Clustering and Non-Gaussian Behavior in Granular Matter

Abstract

We investigate the properties of a model of granular matter consisting of N Brownian particles on a line subject to inelastic mutual collisions. This model displays a genuine thermodynamic limit for the mean values of the energy and the energy dissipation. When the typical relaxation time τ associated with the Brownian process is small compared with the mean collision time τc the spatial density is nearly homogeneous and the velocity probability distribution is gaussian. In the opposite limit τ ≫ τc one has strong spatial clustering, with a fractal distribution of particles, and the velocity probability distribution strongly deviates from the gaussian one.PACS: 81.05. Rm, 05.20.Dd, 05.40.+j In the past few years granular materials have become an intriguing subject of research [1] - [7], since they pose novel questions and challenges to the theorists and experimentalists. The constituting elements of such materials are solid particles, whose size may range from few microns to few centimeters, and which are subject to nonconservative contact forces such as friction and cohesion.Their collective behavior is peculiar and different from other forms of matter, such as solids, liquids or gases, and the ordinary statistical mechanical approach, which successfully deals with large assemblies of microscopic particles is not adequate.Generally speaking granular materials cannot be described as equilibrium systems neither from the configurational point of view nor from the dynamical point of view. It is known in fact that these systems remain easily trapped in some metastable configurations which can last for long time intervals unless they are shaken or perturbed [2]. On the other hand while in equilibrium statistical mechanics the kinetic energy per particle is proportional to the temperature and the velocities are gaussianly distributed, in the systems we consider the tails of the distribution deviate from the Maxwell law [8] . This phenomenon is accompanied by a pronounced clustering of the particles [3,4] or inelastic collapse [6].Several approaches have been proposed for the study of the so-called "granular gases" [7,9]. One crucial difference between ordinary gases and granular media is represented by the intrinsic inelasticity of the interactions among the grains, which makes any theory based on energy conservation , e.g. for ideal gases, not suitable.In the present work we study a one dimensional mechanical model, in the spirit of the one recently introduced by Kadanoff and coworkers [9], but containing some important differences regarding the energyexchange process. We consider N identical particles on a circle of length L [10] obeying to the following equations:where, 1 ≤ i ≤ N , T F is the temperature of a microscopic medium that we discuss below, τ is the relaxation time, in absence of collisions, and f i (t) is a standard white noise with zero average and variance < f i (t)f j (t ′ ) >= δ ij δ(t − t ′ ). In addition the particles are subject to inelastic collisions according to the rulewhere r is the restit...

Transport coefficients for granular media from molecular dynamics simulations

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“…Among these assumptions are that velocity distributions are Gaussian and that the mean energy is shared equally among the various degrees of freedom. However, recent numerical and theoretical research indicates that excited inelastic hard spheres can exhibit non-Gaussian velocity distributions [4][5][6][7][8][9][10], though the predicted velocity distributions differ considerably from each other. Some thermodynamic descriptions of granular media make use of the concept of entropy [11], or separate the dissipative degrees of freedom from conservative ones [3].…”

“…Assuming a Gaussian velocity distribution for a nearly homogeneous granular medium, Puglisi et al [7,8] predict a non-Gaussian velocity distribution due to clustering: a superposition of Gaussian velocity distributions with different widths. For inelastic particles this leads to high velocity tails in the velocity distribution which decrease more slowly than a Gaussian function but faster than an exponential.…”

Temperature probes in binary granular gases

“…In experiments the energy is typically supplied at the boundaries, leading the system to a heterogeneous stationary state [3,4,5,18]. In order to avoid the complication of strong temperature heterogeneities, we will use a homogeneous driving in the form of a "thermostat": in this mechanism (which recently has attracted the attention of many theorists [19,20,21,22,23,24,25]), the particles are submitted, between collisions, to a random force in the form of an uncorrelated white noise (e.g. Gaussian) with the possible addition of a viscous term.…”