording to the eqution dt g t tt dt g t. 2.5 The cooling rte (t) depends on time only through T g (t) nd is determined from the second moment of the Boltzmnn eqution for the gs 8, t1 2 g n g d1 g v g t 2d d1/2 d3 2 dv dv 1 g 3 vv 1. 2.6 The solution to Eq. 2.5 is T g tt g 0 1 0 2 2 t, 2.7 showing tht the temperture of the gs decreses s t 2 for lrge times if the system remins in the HCS. The explicit form of will be not given here, but it is known in the so-clled first Sonine pproximtion 1,9. In Ref. 7 it ws shown tht the Boltzmnn-Lorentz eqution reduces to Fokker-Plnck eqution for symptoticlly smll, t v Fr,v,t e t v v k BT g t m vfr,v,t, 2.8 where e (t) is the sme friction coefficient s for elstic collisions, except s function of the time-dependent temperture T g (t), e t 4d1/2 d1 0 n dd/2 g v g t, 2.9 where v g (t) is defined in terms of T g (t) following Eq. 2.4 bove. All effects of inelstic collisions mong the gs prticles in Eq. 2.8 pper through the time dependence of T g (t), while the inelsticity of collisions between the tgged prticle nd gs prticles mnifests itself only through the prmeter (1)/2. The Fokker-Plnck eqution 2.8 cn be mpped onto the corresponding eqution for elstic collisions using the dimensionless vribles v* v v 0 t, where we hve introduced r*1 et v 0 t r, t*1 0 t dte t, 2.10

7176 BREY, RUIZ-MONTERO, GARCÍA-ROJO, AND DUFTY PRE 60 v 0 t 1 1/2 2k BT g t m 1/2, t 2 e t. 2.11 Since both (t) nd e (t) re proportionl to T 1/2 g (t), it follows tht is time-independent quntity. The bove definitions pply only for 1, point discussed further in the finl section. Here we only note tht it is necessry condition in the derivtion of Eq. 2.8. In terms of the new vribles, the Fokker Plnck eqution 2.8 becomes t * v* F*r*,v*,t* r* v* v* 1 F*r*,v*,t*, 2.12 2 v* with the scled probbility density F* given by v 2d 0 t F*r*,v*,t* 1 e t Fr,v,t, 2.13 d which is lso normlized to unity. Eqution 2.12 is the sme s the dimensionless form of the Fokker-Plnck eqution for elstic collisions ( g 1). Consequently, the physicl properties of mssive tgged prticle with inelstic collisions moving in gs in the HCS re the sme s those for n elstic prticle in n equilibrium gs; the only differences re the relevnt spce nd time scles. For exmple, t* e t for elstic collisions while for inelstic collisions the reltionship is t* 1 ln 1 0 2 t, 2.14 nd the time scle is stretched logrithmiclly. The generl solution of Eq. 2.12 to the initil vlue problem for n unbounded system is well known 10: F*r*,v*,t* dr* dv* G*r*,v*,t*;r*,v*,0 with G*r*,v*,t*;r*,v*,0 et*bt* 2 F*r*,v*,0, d/2 expet*r*r* ct* v*v* 2 bt*v*e t* v* 2. 2.15 2.16 Consider first sptilly homogeneous initil stte for the tgged prticle F*(r*,v*,0)F*(v*,0). Then Eq. 2.15 simplifies to F*v*,t* bt* d/2 dv* expbt* v*e t* v* 2 F*v*,0. For t*1, b(t*) 1 nd, consequently, F*v*,t* F M * v* 1 * d/2 e v*2, 2.18 2.19 where * is the volume of the system mesured in the reduced length scle defined in Eq. 2.10. Thus, for generl homogeneous initil conditions, the distribution function pproches exponentilly fst sttionry Mxwellin distribution in the reduced units. In terms of the originl vribles this result is Fv,t F M v,t 1 d/2 m e 2k B T t mv2 /2k B T (t), 2.20 with the temperture prmeter T (t) given by T tt g t 1. 2.21 Interestingly, the tgged prticle pproches homogeneous cooling stte with scling form similr to Eq. 2.4 for the gs, but with two importnt differences. First, the cooling stte for the tgged prticle is Gussin while tht for the gs is not. Second, the cooling temperture for the tgged prticle is different from tht for the gs, lthough the cooling rtes re the sme. To elborte on this ltter feture, it is convenient to define more generlly the kinetic temperture for the tgged prticle by d 2 k BTt dr dv 1 2 mv2 Fr,v,t k BT g t 1 dr* dv* v* 2 F*r*,v*,t. Evluting the integrl using Eq. 2.15 gives Tt T g t 1 T0 e2t* T g 0 1. 2.22 2.23 In the bove expressions we hve defined 1 et* 2t*2ct*, bt* 1 1e 2t*, ct* 1et* 1e t*. 2.17 The pproch of the tgged prticle temperture to the symptotic vlue T (t) is exponentilly fst on the reduced time scle. Note tht T (t)t g (t) even in the cse of elstic collisions for the tgged prticle (1), if the gs prticle collisions re inelstic. Conversely, if gs prticle collisions re elstic but the tgged prticle collisions re inelstic, then 0 nd the tgged prticle pproches the constnt temperture of the gs, but still differs from it by fctor of.

PRE 60 BROWNIAN MOTION IN A GRANULAR GAS 7177 The result in Eq. 2.23 pplies even for rbitrry inhomogeneous sttes of the Brownin prticle. Consider next n initil stte whose velocity distribution is given by the homogeneous cooling Mxwellin but with sptil inhomogeneity, F*r*,v*,0n*r*,0 d/2 e v*2, 2.24 where n*(r*,0) is the probbility density for finding the tgged prticle t position r* t time t0. The corresponding quntity t time t* is obtined by integrtion of F*(r*,v*,t*), s given in Eq. 2.15, with respect to v*, giving d/2 d n*r*,t* dr* where 2l* 2 t* expdr*r* 2 /2l* 2 t*n*r*,0, 2.25 l* 2 t*dt*1e t*. 2.26 It is esily verified tht l* 2 (t*) is the men-squre displcement of the tgged prticle in terms of the dimensionless vribles. Moreover, it follows directly from Eq. 2.25 tht n*(r*,t*) obeys the extended diffusion eqution t * n*r*,t*1et* D* * 2 n*r*,t*, 2.27 with D*1/2. This eqution is exct for ll times if the initil condition hs the ssumed form, nd shows most directly the pproch to hydrodynmic stge, i.e., the usul diffusion eqution pplies exponentilly fst for t*1. The vlue D*1/2 is consistent with the Einstein result D* 1 2d lim t* 1 l* 2 t*. 2.28 t* In terms of the originl vribles, the symptotic diffusion eqution is with t nr,tdt 2 nr,t, Dt D et 1 2. 2.29 2.30 Here D e (t)k B T g (t)/m e (t) is the sme s the diffusion coefficient for elstic collisions, except s function of the time-dependent gs temperture. For given temperture, the diffusion coefficient is seen to be enhnced by fctor of (1) 2, depending only on the inelsticity of gs collisions. For more generl initil conditions, the dynmics is qulittively similr with rpid velocity relxtion followed by sptil diffusion. In the next section, severl predictions from this Fokker-Plnck nlysis re tested ginst direct simultion of the corresponding property from the Boltzmnn- Lorentz eqution. III. MONTE CARLO SIMULATION OF THE BOLTZMANN-LORENTZ EQUATION To test the forml nlysis leding from the Boltzmnn- Lorentz eqution to the Fokker-Plnck, the direct simultion Monte Crlo DSMC method 11 hs been pplied to the former. Specificlly, the predictions of velocity relxtion to Mxwellin t new temperture nd the pproch to diffusion with n enhnced diffusion coefficient hve been investigted for system of hrd spheres with mss rtio 10 2, equl dimeters for ll prticles, nd coefficient of norml restitution for collisions between gs prticles g 0.99. These vlues ssure the necessry condition 1 for 01, s shown in the Appendix. The DSMC for the Boltzmnn eqution with elstic collisions is well described in Ref. 11. Its dpttion to the Boltzmnn-Lorentz eqution with inelstic collisions is strightforwrd. As indicted in Eq. 2.1, the distribution function for the gs, f s, is required for input, nd is tken here to be the homogeneous cooling solution to the Boltzmnn eqution f H. It is known tht this solution is unstble to long-wvelength sptil perturbtions t sufficiently long times, so ny possible interference of relxtion processes by this instbility is not ddressed. However, for the chosen vlue of g the homogeneous cooling stte is stble on the time scles studied 12. In the simultions 10 5 trjectories of tgged prticle hve been generted. Independently of its position, collisions of the tgged prticle lwys took plce with prticles of homogeneous gs whose velocity distribution ws given by Eq. 2.4. As lredy mentioned, the exct form of is not known, nd the expression obtined in the first Sonine pproximtion 1,9 ws used to generte the velocity of the colliding gs prticles. Thus, the trjectories of the gs prticles re not required, which increses gretly the efficiency of the numericl simultion nd voids the introduction of specific boundry conditions the system is formlly considered s infinite. A similr method lredy hs been used to study self-difussion in low-density grnulr flow 6. In typicl run, collisions between the tgged prticle nd the fluid prticles were considered s uncoupled during time step t 0, chosen much smller thn the initil verge collision time of the tgged prticle. This mens tht the position of the tgged prticle ws chnged t constnt velocity for n intervl t 0 between every two pplictions of the collision lgorithm. For the collisions, the velocities of the prticles were generted from the HCS distribution, s mentioned bove, with gs temperture determined from the lw given by Eq. 2.5 nd considered to be constnt in the time intervl t 0. This implies tht t 0 hs to be chosen lso much smller thn the time chrcterizing the cooling of the gs, 2/(0). In the simultions we report here we used t 0 210 4 /(0). The initil condition in ll simultions ws Gussin velocity distribution with the sme temperture s the surrounding fluid. As first test of the Fokker-Plnck limit, the predicted pproch to Gussin cooling stte is studied. The ssoci-

7178 BREY, RUIZ-MONTERO, GARCÍA-ROJO, AND DUFTY PRE 60 FIG. 1. Plot of the rtio of the tgged prticle temperture to the surrounding gs temperture s function of the dimensionless reduced time t* defined in Eq. 2.10. The rtio of gs to tgged prticle mss is 10 2 nd the coefficient of restitution for the gs collisions is g 0.99. The solid lines re the predictions from the Fokker-Plnck eqution nd the symbols from the DSMC method of the Boltzmnn-Lorentz eqution. ted scling velocity is defined in terms of the temperture T(t) of the tgged prticle, chrcterizing its kinetic energy. It is expected tht this temperture pproches n symptotic vlue proportionl to the gs temperture T g (t), ccording to Eq. 2.23. Figure 1 shows the time evolution of the rtio T(t)/T g (t) s obtined from the numericl simultion of the Boltzmnn-Lorentz eqution for 0.5, 0.7, nd 0.99. In ll cses it is seen tht the rtio evolves from the unit initil condition until reching stedy vlue. Also plotted is the theoreticl prediction. Figure 2 shows detiled comprison of the numericl symptotic vlues with Eq. 2.21 over this rnge of vlues of. The greement is very good considering tht corrections to the Fokker Plnck limit re of order FIG. 2. Asymptotic vlue of the rtio between the tgged prticle temperture T nd the gs temperture T g s function of the coefficient of norml restitution for collisions between the tgged prticle nd the fluid prticles. The mss rtio nd the coefficient of restitution for gs collisions re the sme s in Fig. 1. The solid line is from the Fokker-Plnck eqution nd the dots from the numericl simultion of the Boltzmnn-Lorentz eqution. FIG. 3. Time evolution of the reduced fourth velocity moment of the tgged prticle distribution. Here time is mesured in units of /v 0, where is the men free pth. The continuous line corresponds to 0.95 nd the dshed line to 0.5. The stright lines represent the fourth moment of the HCS distribution for the fluid t 0.95 continuous nd 0.5 dshed. 1/2. Interestingly, we hve found tht the greement between theory nd simultions improves significntly if the mss rtio nd g re chnged to reduce the vlue of. The Gussin chrcter of the tgged prticle distribution cn be studied vi its fourth moment, or the normlized expression 3v 4 /5v 2 2, which hs the vlue unity for Gussin. Here it is v n 1 dr dv v n Fr,v,t. 3.1 Figure 3 shows the simultion vlues for the normlized moment s function of time for 0.5 nd 0.99. Also shown re the corresponding results for the gs distribution, i.e., for the velocity distribution in Eq. 2.4. The ltter shows significnt devitions from unity wheres the tgged prticle results confirm the Gussin even t strong dissiption. Although we hve plotted in the figure the results corresponding to two extreme vlues of, similr behvior hs been obtined lso for severl intermedite vlues. It could be climed tht the Gussin distribution observed in the simultions is in some wy influenced by the initil Gussin distribution. To clrify this point we hve lso considered initil distributions fr from the Gussin, nmely uniform distribution with zero men. It ws observed tht the distribution evolved towrds Gussin very fst, indicting tht the use of n initil Gussin does not limit t ll the results presented here. The pproch to diffusion cn be studied vi the mensqure displcement of the tgged prticle. In Fig. 4, comprison of the simultions with the universl form given by Eq. 2.26 for 0.5, 0.7, nd 0.99 is presented. The results confirm both the pproch to diffusion nd the predicted limiting form with D*1/2. Figure 5 shows more detiled comprison with the predicted diffusion coefficient over wide rnge of vlues. Agin, the greement is very good, since the discrepncies re smller thn 2%.

PRE 60 BROWNIAN MOTION IN A GRANULAR GAS 7179 FIG. 4. Time evolution of the men-squre displcement of the tgged prticle for three different vlues of the coefficient of restitution for collisions between the tgged prticle nd the gs prticles. Time nd length re mesured in the reduced dimensionless units defined in the min text. The solid line is the theoreticl prediction given by Eq. 2.26. IV. DISCUSSION The Fokker-Plnck eqution, obtined from the Boltzmnn-Lorentz eqution in the limit of smll rtio of gs to tgged prticle mss, llows n exct nlysis of the tgged prticle dynmics. The surrounding gs is ssumed to be in the homogeneous cooling stte. In prticulr, the study shows the cler seprtion of time scles required for hydrodynmic description. For tgged prticle dynmics the ltter refers to diffusion. The objective here hs been to confirm the vlidity of the symptotic nlysis leding to the Fokker-Plnck limit by direct Monte Crlo simultion of the Boltzmnn-Lorentz eqution. The excellent greement found for the vrious properties studied provides convincing evidence for the vlidity of the Fokker-Plnck eqution nd its exct consequences. The seprtion of time scles between microscopic kinetic nd diffusive modes is quite cler, once the complictions due to cooling hve been suppressed by the chnge of FIG. 5. Comprison between the numericl vlue of the reduced self-diffusion coefficient obtined from the Boltzmnn-Lorentz eqution dots nd the predicted vlue D*1/2 solid line. FIG. 6. Distribution function of the tgged prticle normlized by the Gussin for three different vlues of the restitution coefficient. For the three of them it is 1, nd the Fokker-Plnck eqution is not expected to hold. The velocity is mesured in the reduced units defined in Eq. 2.10. vribles in Eq. 2.10. In the new vribles, the description is independent of nd, therefore, pplies for rbitrry degree of dissiption for the tgged prticle. However, the gs is implicitly restricted to wek dissiption due to the condition 1, where is defined by Eq. 2.11. The explicit form for given in the Appendix shows tht this condition requires 1 2 g. The origin of this condition is requirement tht the rtio of tgged prticle temperture to gs temperture does not grow in time, since it ppers s fctor of in the symptotic nlysis 7. The relevnce of this condition for the Fokker-Plnck limit is estblished in Fig. 6 showing the distribution function F normlized by the Gussin F M for stronger gs dissiption, g 0.95, nd for severl vlues of the restitution coefficient for collisions between the tgged prticle nd the fluid, 0.99, 0.9, nd 0.8. The corresponding vlues for re 1.73, 1.81, nd 1.915, respectively. The devitions from Mxwellin re now lrge nd the Fokker-Plnck description, which predicts Mxwellin velocity distribution, is no longer vlid. Let us stress tht the restriction 1 ffects only the Fokker-Plnck limit nd not the more generl picture of pproch to diffusion, s described by the Boltzmnn-Lorentz eqution. This hs been demonstrted in Ref. 6, where the men-squre displcement hs been simulted for mechniclly identicl prticles for 0.6 g 1, which includes conditions of strong gs prticle dissiption nd consequently 1. All our previous nlysis is bsed on the ccurcy of the Boltzmnn-Lorentz eqution to describe the time evolution of tgged prticle in low-density gs, when ll the collisions re inelstic. For the prticulr cse of self-diffusion the vlidity of the kinetic eqution hs been explicitly shown in Ref. 6, by compring its predictions with the results obtined from moleculr-dynmics simultion of system of inelstic disks. Here we present some dditionl evidence when the Brownin limit of mssive prticle is considered. From computtionl point of view, the study of time evolution of mssive prticle in bth, whose stte must not be perturbed by the motion of the prticle, is very demnding simultion, requiring lot of computing time in order to get

7180 BREY, RUIZ-MONTERO, GARCÍA-ROJO, AND DUFTY PRE 60 FIG. 7. Time evolution of the men-squre displcement of tgged disk in low-density gs of inelstic prticles, using moleculr dynmics. The mss rtio is 210 2 nd two vlues for the coefficient of restitution for collisions of the tgged prticles hve been considered: 0.9 squres nd 0.8 circles. The coefficient of restitution for collisions mong fluid prticles is g 0.99. Quntities re mesured in the dimensionless reduced units defined in the min text. The solid line is the theoreticl prediction from the Fokker-Plnck description, Eq. 2.26. significnt level of sttistics. We hve simulted system of hrd disks in squre domin with periodic boundry conditions. The gs ws composed of 3025 prticles nd the results we present re verges over 500 trjectories of the Brownin prticle, ech of them generted by different computer run. Agin the tgged nd gs prticles were of the sme size. The initil condition ws generted by running the gs, without tgged prticle, for period of time lrge enough to llow the system to rech the homogeneous cooling stte. Then, one of the gs prticles, rndomly selected, ws substituted by the tgged prticle, i.e., its mss ws chnged to the new vlue. The simultion technique ws bsed on the event driven lgorithm 13. Figure 7 shows the results obtined for the time evolution of the men-squre displcement of the tgged prticle in system with g 0.99 nd mss rtio 0.02. The density of the gs is n 2 1.32210 3, which corresponds to solid frction 1.03910 3. Two vlues of the restitution coefficient for collisions of the tgged prticle hve been considered, 0.9 nd 0.8. The solid line is the prediction given by Eq. 2.26. Agin good greement is obtined, providing very strong test on the vlidity of the kinetic eqution even before the system reches the hydrodynmic regime. Similr results were obtined for other vlues of the prmeters of the system, nmely those considered in the DSMC results reported in the preceding section. In the figure it is observed tht the discrepncy between theory nd simultion increses with time. This is sttistics effect which decreses s the number of trjectories increses. This is more clerly seen in Fig. 8, where we hve represented the time evolution of the temperture of the tgged prticle for the simultions corresponding to 0.9 in Fig. 7. Although the results re consistent with the predictions from the Fokker-Plnck eqution, the fluctutions re too lrge to mke detiled quntittive comprison. This is not the cse FIG. 8. Time evolution of the temperture of the Brownin prticle for the sme system s considered in Fig. 7 with 0.9. The temperture is scled with the time-dependent temperture of the gs. The solid line is the theoreticl prediction given by Eq. 2.23. Time is mesured in the reduce unit defined in the text. for the men-squre displcement, s seen in Fig. 7. In this sense, when compring the results obtined by using the DSMC method with those from MD, we must tke into ccount tht the number of trjectories we hve considered in the former is 200 times lrger thn in the ltter. Incresing the number of MD trjectories by such fctor would imply time too long for prcticl computer simultions. In summry, forml limit of the Boltzmnn-Lorentz eqution for tgged prticle with mss lrge compred to prticles of the surrounding gs leds to Fokker-Plnck kinetic eqution. This eqution cn be mpped onto the corresponding eqution for elstic collisions by chnge of spce nd time scles whose exct solution is known. In prticulr, this eqution demonstrtes the seprtion of time scles ssocited with the rpid trnsition from complex initil trnsients to hydrodynmics diffusion. The property occurs independent of the degree of dissiption in the collisions between the hevy prticle nd surrounding gs prticles. In the present pper, the forml nlysis leding to the Fokker- Plnck eqution hs been verified by direct simultion of the Boltzmnn-Lorentz eqution. Finlly, the vlidity of the Boltzmnn-Lorentz eqution itself hs been confirmed by selected moleculr-dynmics simultions. ACKNOWLEDGMENTS The reserch of J.J.B., M.J.R-M., nd R.G-R. ws prtilly supported by the Dirección Generl de Investigción CientíficyTécnic Spin through the grnt PB98-1124. The reserch of J.W.D ws supported in prt by NSF Grnt No. PHY 9722133. APPENDIX: DETAILED FORM FOR The Fokker-Plnck limit leding to Eq. 2.8 is restricted to smll mss rtio nd 1. The prmeter is defined by Eq. 2.11, with the cooling rte nd friction coefficient e given by Eqs. 2.6 nd 2.9, respectively,

PRE 60 BROWNIAN MOTION IN A GRANULAR GAS 7181 t 2 e t 1 g 2 d/2 16 d3 2 1 g 2 d1 g 42 0 d1 g 0 dv dv 1 g 3 vv 1 1 3 32 c* g. A1 In the third line the integrl hs been performed using n pproximte solution to the Boltzmnn eqution obtined by expnding (v) in Sonine polynomils nd retining the first correction to the Gussin 1,9. This pproximtion hs been tested vi Monte Crlo simultion nd is ccurte for the relevnt velocities within few percent 12. The contribution c*( g ) bove is due to this first correction nd is given by 321 g 12 2 g c* g 924d8d41 g 30 2 g 1 g. A2 The condition 1 restricts the rnge of g to tht for wek dissiption in the gs since must be smll. The detiled vlues cn be djusted by considering prticles of very different sizes. For simplicity, the sizes hve been tken equl in the simultions reported in Sec. III. Then, for the chosen vlues in the DSMC method in Sec. III, g 0.99 nd 10 2, the bove expression for simplifies to 0.704 1, A3 where we hve lso prticulrized for d3. Clerly 1 for ll 01 nd the Fokker-Plnck eqution is expected to hold for rbitrry inelsticity of the collisions of the tgged prticle with the gs prticles. In the reported MD simultions, we used hrd disks (d 2), 210 2, nd g 0.99. This leds to 0.351 1, A4 nd, gin, the condition 1 is fulfilled for rbitrry vlues of. When crrying out the simultions, we hve observed tht the greement with the theoreticl predictions inproves s the vlue of decreses, lthough the vlidity of the theory requires just 1. 1 A. Goldshtein nd M. Shpiro, J. Fluid Mech. 282, 75 1995. 2 N. Sel nd I. Goldhirsch, J. Fluid Mech. 361, 411998. 3 J.J. Brey, J.W. Dufty, C-S. Kim, nd A. Sntos, Phys. Rev. E 58, 4638 1998. 4 V. Grzo nd J.W. Dufty, Phys. Rev. E 59, 5895 1999. 5 M-L. Tn nd I. Goldhirsch, Phys. Rev. Lett. 81, 3022 1998; L.P. Kdnoff, Rev. Mod. Phys. 71, 435 1999. 6 J.J. Brey, M.J. Ruiz-Montero, D. Cubero, nd R. Grci-Rojo, Phys. Fluids to be published. 7 J.J. Brey, J.W. Dufty, nd A. Sntos, J. Stt. Phys. 97, 281 1999. 8 J.J. Brey, J.W. Dufty, nd A. Sntos, J. Stt. Phys. 87, 1051 1997. 9 T.P.C. vn Noije nd M.H. Ernst, Grnulr Mtter 1, 57 1998. 10 J. A. McLennn, Introduction to Nonequilibrium Sttisticl Mechnics Prentice-Hll, Englewood Cliffs, NJ, 1989. 11 G. Bird, Moleculr Gs Dynmics nd the Direct Simultion of Gs Flows Clrendon Press, Oxford, 1994. 12 J.J. Brey, M.J. Ruiz-Montero, nd D. Cubero, Phys. Rev. E 54, 3664 1996. 13 M.P. Allen nd D.J. Tildesley, Computer Simultion of Liquids Oxford University Press, Oxford, 1987.