Dynamics of a freely evolving, two-dimensional granular medium

Hard-sphere limit of soft-sphere model for granular materials: stiffness dependence of steady granular flow

Dynamical behavior of steady granular flow is investigated numerically in the inelastic hard-sphere limit of the soft-sphere model. We find distinctively different limiting behaviors for the two flow regimes, i.e., the collisional flow and the frictional flow. In the collisional flow, the hard-sphere limit is straightforward; the number of collisions per particle per unit time converges to a finite value and the total contact time fraction with other particles goes to zero. For the frictional flow, however, we demonstrate that the collision rate diverges as the power of the particle stiffness so that the time fraction of the multiple contacts remains finite even in the hard-sphere limit, although the contact time fraction for the binary collisions tends to zero.

Dynamics of freely cooling granular gases

We study dynamics of freely cooling granular gases in two dimensions using large-scale molecular dynamics simulations. We find that for dilute systems the typical kinetic energy decays algebraically with time, E(t) approximately t(-1), and velocity statistics are characterized by a universal Gaussian distribution in the long time limit. We show that in the late clustering regime particles move coherently as typical local velocity fluctuations, Deltav, are small compared with the typical velocity, Deltav/v approximately t(-1/4). Furthermore, locally averaged shear modes dominate over acoustic modes. The small thermal velocity fluctuations suggest that the system can be heuristically described by Burgers-like equations.

Transport theory of granular swarms

The transport of trace granular gas (swarm) in a carrier granular fluid is studied by means of the Boltzmann-Lorentz kinetic equation. Time-dependent perturbation theory is used to follow the evolution of the granular swarm from an arbitrary initial distribution. A nonhydrodynamic extension of the diffusion equation is derived, with transport coefficients that are time dependent and implicitly depend on the wave vector. Transport coefficients of any order are obtained as velocity moments of the solutions of the corresponding kinetic equations derived from the Boltzmann-Lorentz equation. For the special case of the initial distribution of swarm particles, transport coefficients are identified as time derivatives of the moments of the number density. Finally the granular particle transport theory is extended by the introduction of the concept of non-particle-conserving collisions.

Symmetry breaking and coarsening of clusters in a prototypical driven granular gas

Granular hydrodynamics predicts symmetry-breaking instability in a two-dimensional ensemble of nearly elastically colliding smooth hard disks driven, at zero gravity, by a rapidly vibrating sidewall. Supercritical and subcritical symmetry-breaking bifurcations of the stripe state are identified, and the supercritical bifurcation curve is computed. The cluster dynamics proceed as a coarsening process mediated by the gas phase. Well above the bifurcation point the final steady state, selected by coarsening, represents a single strongly localized densely packed "droplet."

Symmetry-breaking instability in a prototypical driven granular gas

Symmetry-breaking instability of a laterally uniform granular cluster (strip state) in a prototypical driven granular gas is investigated. The system consists of smooth hard disks in a two-dimensional box, colliding inelastically with each other and driven, at zero gravity, by a "thermal" wall. The limit of nearly elastic particle collisions is considered, and granular hydrodynamics with the Jenkins-Richman constitutive relations is employed. The hydrodynamic problem is completely described by two scaled parameters and the aspect ratio of the box. Marginal stability analysis predicts a spontaneous symmetry-breaking instability of the strip state, similar to that predicted recently for a different set of constitutive relations. If the system is big enough, the marginal stability curve becomes independent of the details of the boundary condition at the driving wall. In this regime, the density perturbation is exponentially localized at the elastic wall opposite the thermal wall. The short- and long-wavelength asymptotics of the marginal stability curves are obtained analytically in the dilute limit. The physics of the symmetry-breaking instability is discussed.

Diffusion in a granular fluid. II. Simulation

The linear-response description for impurity diffusion in a granular fluid undergoing homogeneous cooling is developed in the preceding paper. The formally exact Einstein and Green-Kubo expressions for the self-diffusion coefficient are evaluated there from an approximation to the velocity autocorrelation function. These results are compared here to those from molecular-dynamics simulations over a wide range of density and inelasticity, for the particular case of self-diffusion. It is found that the approximate theory is in good agreement with simulation data up to moderate densities and degrees of inelasticity. At higher density, the effects of inelasticity are stronger, leading to a significant enhancement of the diffusion coefficient over its value for elastic collisions. Possible explanations associated with an unstable long wavelength shear mode are explored, including the effects of strong fluctuations and mode coupling.

Diffusion in a granular fluid. I. Theory

Many important properties of granular fluids can be represented by a system of hard spheres with inelastic collisions. Traditional methods of nonequilibrium statistical mechanics are effective for analysis and description of the inelastic case as well. This is illustrated here for diffusion of an impurity particle in a fluid undergoing homogeneous cooling. An appropriate scaling of the Liouville equation is described such that the homogeneous cooling ensemble and associated time correlation functions map to those of a stationary state. In this form the familiar methods of linear response can be applied, leading to Green-Kubo and Einstein representations of diffusion in terms of the velocity and mean-square displacement correlation functions. These correlation functions are evaluated approximately using a cumulant expansion and from kinetic theory, providing the diffusion coefficient as a function of the density and the restitution coefficients. Comparisons with results from molecular-dynamics simulation are given in the following companion paper.

Spontaneous air-driven separation in vertically vibrated fine granular mixtures

We report the observation of the spontaneous separation of vertically vibrated mixtures of fine bronze and glass spheres of similar diameters. At low frequencies and at sufficient vibrational amplitudes, a sharp boundary forms between a lower region of glass and an upper region of the heavier bronze. The boundary undergoes various oscillations, including periodic tilting motion, but remains extremely sharp. At higher frequencies, the bronze separates as a mid-height layer between upper and lower glass regions, and the oscillations are largely absent. The mechanism responsible for the separation can be traced to the effect of air on the granular motion.

Dynamics of rarefied granular gases

This paper presents quite general bidimensional gas-dynamic equations--derived from kinetic theory-which include the fourth cumulant kappa(r,t) as a dynamic field. The dynamics describes a low-density system of inelastic hard spheres (disks) with normal restitution coefficient r. Two illustrative examples are given and the role of kappa in them is discussed. Our general gas-dynamic equations would deal with 9 hydrodynamic fields (which corresponds to 14 in three-dimension). These fields are the standard hydrodynamic fields plus the components p(ij) of the traceless part of the pressure tensor, the energy flux vector Q and the fourth cumulant kappa. The present formulation requires no constitutive equations. The two examples are: the well-known homogeneous cooling state and a system, with and without gravity, steadily heated by two parallel walls. In the first case, the dynamics yield a description of the homogeneous cooling state consistent with known results adding extra details mainly about the transient time behavior. The steadily heated system kept in a static state gives rise to quite simple but nontrivial equations. In the case with gravity, it is shown that when kappa is included as a dynamic field, the formalism leads to a non-Fourier law already to first order in dissipation. Setting gravity g=0 a perturbative solution is shown and favorably compared with observations obtained from molecular dynamics (MD). In both cases, with and without gravity, kappa is not homogeneous. An analytic extension suggests a divergent situation for a small negative value of q, which originates in the unavoidable extension of the formalism to exothermic collisions associated with a restitution coefficient larger than one. This divergent behavior is observed in MD.

Symmetry-breaking instability and strongly peaked periodic clustering states in a driven granular gas

An ensemble of inelastically colliding grains driven by a horizontally vibrating wall in two dimensions exhibits clustering. Working in the limit of nearly elastic collisions and employing granular hydrodynamics, we predict, by a marginal stability analysis, a spontaneous symmetry breaking of the laterally uniform clustering state. Two-dimensional steady-state solutions found numerically describe laterally periodic clustering states. Well within the instability region the density of these states is strongly peaked, with most of the granulate located in "density islands." Time-dependent granular hydrodynamic simulations show that strongly peaked states can develop from small-amplitude single-mode density perturbations.

Characterization of the stationary states of a dilute vibrofluidized granular bed

This paper reports two phenomena in an event driven simulation of a dilute vibrofluidized granular material in two dimensions. Both phenomena show inhomogeneity in the horizontal direction. They are convection rolls similar to the Rayleigh-Bénard thermal convection in fluids, and a clustering instability, where the bed spontaneously phase separates into coexisting dense and dilute regions. Detailed investigations show that these are different from the known instabilities in a vibrated granular medium. Characterization of these instabilities is carried out with a phase diagram using suitable parameters from the kinetic theory of vibrofluidized beds.

Bifurcations of a driven granular system under gravity

The molecular dynamics study on the granular bifurcation in a simple model is presented. The model consists of hard disks, which undergo inelastic collisions; the system is under the uniform external gravity and is driven by the heat bath. The competition between the two effects, namely, the gravitational force and the heat bath, is carefully studied. We found that the system shows three phases, namely, the condensed phase, the locally fluidized phase, and the granular turbulent phase, upon increasing the external control parameter. We conclude that the transition from the condensed phase to the locally fluidized phase is distinguished by the existence of fluidized holes, and the transition from the locally fluidized phase to the granular turbulent phase is understood by the destabilization transition of the fluidized holes due to mutual interference.

Inelastic collapse in simple shear flow of a granular medium

Previous investigations have shown that inelastic collapse is a common feature of inelastic, hard-sphere simulations of nondriven (or unforced) flows, provided that the coefficient of restitution is small enough. The focus of the current effort is on a driven system, namely, simple shear flow. Two-dimensional, hard-sphere simulations have been carried out over a considerable range of restitution coefficients (r), solids fractions (nu), and numbers of particles (N). The results indicate that inelastic collapse is an integral feature of the sheared system. Similar to nondriven systems, this phenomenon is characterized by a string of particles engaging in numerous, repeated collisions just prior to collapse. The collapsed string is typically oriented along a 135 degrees angle from the streamwise direction. Inelastic collapse is also found to be more likely in systems with lower r, higher nu, and higher N, as is true for unforced systems. Nonetheless, an examination of the boundary between the collapsed and noncollapsed states reveals that the sheared system is generally more "resistant" to inelastic collapse than its nondriven counterpart. Furthermore, a dimensionless number V* is identified that represents the magnitude of the initial fluctuating velocities relative to that of a characteristic steady-state velocity (i.e., the product of shear rate and particle diameter). For values of V*>>O(1), the transient portion of the simulation is found to be more reminiscent of a nondriven system (i.e., isotropic particle bunching is observed instead of diagonal particle bands).

Model for the atomic-scale structure of the homogeneous cooling state of granular fluids

It is proposed that the equilibrium generalized mean spherical model of fluid structure may be extended to nonequilibrium states with equation of state information used in equilibrium replaced by an exact condition on the two-body distribution function. The model is applied to the homogeneous cooling state of granular fluids, and upon comparison to molecular-dynamics simulations, is found to provide an accurate picture of the pair distribution function.

Lyapunov spectrum of granular gases

We calculate and study the Lyapunov spectrum of a granular gas maintained in a steady state by an isokinetic thermostat. Considering restitution coefficients greater than unity allows us to show that the spectra change smoothly and continuously at equilibrium. The shearing instability of the granular gas, however, provokes an abrupt change in the structure of the spectrum. The relationship between various physically relevant quantities and the energy dissipation rate differs from previously studied nonequilibrium steady states.

Dynamics of electrostatically driven granular media: effects of humidity

We performed experimental studies of the effect of humidity on the dynamics of electrostatically driven granular materials. Both conducting and dielectric particles undergo a phase transition from an immobile state (granular solid) to a fluidized state (granular gas) with increasing applied field. Spontaneous precipitation of solid clusters from the gas phase occurs as the external driving is decreased. The clustering dynamics in conducting particles is primarily controlled by screening of the electric field but is aided by cohesion due to humidity. It is shown that humidity effects dominate the clustering process with dielectric particles.

Reverse Brazil nut problem: competition between percolation and condensation

In the Brazil nut problem (BNP), hard spheres with larger diameters rise to the top. There are various explanations (percolation, reorganization, convection), but a broad understanding or control of this effect is by no means achieved. A theory is presented for the crossover from BNP to the reverse Brazil nut problem based on a competition between the percolation effect and the condensation of hard spheres. The crossover condition is determined, and theoretical predictions are compared to molecular dynamics simulations in two and three dimensions.

Statistical mechanics of fluidized granular media: short-range velocity correlations

A statistical mechanical study of fluidized granular media is presented. Using a special energy injection mechanism, homogeneous fluidized stationary states are obtained. Molecular dynamics simulations and theoretical analysis of the inelastic hard-disk model show that there is a large asymmetry in the two-particle distribution function between pairs that approach and separate. Large velocity correlations appear in the postcollisional states due to the dissipative character of the collision rule. These correlations can be well-characterized by a state dependent pair correlation function at contact. It is also found that velocity correlations are present for pairs that are about to collide. Particles arrive at collisions with a higher probability that their velocities are parallel rather than antiparallel. These dynamical correlations lead to a decrease of the pressure and of the collision frequency as compared to their Enskog values. A phenomenological modified equation of state is presented.

Nonlinear hydrodynamical approach to granular materials

We propose a nonlinear hydrodynamical model of granular materials. We show how this model describes the formation of a sandpile from a homogeneous distribution of material under gravity, and then discuss a simulation of a rotating sandpile which shows, in qualitative agreement with experiment, a static and dynamic angle of repose.

Grain dynamics in a two-dimensional granular flow

We have used particle tracking methods to study the dynamics of individual balls comprising a granular flow in a small-angle two-dimensional funnel. We statistically analyze many ball trajectories to examine the mechanisms of shock propagation. In particular, we study the creation of, and interactions between, shock waves. We also investigate the role of granular temperature and draw parallels to traffic flow dynamics.

Dynamics of inelastic collapse

We consider a particle randomly accelerated by Gaussian white noise on the half-line x>0. The collisions of the particle with the wall at x=0 are inelastic. The velocities just before and after reflection are related by v(f)=-rv(i), where r is the coefficient of restitution. Cornell, Swift, and Bray have shown that for r<r(c)=e(-pi/sqrt[3])=0.163, there is inelastic collapse, i.e., boundary collisions localize the particle at x=0. The probability that the particle is not yet absorbed at the boundary after a time t decays, for long times, as t(-straight theta(r)). The exponent straight theta(r) is calculated exactly. We also consider the case of "partial-survival" boundary conditions, i.e., elastic reflection with probability p and absorption with probability 1-p, and derive the analogous persistence exponent straight phi(p). The exact exponents satisfy the Swift-Bray conjecture straight theta(r)=straight phi(r(2straight theta(r))).

Instability of symmetric couette flow in a granular gas: hydrodynamic field profiles and transport

We investigated the inelastic hard-disk gas sheared by two parallel bumpy walls (Couette flow). In our molecular dynamic simulations we found a sensitivity to the asymmetries of the initial particle positions and velocities and an asymmetric stationary state, where the deviation from (anti)symmetric hydrodynamic fields is stronger as the normal restitution coefficient decreases. For better understanding of this sensitivity we carried out a linear stability analysis of the former kinetic theoretical solution [J. T. Jenkins and M. W. Richman, J. Fluid. Mech. 171, 53 (1986)] and found it to be unstable. The effect of this asymmetry on the self-diffusion coefficient is also discussed.

Nonlinear analysis of the shearing instability in granular gases

It is known that a finite-size homogeneous granular fluid develops a hydrodynamiclike instability when dissipation crosses a threshold value. This instability is analyzed in terms of modified hydrodynamic equations: first, a source term is added to the energy equation which accounts for the energy dissipation at collisions and the phenomenological Fourier law is generalized according to previous results. Second, a rescaled time formalism is introduced that maps the homogeneous cooling state into a nonequilibrium steady state. A nonlinear stability analysis of the resulting equations is done which predicts the appearance of flow patterns. A stable modulation of density and temperature is produced that does not lead to clustering. Also a global decrease of the temperature is obtained, giving rise to a decrease of the collision frequency and dissipation rate. Good agreement with molecular dynamics simulations of inelastic hard disks is found for low dissipation.

Thermal convection in fluidized granular systems

Thermal convection is observed in molecular dynamic simulations of a fluidized granular system of nearly elastic hard disks moving under gravity, inside a square box. Boundaries introduce no shearing or time dependence, but the energy injection comes from a slip (shear-free) thermalizing base. The top wall is perfectly elastic and lateral boundaries are either elastic or periodic. The spontaneous temperature gradient appearing in the system due to the inelastic collisions, combined with gravity, produces a buoyancy force that, when dissipation is large enough, triggers convection.

Hydrodynamic theory for granular gases

A granular gas subjected to a permanent injection of energy is described by means of hydrodynamic equations derived from a moment expansion method. The method uses as reference function not a Maxwellian distribution f(M) but a distribution f(0)=Phif(M), such that Phi adds a fourth cumulant kappa to the velocity distribution. The formalism is applied to a stationary conductive case showing that the theory fits extraordinarily well the results coming from our Newtonian molecular dynamic simulations once we determine kappa as a function of the inelasticity of the particle-particle collisions. The shape of kappa is independent of the size N of the system.

Potential energy in a three-dimensional vibrated granular medium measured by NMR imaging

Fast NMR imaging was used to measure the density profile of a three-dimensional granular medium fluidized by vertical vibrations of the container. For container acceleration much larger than gravity, the rise in center of mass of the granular medium is found to scale as v(alpha)(0)/N(beta)(l) with alpha = 1.0+/-0.2 and beta = 0.5+/-0.1, where v(0) is the vibration velocity, and N(l) is the number of layers of grains in the container. This value for alpha is significantly less than found previously for experiments and simulations in one dimension ( alpha = 2) and two dimensions ( alpha = 1.3-1.5).

Two-dimensional granular gas of inelastic spheres with multiplicative driving

We study a two-dimensional granular gas of inelastic spheres subject to multiplicative driving proportional to a power |v(x)|(delta) of the local particle velocity v(x). The steady state properties of the model are examined for different values of delta, and compared with the homogeneous case delta = 0. A driving linearly proportional to v(x) seems to reproduce some experimental observations which could not be reproduced by a homogeneous driving. Furthermore, we obtain that the system can be homogenized even for strong dissipation, if a driving inversely proportional to the velocity is used.

Self-diffusion in granular gases

The coefficient of self-diffusion for a homogeneously cooling granular gas changes significantly if the impact-velocity dependence of the restitution coefficient epsilon is taken into account. For the case of a constant epsilon the particles spread logarithmically slowly with time, whereas a velocity-dependent coefficient yields a power law time dependence. The impact of the difference in these time dependences on the properties of a freely cooling granular gas is discussed.

Granular collapse as a percolation transition

Inelastic collapse is found in a two-dimensional system of inelastic hard disks confined between two walls which act as an energy source. As the coefficient of restitution is lowered, there is a transition between a state containing small collapsed clusters and a state dominated by a large collapsed cluster. The transition is analogous to that of a percolation transition. At the transition the number of clusters n(s) of size s scales as n(s) approximately s(-tau) with tau approximately equal to 2.7.

Geometrical characterization of hard-sphere systems

By using molecular dynamics simulations on a large number of hard spheres and the Voronoï tessellation we characterize hard-sphere systems geometrically at any packing fraction eta along the different branches of the phase diagram. Crystallization of disordered packings occurs only for a small range of packing fraction. For the other packing fractions the system behaves as either a fluid (stable or metastable) or a glass. We have studied the evolution of the statistics of the Voronoï tessellation during crystallization and characterized the apparition of order by an order parameter (Q(6)) built from spherical harmonics.

Velocity distributions and density fluctuations in a granular gas

Velocity distributions in a vibrated granular monolayer are investigated experimentally. Non-Gaussian velocity distributions are observed at low vibration amplitudes but cross over smoothly to Gaussian distributions as the amplitude is increased. Cross-correlations between fluctuations in density and temperature are present only when the velocity distributions are strongly non-Gaussian. Confining the expansion of the granular layer results in non-Gaussian velocity distributions that persist to high vibration amplitudes.

Fluctuations in granular media

Dense slowly evolving or static granular materials exhibit strong force fluctuations even though the spatial disorder of the grains is relatively weak. Typically, forces are carried preferentially along a network of "force chains." These consist of linearly aligned grains with larger-than-average force. A growing body of work has explored the nature of these fluctuations. We first briefly review recent work concerning stress fluctuations. We then focus on a series of experiments in both two- and three-dimension [(2D) and (3D)] to characterize force fluctuations in slowly sheared systems. Both sets of experiments show strong temporal fluctuations in the local stress/force; the length scales of these fluctuations extend up to 10(2) grains. In 2D, we use photoelastic disks that permit visualization of the internal force structure. From this we can make comparisons to recent models and calculations that predict the distributions of forces. Typically, these models indicate that the distributions should fall off exponentially at large force. We find in the experiments that the force distributions change systematically as we change the mean packing fraction, gamma. For gamma's typical of dense packings of nondeformable grains, we see distributions that are consistent with an exponential decrease at large forces. For both lower and higher gamma, the observed force distributions appear to differ from this prediction, with a more Gaussian distribution at larger gamma and perhaps a power law at lower gamma. For high gamma, the distributions differ from this prediction because the grains begin to deform, allowing more grains to carry the applied force, and causing the distributions to have a local maximum at nonzero force. It is less clear why the distributions differ from the models at lower gamma. An exploration in gamma has led to the discovery of an interesting continuous or "critical" transition (the strengthening/softening transition) in which the mean stress is the order parameter, and the mean packing fraction, gamma, must be adjusted to a value gamma(c) to reach the "critical point." We also follow the motion of individual disks and obtain detailed statistical information on the kinematics, including velocities and particle rotations or spin. Distributions for the azimuthal velocity, V(theta), and spin, S, of the particles are nearly rate invariant, which is consistent with conventional wisdom. Near gamma(c), the grain motion becomes intermittent causing the mean velocity of grains to slow down. Also, the length of stress chains grows as gamma-->gamma(c). The 3D experiments show statistical rate invariance for the stress in the sense that when the power spectra and spectral frequencies of the stress time series are appropriately scaled by the shear rate, Omega, all spectra collapse onto a single curve for given particle and sample sizes. The frequency dependence of the spectra can be characterized by two different power laws, P proportional, variant omega(-alpha), in the high and low frequency regimes: alpha approximately 2 at high omega; alpha<2 at low omega. The force distributions computed from the 3D stress time series are at least qualitatively consistent with exponential fall-off at large stresses. (c) 1999 American Institute of Physics.

Cluster-growth in freely cooling granular media

When dissipative particles are left alone, their fluctuation energy decays due to collisional interactions, clusters build up and grow with time until the system size is reached. When the effective dissipation is strong enough, this may lead to the "inelastic collapse," i.e., the divergence of the collision frequency of some particles. The cluster growth is an interesting physical phenomenon, whereas the inelastic collapse is an intrinsic effect of the inelastic hard sphere (IHS) model used to study the cluster growth-involving only a negligible number of particles in the system. Here, we extend the IHS model by introducing an elastic contact energy and the related contact duration t(c). This avoids the inelastic collapse and allows to examine the long-time behavior of the system. For a quantitative description of the cluster growth, we propose a burning-like algorithm in continuous space, that readily identifies all particles that belong to the same cluster. The criterion for this is here chosen to be only the particle distance. With this method we identify three regimes of behavior. First, for short times a homogeneous cooling state (HCS) exists, where a mean-field theory works nicely, and the clusters are tiny and grow very slowly. Second, at a certain time which depends on the system's properties, cluster growth starts and the clusters increase in size and mass until, in the third regime, the system size is reached and most of the particles are collected in one huge cluster. (c) 1999 American Institute of Physics.

Origin of density clustering in a freely evolving granular gas

The physical mechanisms leading to the development of density inhomogeneities in a freely evolving low density granular gas are investigated. By means of the direct simulation Monte Carlo method, numerical solutions of the inelastic Boltzmann equation are constructed for both a perturbed system and also for an initially homogeneous state. Analysis of the Fourier components of the fields indicates that the nonlinear coupling contributions of the transversal velocity play a crucial role in the initial setup of clustering. A simple hydrodynamic model is proposed to describe what is observed in the simulations. Finally, the nature of the inhomogeneous state is briefly discussed.

Dynamics of spherical particles on a surface: collision-induced sliding and other effects

We present a model for the motion of hard spherical particles on a two-dimensional surface. The model includes both the interaction between the particles via collisions and the interaction of the particles with the substrate. We analyze in detail the effects of sliding and rolling friction, which are usually overlooked. It is found that the properties of this particulate system are influenced significantly by the substrate-particle interactions. In particular, sliding of the particles relative to the substrate after a collision leads to considerable energy loss for common experimental conditions. The presented results provide a basis that can be used to realistically model the dynamical properties of the system, and provide further insight into density fluctuations and related phenomena of clustering and structure formation.

Granular cooling of hard needles

We have developed a kinetic theory of hard needles undergoing binary collisions with loss of energy due to normal and tangential restitution. In addition, we have simulated many particle systems of granular hard needles. The theory, based on the assumption of a homogeneous cooling state, predicts that granular cooling of the needles proceeds in two stages: An exponential decay of the initial configuration to a state where translational and rotational energies take on a time independent ratio (different from unity), followed by an algebraic decay of the total kinetic energy of approximately t(-2). The simulations support the theory very well for low and moderate densities. For higher densities, we have observed the onset of the formation of clusters and shear bands.

Kinetic approach to granular gases

We address the problem of the so-called "granular gases," i.e., gases of massive particles in rapid movement undergoing inelastic collisions. We introduce a class of models of driven granular gases for which the stationary state is the result of the balance between the dissipation and the random forces which inject energies. These models exhibit a genuine thermodynamic limit, i.e., at fixed density the mean values of kinetic energy and dissipated energy per particle are independent of the number N of particles, for large values of N. One has two regimes: when the typical relaxation time tau of the driving Brownian process is small compared with the mean collision time tau(c) the spatial density is nearly homogeneous and the velocity probability distribution is Gaussian. In the opposite limit tau>>tau(c) one has strong spatial clustering, with a fractal distribution of particles, and the velocity probability distribution strongly deviates from the Gaussian one. Simulations performed in one and two dimensions under the Stosszahlansatz Boltzmann approximation confirm the scenario. Furthermore, we analyze the instabilities bringing to the spatial and the velocity clusterization. Firstly, in the framework of a mean-field model, we explain how the existence of the inelasticity can lead to a spatial clusterization; on the other hand, we discuss, in the framework of a Langevin dynamics treating the collisions in a mean-field way, how a non-Gaussian distribution of velocity can arise. The comparison between the numerical and the analytical results exhibits an excellent agreement.