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Noname manuscript No. (will be inserted by the editor) Dynamics of a massive intruder in a homogeneously driven granular fluid A. Puglisi · A. Sarracino · G. Gradenigo · D. Villamaina 2 1 0 Received: date/Accepted: date 2 n Abstract A massive intruder in a homogeneously driven granular fluid, in a J diluteconfigurations,performsamemory-lessBrownianmotionwithdragand 9 temperature simply related to the average density and temperature of the fluid.Atvolumefraction∼10−50%theintruder’svelocitycorrelateswiththe ] localfluidvelocityfield:suchsituationisapproximatelydescribedbyasystem t f of coupled linear Langevin equations equivalent to a generalized Brownian o motion with memory. Here one may verify the breakdownof the Fluctuation- s . Dissipation relation and the presence of a net entropy flux - from the fluid to t a the intruder - whose fluctuations satisfy the Fluctuation Relation. m - Keywords Granular materials · Non-equilibrium fluctuations d Granular fluids represent a valid benchmark for modern theories of non- n o equilibrium statistical mechanics [9]. Due to dissipative interactions among c the microscopic constituents, energy is not conserved and an external source [ isnecessarytomaintainastationarystate.Theconsequenceisabreakdownof 1 time reversal invariance and the failure of properties such as the Equilibrium v Fluctuation-Dissipationrelation(EFDR)[6].Inrecentyears,asystematicthe- 1 oryfor thedilute limithasbeendeveloped,ingoodagreementwithnumerical 8 simulations[2,1],whileageneralunderstandingofdensegranularfluidsisstill 7 lacking. A common approach is the so-called Enskog correction [2,3], which 1 . reduces the breakdown of Molecular Chaos to a renormalization of the col- 1 lision frequency. In cooling regimes, the Enskog theory may describe strong 0 2 1 Allauthors : CNR-ISCc/oDip.diFisica, v Universita‘degliStudidiRoma”LaSapienza”, i PiazzaleAldoMoro2,I-00185Roma,Italy. X Tel.:+39-0649913508 Fax:+39-064463158 r E-mail:[email protected] a E-mail:[email protected] E-mail:[email protected] E-mail:[email protected] 2 A.Puglisietal. non-equilibriumeffects,duetothe explicitcoolingtime-dependence[22].Nev- ertheless it cannot describe dynamical effects in stationary regimes, such as multiple characteristic times or different decays of response and autocorrelation [5,18]. Here we review a recent model [24] for the dynamics of a massive tracer moving in a gas of smaller granular particles, both coupled to an external bath.Takingasreferencepointthedilute limit,wherethesystemhasaclosed analyticaldescription[23],aLangevinequationlinearlycoupledto afluctuat- inglocalvelocityfieldisproposedasfirstapproximationcapableofdescribing the dense case. Its main features are: i) the decay of correlationand response functions is not simply exponential and shows backscattering [15,4] and ii) the EFDR [10,13] of the first and second kind do not hold. In such a model, detailedbalanceis notnecessarilysatisfied,andafluctuating entropyproduction [25] can be measured, which fairly verifies the Fluctuation Relation [11, 12,13]. Themodelreviewedhereis thefollowing:an“intruder”disc ofmassm = 0 M and radius R, moving in a gas of N granular discs with mass m = m i (i > 0) and radius r, in a two dimensional box of area A = L2. We denote by n = N/A the number density of the gas and by φ the occupied volume fraction,i.e.φ=π(Nr2+R2)/AandwedenotebyV (orv )andv (orv with 0 i i > 0) the velocity vector of the tracer and of the gas particles, respectively. Interactions among the particles are hard-core binary instantaneous inelastic collisions, such that particle i, after a collision with particle j, comes out with a velocity v′ = v −(1+α) mj [(v −v )·nˆ]nˆ where nˆ is the unit i i mi+mj i j vector joining the particles’ centers of mass and α ∈ [0,1] is the restitution coefficient (α = 1 is the elastic case). The mean free path of the intruder is proportional to l = 1/(n(r+R)) and we denote by τ its mean collision 0 c time.Twokinetic temperatures canbe introducedforthe two species:the gas temperature T =mhv2i/2 and the tracer temperature T =MhV2i/2. g tr The equation of motion of the i-th particle reads: m v˙ (t) = −γ v (t)+ i i b i f (t)+ξ (t).Heref (t)istheforcetakingintoaccountthecollisionsofparticle i b i i with other particles, and ξ (t) is a white noise (different for all particles), b ′ ′ with hξ (t)i = 0 and hξ (t)ξ (t)i = 2T γ δ(t−t). The effect of the external b b b b b energy source balances the energy lost in the collisions and a stationary state is attained with m hv2i≤T [27,14,19,16,8] . i i b At low packing fractions, φ<0.1, and in the large mass limit, m/M ≪1, using the Enskog approximation it has been shown [23] that the dynamics of the intruder is described by a linear Langevin equation: MV˙ =−Γ V +E , (1) E E withE awhitenoisewithhE i=0,hE (t)E (t′)i=2δ(t−t′)Γ TE andTE = E E E E E tr tr (γ T +γE1+αT )/Γ is the tracer’s temperature. In this limit the velocity b b g 2 g E autocorrelationfunctionshowsasimpleexponentialdecay,withcharacteristic time M/Γ , where Γ = γ +γE and γE = g2(r+R) 2πmT (1+α) where E E b g g l0 g g2(r+R) is the pair correlation function for a gas paprticle and the intruder Dynamicsofamassiveintruderinahomogeneously drivengranularfluid 3 at contact. Time-reversal and the EFDR, weakly modified for uniform dilute granular gases [17,5,21], become perfectly satisfied for a massive intruder. Asthepackingfractionisincreased,theEnskogapproximationfailsinpre- dicting dynamical properties. In particular, velocity autocorrelation C(t) = hV(t)V(0)i/hV2i and linear response function R(t) = δV(t)/δV(0) show an exponential decay modulated by oscillating functions [4,24]. Moreover violations of the EFDR C(t) = R(t) are observed for α < 1 [18,26]. The Enskog approximation is unable to explain the observed functional forms, because it only modifies by a constantfactor the collisionfrequency [2,23]:a model with morethanonecharacteristictime is needed.Afirstapproximationis givenby an auxiliary field coupled to the intruder’s velocity: MV˙ =−Γ (V −U)+ 2Γ T E (2) E E g V M′U˙ =−Γ′U −Γ V +p2Γ′T E , E b U p where E and E are white noises of unitary variance. Two new parameters V U ′ ′ appear: the mass of the local field M and its drag coefficient Γ . The dilute ′ ′ limit here is obtained for Γ ∼ M → ∞. In such a limit indeed U → 0 and the equation for V comes back in the form discussed above [23]. In such a form (2), the dynamics of the tracer is remarkably simple: indeed V follows a Langevin equation in a Lagrangian frame with respect to a field U, which is the local average velocity field of the gas particles colliding with the tracer. A first justification of this model comes from realizing [24] that it is equivalent to a Generalized Langevin Equation with exponential memory, which is consistent with a typical approximation done for Brownian Motion when, at high densities, the coupling of the intruder with fluid hydrodynamic modes, decayingexponentiallyintime (see[28],Cap.8.6and9.1),mustbetakeninto account.Suchacoupling,whichinprincipleinvolvesacontinuumofmodes,is reduced here to a single dominant mode: this is sufficient to introduce a new non-trivial timescale. The full coupling would reproduce finer features which become relevant at larger densities or larger times, such as long-time power- lawtails.Thefactthatthe “temperature”ofthe localvelocityfieldU isequal to the bath temperature T comes as a consequence of the conservation of b momentum in collisions, implying that the averagevelocity of a groupof par- ticlesisnotchangedbycollisionsamongthemselvesandisonlyaffectedbythe external bath and a (small) number of collisions with outside particles. This scenariois fully consistentwithrecentstudy ofhydrodynamicfluctuations for the velocity field of the same fluid model [7,8]. Astrongerjustificationcomes,however,fromitseffectivnessinreproducing the numerical results, as detailed in [24]. From the simulations it is seen that ′ ′ therelaxationtimeofthelocalfieldτ =M /Γ ,rescaledbythemeancollision U time,increaseswiththepackingfractionandwiththeinelasticity,asexpected. At high densities it appears that Γ′ ∼1/φ, and T ∼T ∼TE, likely due to tr g g stronger correlations among particles. At large φ we observe T > TE, con- tr tr sistent with a smaller dissipation for correlated collisions. Model (2) predicts 4 A.Puglisietal. C =f (t) and R=f (t) with C R f =e−gt[cos(ωt)+a sin(ωt)]. (3) C(R) C(R) ′ ′ Thevariablesg,ω,a anda areknownalgebraicfunctionsofΓ ,T ,Γ ,M C R E g and T . In particular,the ratio a /a =[T −Ω(T −T )]/[T +Ω(T −T )], b C R g b g g b g ′ ′ ′ with Ω =Γ /((Γ +Γ )(Γ M /M −Γ )). Hence, in the elastic (T →T ) as E E E g b ′ wellasinthedilutelimit(Γ →∞),onegetsa =a andrecoverstheEFDR C R C(t)=R(t). Such predictions are all verifiedin numericalsimulations [24]. In particular Fig. 1 depicts correlation and response functions in a dense case (elastic and inelastic): symbols correspond to numerical data and continuous linestheanalyticalcurves.Intheinelasticcase,deviationsfromEFDRR(t)= C(t) are observed. In the inset of Fig. 1 the ratio R(t)/C(t) is also reported. Itisimportanttonoticethatthemainresponsibilityforthebreakdownofthe EFDR is the coupling betweenV andU,indeed Eq.(3) canbe expressedin a ′ different way: R(t)=aC(t)+bhV(t)U(0)i with a=[1−(T −T )Ω /Γ ] and g b a b=(T −T )Ω , where Ω and Ω are knownfunctions of the parameters.At g b b a b equilibrium or in the dilute limit the EFDR is recovered. 4 R(t) & C(t) t/τ=300 t/τc=600 0.1 Σ<>t2 tt//ττcc==24480000 α=0.6 Σ)]/t0 y=cx R(t)/C(t) P(- 0.01 1 Σ)/t-2 1Σ)]/tt 0.90 10 20 30 40 50 α=1.0 log[P(-4 -0.4 -0.2 0Σ/t0.2 0.4 00.5-log[P( t 0.001 30 40 50t/τ 60 70 80 -4 -3 -2 -1 Σ/<0Σ> 1 2 3 4 c t t Fig. 1 (Coloronline).Left:correlationfunctionC(t)(blackcircles)andresponsefunction R(t)(redsquares)forα=1andα=0.6,atφ=0.33.Continuouslinesshowcurvesobtained withEqs.(3).Inset: theratioR(t)/C(t) isreportedinthesamecases.Right:Checkofthe fluctuation relation (5) in the system with α = 0.6 and φ = 0.33. Inset: collapse of the rescaledprobabilitydistributionsofΣt atlargetimesontothelargedeviationfunction. An important independent assessment of model (2) comes from the study of the fluctuating entropyproduction [25] which quantifies the deviation from detailed balance in a trajectory. Given the trajectory in the time interval [0,t], {V(s)}t, and its time-reversed {IV(s)}t ≡ {−V(t−s)}t, the entropy 0 0 0 production for our model takes the form [20] P({V(s)}t) 1 1 t Σ =log 0 ≈Γ − ds V(s)U(s). (4) t P({IV(s)}t0) E(cid:18)Tg Tb(cid:19)Z0 Thisfunctionalvanishesexactlyintheelasticcase,α=1,whereequipartition holds, T = T , and is zero on average in the dilute limit, where hVUi = 0. g b Dynamicsofamassiveintruderinahomogeneously drivengranularfluid 5 Formula (4) reveals that the leading source of entropy production is the energy transferred by the “force” Γ U on the tracer, weighed by the difference E betweentheinversetemperaturesofthetwo“thermostats”.Therefore,tomea- sure entropy production, we need to measure the fluctuations of U: a possible choice is a local average of particles’ velocities in a circle of radius l+R cen- teredonthetracer.Detailsonhowtochooseinareliablewaythecorrectlare given in [24]. Following such procedure, in the case φ=0.33 and α=0.6, we estimate for the correlationlength l∼9r∼6l . Then, measuring the entropy 0 productionfromEq.(4)alongmanytrajectoriesoflengtht,we computedthe probability P(Σ =x) and compared it to P(Σ =−x), in order to verify the t t Fluctuation Relation [11,12,13] P(Σ =x) t log =x. (5) P(Σ =−x) t In the right frame of Fig. 1 the results of this comparison are reported. The main frame confirms that at large times the Fluctuation Relation (5) is well verifiedwithinthestatisticalerrors.TheinsetshowsthecollapseoflogP(Σ )/t t onto the large deviation rate function for large times. Notice that - in formula (4) - a wrong evaluation of the weighing factor (1/T −1/T ) or of the g b “energy injection rate” Γ U(t)V(t) in Eq. (4) could produce a completely E different slope in Fig. 1 (right frame). ′ To conclude this paper, we stress that velocity correlations hV(t)U(t)i between the intruder and the surrounding velocity field are responsible for both the violations of the EFDR and the appearance of a non-zero entropy production,providedthatthe twofieldsareat different temperatures.We also mention that larger violations of EFDR can be observed using an intruder withamassequalorsimilartothatofotherparticles[18],withthe important differencethatinsuchacaseasimple“Langevin-like”modelfortheintruder’s dynamics is not available. Acknowledgements This work is dedicated to the memory of Isaac Goldhirsch, from whom we learned plenty of science, but also a smilingattitude toward serious things. The workissupportedbythe“Granular-Chaos”project,MIURgrantnumberRBID08Z9JE. References 1. Brey,J.J.,Maynar,P.,deSoria,M.I.G.:Fluctuatinghydrodynamicsfordilutegranular gases. Phys.Rev.E79,051,305(2009) 2. 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