Dynamics of a massive intruder in a homogeneously driven granular fluid

Abstract

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 ∼10-50% 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 [9]. 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) [6]. In recent years, a systematic theory for the dilute limit has been developed, in good agreement with numerical simulations [1,2], while a general understanding of dense granular fluids is still lacking. A common approach is the so-called Enskog correction [2,3], 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 [21]. Nevertheless it cannot describe dynamical effects in stationary regimes, such as multiple characteristic times or different decays of response and autocorrelation [5,17].Here we review a recent model [23] 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 [22], 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 [14,4] and (ii) the EFDR [10,13] of the first and second kind do not hold. In such a model, detailed balance is not necessarily satisfied, and a fluctuating entropy production [24] can be measured, which fairly verifies the Fluctuation Relation [11][12][13].The model reviewed here is the following: an "intruder" disc of mass m 0 = M and radius R, moving in a gas of N granular discs with mass m i = m (i > 0) and radius r , in a two dimensional box of area A = L 2 . We denote by n = N /A the number density of the gas and by φ the occupied volume fraction, i.e. φ = π(Nr 2 + R 2 )/A and we denote by V (or v 0 ) and v (or v i with i > 0) the velocity vector of the tracer and of the gas particles, respectively. Interactio...

Temperature in and out of equilibrium: A review of concepts, tools and attempts

“…The validity of an equilibrium-like Fluctuation-Dissipation relation was observed, with an effective temperature which -as orders of magnitude -was "related to the granular temperature", a fact which is reminiscent of the observations discussed in the previous paragraphs, but cannot be fully evaluated since the real granular temperature was not measured. Most recent studies, both theoretical [234,236,116,113,247] and experimental [237], have shown that when the granular is a liquid and not a gas, deviations from the equilibrium Fluctuation-Dissipation relation are observed: most importantly, such deviations are not well described by an effective temperature. In granular liquids, as a matter of fact, granular temperature is much less useful than in gases, and cannot be replaced by some other temperature for the purpose of an effective description.…”

“…A simple example is provided, again, by the case of a massive intruder M m [113,247]. For the purpose of describing the velocity autocorrelation of the tracer and its linear response, the model in Eqs.…”

Field Induced Stationary State for an Accelerated Tracer in a Bath

“…where F ν (v, t|v 0 ) is the conditional velocity distribution of the tracer at time t knowing it had velocity v 0 at time 0, i.e. it is the time-dependent distribution F ν (v, t) with the initial condition F ν (v, 0) = δ(v − v 0 ), which may also be represented using the implicit formulation (15). We may thus define the auxiliary function…”

“…The bulk of ulterior developments on this modified Lorentz problem focused on asymptotic and relaxation properties [8][9][10][11][12][13]. Recently, the model was used as a paradigm for the study of microscopic entropy production and fluctuation relations in non equilibrium systems [14,15], and to discuss the force-velocity relation [16] as measured in experiments [17].…”

Intruder Dynamics in a Frictional Granular Fluid: A Molecular Dynamics Study

“…There also exist a few simulation studies of F d acting on an intruder in a d = 2 granular medium [40][41][42][43]. Bharadwaj et al [42] obtained F d for an immersed cylinder in a stream of solid particles.…”