Long-Time Tails and Cage Effect in Driven Granular Fluids

Abstract

We study the velocity autocorrelation function (VACF) of a driven granular fluid in the stationary state in 3 dimensions (3d). As the critical volume fraction of the glass transition in the corresponding elastic system is approached, we observe pronounced cage effects in the VACF as well as a strong decrease of the diffusion constant depending on the inelasticity. At moderate densities the VACF is shown to decay algebraically in time, like t −3/2 , if momentum is conserved locally and like t −1 , if momentum is not conserved by the driving. A simple scaling argument supports the observed long time tails.PACS numbers: 45.70.-n, 51.20.+d, 47.10.-g Strongly agitated granular fluids have attracted a lot of attention in recent years [1]. Most of the theoretical work which is based on microscopic dynamics has been done for either rather dilute or weakly inelastic sytems, generalising kinetic theory to gases of inelastically colliding particles. The velocity autocorrelation function [2] as well as transport coefficients [3] have been calculated for the homogeneous cooling state, which has also been simulated for a wide range of inelasticities [4].Comparatively few studies have been performed on the stationary state of granular fluids in the moderate or high density regime. This is surprising, given the fact that the corresponding (elastic) molecular fluids have been studied in great detail [5] and revealed several interesting features already in the dynamics of a single tagged particle: backscattering as indicated by a negative velocity autocorrelation, long-time tails due to the coupling of the tagged particle's density to a shear flow and a glass transition at a volume fraction η ≈ 0.58 accompanied by a strong decrease of the diffusion constant as a precursor to structural arrest. It is our aim to understand which of these features pertain to an inelastic gas and how they are destroyed by increasingly more dissipative collisions. This applies in particular to the glass transition, which has been conjectured to be related to the jamming transition in granular matter [6].Several experimental groups have measured the VACF in dense granular flow [7][8][9][10][11][12].The VACF in the steady state of a 3d vibro-fluidized bed [8] was shown to exhibit strong backscattering effects. In 2d vibrated layers, high speed cameras have been used to measure the VACF. Even though long-time tails seem to be beyond the experimental resolution, these experiments give evidence for a nonexponential decay [11]. Caging effects have clearly been seen in air-fluidized beds [13] as well as in sheared granular materials [14]. In recent experiments [12] the development of a plateau in the mean square displacement has been observed, but may be related to crystallization as seen in monodisperse vibrated layers [11,12].Model-We investigate a system of monodisperse hard spheres of diameter a and mass m. The time evolution is governed by instantaneous inelastic two-particle collisions. Given the relative velocity g := v 1 −v 2 , the change of g in ...

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