Dynamics of a massive intruder in a homogeneously driven granular fluid
A massive intruder in a homogeneously driven granular fluid, in dilute configurations, performs a memory-less Brownian motion with drag and temperature simply related to the average density and temperature of the fluid. At volume fraction the intruder’s velocity correlates with the local fluid velocity field: such situation is approximately described by a system of coupled linear Langevin equations equivalent to a generalized Brownian motion with memory. Here one may verify the breakdown of the Fluctuation-Dissipation relation and the presence of a net entropy flux - from the fluid to the intruder - whose fluctuations satisfy the Fluctuation Relation.
Keywords:Granular materials Non-equilibrium fluctuations
Granular fluids represent a valid benchmark for modern theories of non-equilibrium statistical mechanics JNB96b (). Due to dissipative interactions among the microscopic constituents, energy is not conserved and an external source is necessary to maintain a stationary state. The consequence is a breakdown of time reversal invariance and the failure of properties such as the Equilibrium Fluctuation-Dissipation relation (EFDR) GN00 (). In recent years, a systematic theory for the dilute limit has been developed, in good agreement with numerical simulations BP04 (); BMG09 (), while a general understanding of dense granular fluids is still lacking. A common approach is the so-called Enskog correction BP04 (); DS06 (), which reduces the breakdown of Molecular Chaos to a renormalization of the collision frequency. In cooling regimes, the Enskog theory may describe strong non-equilibrium effects, due to the explicit cooling time-dependence SD01 (). Nevertheless it cannot describe dynamical effects in stationary regimes, such as multiple characteristic times or different decays of response and autocorrelation G04 (); PBV07 ().
Here we review a recent model SVGP10 () for the dynamics of a massive tracer moving in a gas of smaller granular particles, both coupled to an external bath. Taking as reference point the dilute limit, where the system has a closed analytical description SVCP10 (), a Langevin equation linearly coupled to a fluctuating local velocity field is proposed as first approximation capable of describing the dense case. Its main features are: i) the decay of correlation and response functions is not simply exponential and shows backscattering OK07 (); FAZ09 () and ii) the EFDR KTH91 (); BPRV08 () of the first and second kind do not hold. In such a model, detailed balance is not necessarily satisfied, and a fluctuating entropy production seifert05 () can be measured, which fairly verifies the Fluctuation Relation Kurchan (); LS99 (); BPRV08 ().
The model reviewed here is the following: an “intruder” disc of mass and radius , moving in a gas of granular discs with mass () and radius , in a two dimensional box of area . We denote by the number density of the gas and by the occupied volume fraction, i.e. and we denote by (or ) and (or with ) 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 , after a collision with particle , comes out with a velocity where is the unit vector joining the particles’ centers of mass and is the restitution coefficient ( is the elastic case). The mean free path of the intruder is proportional to and we denote by its mean collision time. Two kinetic temperatures can be introduced for the two species: the gas temperature and the tracer temperature .
The equation of motion of the -th particle reads: . Here is the force taking into account the collisions of particle with other particles, and is a white noise (different for all particles), with and . The effect of the external energy source balances the energy lost in the collisions and a stationary state is attained with WM96 (); NETP99 (); PLMPV98 (); PEU02 (); GSVP11b () .
At low packing fractions, , and in the large mass limit, , using the Enskog approximation it has been shown SVCP10 () that the dynamics of the intruder is described by a linear Langevin equation:
with a white noise with , and is the tracer’s temperature. In this limit the velocity autocorrelation function shows a simple exponential decay, with characteristic time , where and where is the pair correlation function for a gas particle and the intruder at contact. Time-reversal and the EFDR, weakly modified for uniform dilute granular gases PBL02 (); G04 (); PVTW06 (), become perfectly satisfied for a massive intruder.
As the packing fraction is increased, the Enskog approximation fails in predicting dynamical properties. In particular, velocity autocorrelation and linear response function show an exponential decay modulated by oscillating functions FAZ09 (); SVGP10 (). Moreover violations of the EFDR are observed for PBV07 (); VPV08 (). The Enskog approximation is unable to explain the observed functional forms, because it only modifies by a constant factor the collision frequency BP04 (); SVCP10 (): a model with more than one characteristic time is needed. A first approximation is given by an auxiliary field coupled to the intruder’s velocity:
where and are white noises of unitary variance. Two new parameters appear: the mass of the local field and its drag coefficient . The dilute limit here is obtained for . In such a limit indeed and the equation for comes back in the form discussed above SVCP10 (). In such a form (2), the dynamics of the tracer is remarkably simple: indeed follows a Langevin equation in a Lagrangian frame with respect to a field , which is the local average velocity field of the gas particles colliding with the tracer. A first justification of this model comes from realizing SVGP10 () 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, decaying exponentially in time (see Z01 (), Cap. 8.6 and 9.1), must be taken into account. Such a coupling, which in principle involves a continuum of modes, 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-law tails. The fact that the “temperature” of the local velocity field is equal to the bath temperature comes as a consequence of the conservation of momentum in collisions, implying that the average velocity of a group of particles is not changed by collisions among themselves and is only affected by the external bath and a (small) number of collisions with outside particles. This scenario is fully consistent with recent study of hydrodynamic fluctuations for the velocity field of the same fluid model GSVP11 (); GSVP11b ().
A stronger justification comes, however, from its effectivness in reproducing the numerical results, as detailed in SVGP10 (). From the simulations it is seen that the relaxation time of the local field , rescaled by the mean collision time, increases with the packing fraction and with the inelasticity, as expected. At high densities it appears that , and , likely due to stronger correlations among particles. At large we observe , consistent with a smaller dissipation for correlated collisions. Model (2) predicts and with
The variables , , and are known algebraic functions of , , , and . In particular, the ratio , with . Hence, in the elastic () as well as in the dilute limit (), one gets and recovers the EFDR . Such predictions are all verified in numerical simulations SVGP10 (). In particular Fig. 1 depicts correlation and response functions in a dense case (elastic and inelastic): symbols correspond to numerical data and continuous lines the analytical curves. In the inelastic case, deviations from EFDR are observed. In the inset of Fig. 1 the ratio is also reported. It is important to notice that the main responsibility for the breakdown of the EFDR is the coupling between and , indeed Eq. (3) can be expressed in a different way: with and , where and are known functions of the parameters. At equilibrium or in the dilute limit the EFDR is recovered.
An important independent assessment of model (2) comes from the study of the fluctuating entropy production seifert05 () which quantifies the deviation from detailed balance in a trajectory. Given the trajectory in the time interval , , and its time-reversed , the entropy production for our model takes the form PV09 ()
This functional vanishes exactly in the elastic case, , where equipartition holds, , and is zero on average in the dilute limit, where . Formula (4) reveals that the leading source of entropy production is the energy transferred by the “force” on the tracer, weighed by the difference between the inverse temperatures of the two “thermostats”. Therefore, to measure entropy production, we need to measure the fluctuations of : a possible choice is a local average of particles’ velocities in a circle of radius centered on the tracer. Details on how to choose in a reliable way the correct are given in SVGP10 (). Following such procedure, in the case and , we estimate for the correlation length . Then, measuring the entropy production from Eq. (4) along many trajectories of length , we computed the probability and compared it to , in order to verify the Fluctuation Relation Kurchan (); LS99 (); BPRV08 ()
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 verified within the statistical errors. The inset shows the collapse of onto the large deviation rate function for large times. Notice that - in formula (4) - a wrong evaluation of the weighing factor or of the “energy injection rate” in Eq. (4) could produce a completely different slope in Fig. 1 (right frame).
To conclude this paper, we stress that velocity correlations 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, provided that the two fields are at different temperatures. We also mention that larger violations of EFDR can be observed using an intruder with a mass equal or similar to that of other particles PBV07 (), with the important difference that in such a case a simple “Langevin-like” model for the intruder’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 smiling attitude toward serious things. The work is supported by the “Granular-Chaos” project, MIUR grant number RBID08Z9JE.
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