Molecular dynamics simulations of vibrated granular gases

Non-equilibrium coexistence between a fluid and a hotter or colder crystal of granular hard disks

Non-equilibrium phase coexistence is commonly observed in both biological and artificial systems, yet understanding it remains a significant challenge. Unlike equilibrium systems, where free energy provides a unifying framework, the absence of such a quantity in non-equilibrium settings complicates their theoretical understanding. Granular materials, driven out of equilibrium by energy dissipation during collisions, serve as an ideal platform to investigate these systems, offering insights into the parallels and distinctions between equilibrium and non-equilibrium phase behavior. For example, the coexisting dense phase is typically colder than the dilute phase, a result usually attributed to greater dissipation in denser regions. In this article, we demonstrate that this is not always the case. Using a simple numerical granular model, we show that a hot solid and a cold liquid can coexist in granular systems. This counterintuitive phenomenon arises because the collision frequency can be lower in the solid phase than in the liquid phase, consistent with equilibrium results for hard-disk systems. We further demonstrate that kinetic theory can be extended to accurately predict phase temperatures even at very high packing fractions, including within the solid phase. Our results highlight the importance of collisional dynamics and energy exchange in determining phase behavior in granular materials, offering new insights into non-equilibrium phase coexistence and the complex physics underlying granular systems.

Ensemble-Based Approaches Ensure Reliability and Reproducibility

It is increasingly widely recognized that ensemble-based approaches are required to achieve reliability, accuracy, and precision in molecular dynamics calculations. The purpose of the present article is to address a frequently raised question: what is the optimal way to perform ensemble simulation to calculate quantities of interest?

Geometry-controlled phase transition in vibrated granular media

We report experiments on the dynamics of vibrated particles constrained in a two-dimensional vertical container, motivated by the following question: how to get the most out of a given external vibration to maximize internal disorder (e.g. to blend particles) and agitation (e.g. to absorb vibrations)? Granular media are analogs to classical thermodynamic systems, where the injection of energy can be achieved by shaking them: fluidization arises by tuning either the amplitude or the frequency of the oscillations. Alternatively, we explore what happens when another feature, the container geometry, is modified while keeping constant the energy injection. Our method consists in modifying the container base into a V-shape to break the symmetries of the inner particulate arrangement. The lattice contains a compact hexagonal solid-like crystalline phase coexisting with a loose amorphous fluid-like phase, at any thermal agitation. We show that both the solid-to-fluid volume fraction and the granular temperature depend not only on the external vibration but also on the number of topological defects triggered by the asymmetry of the container. The former relies on the statistics of the energy fluctuations and the latter is consistent with a two-dimensional melting transition described by the KTHNY theory.

Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties-A Review

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures Ti (measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows.

Dynamics of a vibration-driven single disk

Granular particles exhibit rich collective behaviors on vibration beds, but the motion of an isolated particle is not well understood even for uniform particles with a simple shape such as disks or spheres. Here we measured the motion of a single disk confined to a quasi-two-dimensional horizontal box on a vertically vibrating stage. The translational displacements obey compressed exponential distributions whose exponent [Formula: see text] increases with the frequency, while the rotational displacements exhibit unimodal distributions at low frequencies and bimodal distributions at high frequencies. During short time intervals, the translational displacements are subdiffusive and negatively correlated, while the rotational displacements are superdiffusive and positively correlated. After prolonged periods, the rotational displacements become diffusive and their correlations decay to zero. Both the rotational and the translational displacements exhibit white noise at low frequencies, and blue noise for translational motions and Brownian noise for rotational motions at high frequencies. The translational kinetic energy obeys Boltzmann distribution while the rotational kinetic energy deviates from it. Most energy is distributed in translational motions at low frequencies and in rotational motions at high frequencies, which violates the equipartition theorem. Translational and rotational motions are not correlated. These experimental results show that the random diffusion of such driven particles is distinct from thermal motion in both the translational and rotational degrees of freedom, which poses new challenges to theory. The results cast new light on the motion of individual particles and the collective motion of driven granular particles.

Energy nonequipartition in a collisional model of a confined quasi-two-dimensional granular mixture

A collisional model of a confined quasi-two-dimensional granular mixture is considered to analyze homogeneous steady states. The model includes an effective mechanism to transfer the kinetic energy injected by vibration in the vertical direction to the horizontal degrees of freedom of grains. The set of Enskog kinetic equations for the velocity distribution functions of each component is derived first to analyze the homogeneous state. As in the one-component case, an exact scaling solution is found where the time dependence of the distribution functions occurs entirely through the granular temperature T. As expected, the kinetic partial temperatures T_{i} of each component are different and, hence, energy equipartition is broken down. In the steady state, explicit expressions for the temperature T and the ratio of partial kinetic temperatures T_{i}/T_{j} are obtained by considering Maxwellian distributions defined at the partial temperatures T_{i}. The (scaled) granular temperature and the temperature ratios are given in terms of the coefficients of restitution, the solid volume fraction, the (scaled) parameters of the collisional model, and the ratios of mass, concentration, and diameters. In the case of a binary mixture, the theoretical predictions are exhaustively compared with both direct simulation Monte Carlo and molecular dynamics simulations with a good agreement. The deviations are identified to be originated in the non-Gaussianity of the velocity distributions and on microsegregation patterns, which induce spatial correlations not captured in the Enskog theory.

Blast Dynamics in a Dissipative Gas

The blast caused by an intense explosion has been extensively studied in conservative fluids, where the Taylor-von Neumann-Sedov hydrodynamic solution is a prototypical example of self-similarity driven by conservation laws. In dissipative media, however, energy conservation is violated, yet a distinctive self-similar solution appears. It hinges on the decoupling of random and coherent motion permitted by a broad class of dissipative mechanisms. This enforces a peculiar layered structure in the shock, for which we derive the full hydrodynamic solution, validated by a microscopic approach based on molecular dynamics simulations. We predict and evidence a succession of temporal regimes, as well as a long-time corrugation instability, also self-similar, which disrupts the blast boundary. These generic results may apply from astrophysical systems to granular gases, and invite further cross-fertilization between microscopic and hydrodynamic approaches of shock waves.

Generalized transport coefficients for inelastic Maxwell mixtures under shear flow

The Boltzmann equation framework for inelastic Maxwell models is considered to determine the transport coefficients associated with the mass, momentum, and heat fluxes of a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. The Boltzmann equation is solved by means of a Chapman-Enskog-type expansion around the (local) shear flow distributions f(r)(0) for each species that retain all the hydrodynamic orders in the shear rate. Due to the anisotropy induced by the shear flow, tensorial quantities are required to describe the transport processes instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled equations, which can be analytically solved as functions of the shear rate a, the coefficients of restitution α(rs), and the parameters of the mixture (masses, diameters, and composition). Since the reference distribution functions f(r)(0) apply for arbitrary values of the shear rate and are not restricted to weak dissipation, the corresponding generalized coefficients turn out to be nonlinear functions of both a and α(rs). The dependence of the relevant elements of the three diffusion tensors on both the shear rate and dissipation is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coefficients from their forms for vanishing shear rates is in general significant. A comparison with the previous results obtained analytically for inelastic hard spheres by using Grad's moment method is carried out, showing a good agreement over a wide range of values for the coefficients of restitution. Finally, as an application of the theoretical expressions derived here for the transport coefficients, thermal diffusion segregation of an intruder immersed in a granular gas is also studied.

Anomalous transport of impurities in inelastic Maxwell gases

A mixture of dissipative hard grains generically exhibits a breakdown of kinetic energy equipartition. The undriven and thus freely cooling binary problem, in the tracer limit where the density of one species becomes minute, may exhibit an extreme form of this breakdown, with the minority species carrying a finite fraction of the total kinetic energy of the system. We investigate the fingerprint of this non-equilibrium phase transition, akin to an ordering process, on transport properties. The analysis, performed by solving the Boltzmann kinetic equation from a combination of analytical and Monte Carlo techniques, hints at the possible failure of hydrodynamics in the ordered region. As a relevant byproduct of the study, the behaviour of the second- and fourth-degree velocity moments is also worked out.

Energy non-equipartition in strongly convective granular systems

Using positron emission particle tracking, the effects of convective motion and the resulting segregative behaviour on the partition of kinetic energy between the components of a bidisperse granular system are, for the first time, systematically investigated. It is found that the distribution of energy between the two system components, which are equal in size but differ in their material properties, is strongly dependent on the degree of segregation observed in the granular bed. The results obtained demonstrate that the difference in energy obtained by dissimilar particle species is not an innate property of the materials in question, but can in fact be altered through variation of the relevant system parameters. The existence of a relationship between the convective and segregative properties of a granular system and the degree of energy equipartition within the system implies the possibility of extending existing theory into the convective regime. Thus, our findings represent an incremental step towards the definition of a granular analogy to temperature that can be applied to more generalised systems and, through this, an improved understanding of inhomogeneous granular systems in general.

Transport coefficients for driven granular mixtures at low density

The transport coefficients of a granular binary mixture driven by a stochastic bath with friction are determined from the inelastic Boltzmann kinetic equation. A normal solution is obtained via the Chapman-Enskog method for states near homogeneous steady states. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. They are given in terms of the solutions of a set of coupled linear integral equations. As in the monocomponent case, since the collisional cooling cannot be compensated for locally by the heat produced by the external driving, the reference distributions (zeroth-order approximations) f(i)((0)) (i=1,2) for each species depend on time through their dependence on the pressure and the temperature. Explicit forms for the diffusion transport coefficients and the shear viscosity coefficient are obtained by assuming the steady-state conditions and by considering the leading terms in a Sonine polynomial expansion. A comparison with previous results obtained for granular Brownian motion and by using a (local) stochastic thermostat is also carried out. The present work extends previous theoretical results derived for monocomponent dense gases [Garzó, Chamorro, and Vega Reyes, Phys. Rev. E 87, 032201 (2013)] to granular mixtures at low density.

Boltzmann statistics in a three-dimensional vibrofluidized granular bed: idealizing the experimental system

The importance of the method of energy injection into a vertically vibrated granular bed was investigated using positron emission particle tracking. A comparison was made between two experimental systems. The first was driven by a flat base, the second by a base comprising a monolayer of spheres, each free to move in three dimensions but constrained to the bottom of the system, effectively randomizing energy transfer into the bed. The latter system exhibited more Gaussian velocity distributions, a more isotropic temperature, and a better approximation of molecular chaos than the former, behavior more akin to the idealized situation of white noise forcing than is generally observed in an experimental system.

Two types of oscillation in unequally compartmentalized granular gases

Granular oscillation in an unequally two-compartmentalized system is investigated experimentally, yielding two time periods: τ(LS) represents the time period for particles moving from the large to the small compartment, and τ(SL) is the time needed for movement in a reverse direction. We construct the phase diagram for the unequal system, discovering that there exist two different granular oscillation states. In the GOI state, τ(LS)>τ(SL) is observed, and (τ(LS)-τ(SL)) increases as the difference between the two compartment widths is enlarged. In contrast, τ(LS)<τ(SL) is found in the GOII state, where the total number of particles is considerably lower than that in the GOI state. A flux model, accompanied by a proposed generalized granular temperature, is adopted to quantitatively calculate the observed time periods for both granular oscillation states. We find that the proposed temperature in each compartment is crucial in discriminating τ(LS) from τ(SL), and this is supported by the molecular dynamics analysis of mean kinetic energy of particles in a single compartment.

Segregation of an intruder in a heated granular dense gas

A recent segregation criterion [Phys. Rev. E 78, 020301(R) (2008)] based on the thermal diffusion factor Λ of an intruder in a heated granular gas described by the inelastic Enskog equation is revisited. The sign of Λ provides a criterion for the transition between the Brazil-nut effect (BNE) and the reverse Brazil-nut effect (RBNE). The present theory incorporates two extra ingredients not accounted for by the previous theoretical attempt. First, the theory is based upon the second Sonine approximation to the transport coefficients of the mass flux of the intruder. Second, the dependence of the temperature ratio (intruder temperature over that of the host granular gas) on the solid volume fraction is taken into account in the first and second Sonine approximations. In order to check the accuracy of the Sonine approximation considered, the Enskog equation is also numerically solved by means of the direct simulation Monte Carlo method to get the kinetic diffusion coefficient D(0). The comparison between theory and simulation shows that the second Sonine approximation to D(0) yields an improvement over the first Sonine approximation when the intruder is lighter than the gas particles in the range of large inelasticity. With respect to the form of the phase diagrams for the BNE-RBNE transition, the kinetic theory results for the factor Λ indicate that while the form of these diagrams depends sensitively on the order of the Sonine approximation considered when gravity is absent, no significant differences between both Sonine solutions appear in the opposite limit (gravity dominates the thermal gradient). In the former case (no gravity), the first Sonine approximation overestimates both the RBNE region and the influence of dissipation on thermal diffusion segregation.

Impurity in a sheared inelastic Maxwell gas

The Boltzmann equation for inelastic Maxwell models is considered in order to investigate the dynamics of an impurity (or intruder) immersed in a granular gas driven by a uniform shear flow. The analysis is based on an exact solution of the Boltzmann equation for a granular binary mixture. It applies for conditions arbitrarily far from equilibrium (arbitrary values of the shear rate a) and for arbitrary values of the parameters of the mixture (particle masses m(i), mole fractions x(i), and coefficients of restitution α(ij)). In the tracer limit where the mole fraction of the intruder species vanishes, a nonequilibrium phase transition takes place. We thereby identify ordered phases where the intruder bears a finite contribution to the properties of the mixture, in a region of parameter space that is worked out in detail. These findings extend previous results obtained for ordinary Maxwell gases, and further show that dissipation leads to new ordered phases.

Cooling rates and energy partition in inhomogeneous fluidized granular mixtures

The local cooling rates of the components of a vibrated binary granular mixture in a steady state are investigated. The accuracy of the expression obtained by assuming a local homogeneous cooling state distribution of the gas is analyzed by comparing it with molecular dynamics simulation results. A good agreement is observed. Also, the profiles of the partial temperatures are compared with the theoretical prediction following from the application of the Chapman-Enskog method to solve the kinetic Enskog equations of the mixture. In this case, the agreement is satisfactory if the boundary layers near the walls are excluded. The implications of the results are discussed.

Evolution of the initial hole in vertically vibrated shear thickening fluids

The apparent viscosity of shear thickening fluid (STF) changes dramatically with the applied shear rate, which is a typical rheological property of STF. Such a rheological property affects the vertically vibrated dynamic property of STF. In order to get a better understanding of the vertically vibrated dynamic properties of STF, the surface instabilities in vertically vibrated STF, which was prepared by suspending polymethylmethacrylate particles in ethylene glycol, are investigated. Above a critical driving acceleration, the surface instability transforms from the disappearance to the fission of the initial hole, which is produced by applying a finite perturbation to the surface. The time required for the initial hole to disappear can be affected by the driving acceleration, vibration frequency, volume fraction, thickness, and shape and size of the perturbation. A possible model is proposed, and the expressions of hydrostatic force and viscous dissipative force are employed to clarify the relationship between disappearance of the hole and shear thickening effect. The fission and spreading follow a hexagonal arrangement. At a higher acceleration, the holes cover the entire surface in a state of disorder. The mechanism for the initial hole's evolution in vertically vibrated shear thickening fluids is discussed.

Diffusivity and weak clustering in a quasi-two-dimensional granular gas

We present results from a detailed simulation of a quasi-two-dimensional dissipative granular gas, kept in a noncondensed steady state via vertical shaking over a rough substrate. This gas shows a weak power-law decay in the tails of its pair distribution functions, indicating clustering. This clustering depends monotonically on the dissipation coefficient and disappears when the sphere-sphere collisions are conservative. Clustering is also sensitive to the packing fraction. This gas also displays the standard nonequilibrium characteristics of similar systems, including non-Maxwellian velocity distributions. The diffusion coefficients are calculated over all the conditions of the simulations, and it is found that diluted gases are more diffusive for smaller restitution coefficients.

Crystallization of hard disks induced by a temperature gradient

While uniform temperature has no effect on equilibrium properties of hard-core systems, its gradient might substantially change their behavior. In particular, in hard-disk system subject to temperature difference ΔT disks are repelled from the hot boundary of the system and accumulate at the cold one. Using event-driven molecular dynamics simulations we show that for sufficiently large ΔT or coverage ratio ρ∗, crystal forms at the cold boundary. In this spatially inhomogeneous system a significant decrease of diffusivity of disks clearly marks the stationary interface between liquid and crystal. Such a behavior is also supported through calculation of the radial distribution function and the bond order parameter. Simulations show that for this nonequilibrium system the equipartition of energy holds and velocity obeys the Boltzmann distribution.

Particle kinematics in a dilute, three-dimensional, vibration-fluidized granular medium

We report an experimental study of particle kinematics in a three-dimensional system of inelastic spheres fluidized by intense vibration. The motion of particles in the interior of the medium is tracked by high-speed video imaging, yielding a spatially resolved measurement of the velocity distribution. The distribution is wider than a Gaussian and broadens continuously with increasing volume fraction. The deviations from a Gaussian distribution for this boundary-driven system are different in sign and larger in magnitude than predictions for homogeneously driven systems. We also find correlations between velocity components which grow with increasing volume fraction.

Clustering and phases of compartmentalized granular gases

This paper experimentally investigates the clustering conditions for compartmentalized monodisperse granular gases, determining the critical particle number and condensation granular temperature at the gas-clustering transition. When one heavier intruding particle is added to a monodisperse gas, it is found that the condensation temperature decreases with the ratio of the mass of the intruding particle to that of the background particle. This phenomenon can be mathematically characterized by a proposed linear relation, which is reminiscent of a relation between the freezing point depression for a solvent and the concentration of an added solute. Finally we perform various tests by changing the numbers of two types of particles in bidisperse granular mixtures to construct the phase diagrams, which present the range of the five different states, namely, homogeneous gas, unstable-gas, one-clustering, two-clustering, and granular oscillation states.

Heating mechanism affects equipartition in a binary granular system

Two species of particles in a binary granular system typically do not have the same mean kinetic energy, in contrast to the equipartition of energy required in equilibrium. We investigate the role of the heating mechanism in determining the extent of nonequipartition of kinetic energy. In most experiments, different species are unequally heated at the boundaries. We show by event-driven simulations that differential boundary heating affects nonequipartition even in the bulk of the system. This conclusion is fortified by studying numerical and solvable stochastic models without spatial degrees of freedom. In both cases, even in the limit where heating events are rare compared to collisions, the effect of the heating mechanism persists.

Correlation and response in a driven dissipative model

We consider a simple dissipative system with spatial structure in contact with a heat bath. The system always exhibits correlations except in the cases of zero and maximal dissipation. We explicitly calculate the correlation function and the nonlocal response function of the system and show that they have the same spatial dependence.

Hydrodynamic profiles for an impurity in an open vibrated granular gas

The hydrodynamic state of an impurity immersed in a low density granular gas is analyzed. Explicit expressions for the temperature and density fields of the impurity in terms of the hydrodynamic fields of the gas are derived. It is shown that the ratio between the temperatures of the two components, measuring the departure from the energy equipartition, only depends on the mechanical properties of the particles, being therefore constant in the bulk of the system. This ratio plays an important role in determining the density profile of the intruder and its position with respect to the gas, since it determines the sign of the pressure diffusion coefficient. The theoretical predictions are compared with molecular dynamics simulation results for the particular case of the steady state of an open vibrated granular system in the absence of macroscopic fluxes, and a satisfactory agreement is found.

Velocity distributions in dilute granular systems

We investigate the idea that velocity distributions in granular gases are determined mainly by eta, the coefficient of restitution and q, which measures the relative importance of heating (or energy input) to collisions. To this end, we study by numerical simulation the properties of inelastic gases as functions of eta, concentration phi, and particle number N with various heating mechanisms. For a wide range of parameters, we find Gaussian velocity distributions for uniform heating and non-Gaussian velocity distributions for boundary heating. Comparison between these results and velocity distributions obtained by other heating mechanisms and for a simple model of a granular gas without spatial degrees of freedom, shows that uniform and boundary heating can be understood as different limits of q, with q>>1 and q < or approximately 1 respectively. We review the literature for evidence of the role of q in the recent experiments.

Velocity distributions of granular gases with drag and with long-range interactions

We study velocity statistics of electrostatically driven granular gases. For two different experiments, (i) nonmagnetic particles in a viscous fluid and (ii) magnetic particles in air, the velocity distribution is non-Maxwellian, and its high-energy tail is exponential, P(upsilon) approximately exp(-/upsilon/). This behavior is consistent with the kinetic theory of driven dissipative particles. For particles immersed in a fluid, viscous damping is responsible for the exponential tail, while for magnetic particles, long-range interactions cause the exponential tail. We conclude that velocity statistics of dissipative gases are sensitive to the fluid environment and to the form of the particle interaction.

Brazil nut effect and excluded volume attraction in vibrofluidized granular mixtures

A two dimensional bi-disperse vibrofluidized granular mixture is studied in the rapid flow regime, where particle interactions occur due to instantaneous collisions. Both experiments and simulations are carried out, and these show the existence of two phenomena which have been observed only in very dense granular flows or in equilibrium systems. The Brazil nut phenomenon, which involves the rise of larger particles in a granular mixture upon vibration, has been observed in dense systems due to the percolation of small particles though the interstitial spaces between the large particles, or due to convection rolls. In the present case, where neither effect is present, it is observed that the fluidization of the smaller particles by vibration results in an exponentially decaying density profile, at heights large compared to the particle diameter, and thereby a pressure field that decreases with height. The larger particles, suspended in this decaying pressure field, experience a larger pressure at the bottom and a smaller pressure on top, and they rise to a height where the net force caused by the decreasing pressure is balanced by the weight of the particle. An attractive force between the large particles, similar to the entropic attraction effect in mixtures of colloids and polymers, is also observed in this nonequilibrium system, because when the distance between the large particles is less than the small particle diameter, the pressure between the large particles is smaller than that on the outside. Analytical results are derived for each of these effects, and these are in agreement with the experimental and simulation results.

Local equation of state and velocity distributions of a driven granular gas

We present event-driven simulations of a granular gas of inelastic hard disks with incomplete normal restitution in two dimensions between vibrating walls (without gravity). We measure hydrodynamic quantities such as the stress tensor, density and temperature profiles, as well as velocity distributions. Relating the local pressure to the local temperature and local density, we construct a local constitutive equation. For strong inelasticities the local constitutive relation depends on global system parameters, like the volume fraction and the aspect ratio. For moderate inelasticities the constitutive relation is approximately independent of the system parameters and can hence be regarded as a local equation of state, even though the system is highly inhomogeneous with heterogeneous temperature and density profiles arising as a consequence of energy injection. With respect to local velocity distributions we find that they do not scale with the square root of the local granular temperature. Moreover the high-velocity tails are different for the distribution of the x and the y components of the velocity, and even depend on the position in the sample, the global volume fraction, and the coefficient of restitution.

Crucial role of sidewalls in velocity distributions in quasi-two-dimensional granular gases

Our experiments and three-dimensional molecular dynamics simulations of particles confined to a vertical monolayer by closely spaced frictional walls (sidewalls) yield velocity distributions with non-Gaussian tails and a peak near zero velocity. Simulations with frictionless sidewalls are not peaked. Thus interactions between particles and their containers are an important determinant of the shape of the distribution and should be considered when evaluating experiments on a constrained monolayer of particles.

Anisotropic energy distribution in three-dimensional vibrofluidized granular systems

We examine the energy flows in a three-dimensional model of a granular system consisting of N inelastic hard spheres contained in an open cylinder of radius R under the influence of gravity. Energy is supplied to the system in the vertical direction by a vibrating base and is transferred to the perpendicular directions through particle-particle collisions. We examine how the local and global dissipation of energy by particle-particle and particle-wall collisions depends on the number of particles, the velocity of the vibrating base, and the restitution coefficients.

Thermal convection in monodisperse and bidisperse granular gases: a simulation study

We present results of a simulation study of inelastic hard disks vibrated in a vertical container. An event-driven molecular dynamics method is developed for studying the onset of convection. Varying the relevant parameters (inelasticity, number of layers at rest, intensity of the gravity) we are able to obtain a qualitative agreement of our results with recent hydrodynamical predictions. Increasing the inelasticity, a first continuous transition from the absence of convection to one convective roll is observed, followed by a discontinuous transition to two convective rolls, with hysteretic behavior. At fixed inelasticity and increasing gravity, a transition from no convection to one roll can be evidenced. If the gravity is further increased, the roll is eventually suppressed. Increasing the number of monolayers the system eventually localizes mostly at the bottom of the box: in this case multiple convective rolls as well as surface waves appear. We analyze the density and temperature fields and study the existence of symmetry breaking in these fields in the direction perpendicular to the injection of energy. We also study a binary mixture of grains with different properties (inelasticity or diameters). The effect of changing the properties of one of the components is analyzed, together with density, temperature, and temperature ratio fields. Finally, the presence of a low fraction of quasielastic impurities is shown to determine a sharp transition between convective and nonconvective steady states.

Thermalization of an anisotropic granular particle

We investigate the dynamics of a needle in a two-dimensional bath composed of thermalized point particles. Collisions between the needle and points are inelastic and characterized by a normal restitution coefficient alpha<1. By using the Enskog-Boltzmann equation, we obtain analytical expressions for the translational and rotational granular temperatures of the needle and show that these are, in general, different from the bath temperature. The translational temperature always exceeds the rotational one, though the difference decreases with increasing moment of inertia. The predictions of the theory are in very good agreement with numerical simulations of the model.

Steady-state velocity distributions of an oscillated granular gas

We use a three-dimensional molecular dynamics simulation to study the single particle distribution function of a dilute granular gas driven by a vertically oscillating plate at high accelerations (15g-90g). We find that the density and the temperature fields are essentially time-invariant above a height of about 40 particle diameters, where typically 20% of the grains are contained. These grains form the nonequilibrium steady-state granular gas with a Knudsen number unity or greater. In the steady-state region, the probability distribution function of the horizontal velocity c(x) (scaled by the local horizontal temperature) is found to be nearly independent of height, even though the hydrodynamic fields vary with height. We find that the high energy tails of the distribution function are described by a stretched exponential approximately exp(-Bcalphax), where alpha depends on the restitution coefficient e and falls in the range 1.2<alpha<1.6. However, alpha does not vary significantly for a wide range of friction coefficient values. We find that the distribution function of a frictionless inelastic hard sphere model can be made similar to that of a frictional model by adjusting e. However, there is no single value of e that mimics the frictional model over a range of heights.

Fluid-like behavior of a one-dimensional granular gas

We study the properties of a one-dimensional (1D) granular gas consisting of N hard rods on a line of length L (with periodic boundary conditions). The particles collide inelastically and are fluidized by a heat bath at temperature Tb and viscosity gamma. The analysis is supported by molecular dynamics simulations. The average properties of the system are first discussed, focusing on the relations between granular temperature Tg=mv2, kinetic pressure, and density rho=N/L. Thereafter, we consider the fluctuations around the average behavior obtaining a slightly non-Gaussian behavior of the velocity distributions and a spatially correlated velocity field; the density field displays clustering: this is reflected in the structure factor which has a peak in the k approximately 0 region suggesting an analogy between inelastic hard core interactions and an effective attractive potential. Finally, we study the transport properties, showing the typical subdiffusive behavior of 1D stochastically driven systems, i.e., </x(t)-x(0)/2> approximately Dt(1/2), where D for the inelastic fluid is larger than the elastic case. This is directly related to the peak of the structure factor at small wave vectors.

Simulation study on kinetic temperatures of vibrated binary granular mixtures

We study the unequal kinetic temperatures in two-dimensional binary granular mixtures heated by vertical vibrations under gravity by molecular dynamics simulations. By introducing several variable parameters describing particle and composition properties of mixtures, the coexistence of two different kinetic temperatures for the two components is exhibited. In dilute situations, our numerical results confirm the overall experimental measurements by Feitosa and Menon recently [Phys. Rev. Lett. 88, 198301 (2002)]. For example, the temperature ratio along the vertical direction is constant in the bulk of a system, and is insensitive to external driving or relative concentration of the two components, but mainly influenced by their masses. However, outside of the dilute situations, especially as the number density becomes high enough where the spatial and momentum correlations become salient, the ratio profiles show some different characteristics. Generally, the plateau shape along the vertical direction is broken, and instead, a saddlelike shape appears. Furthermore, the physical implications are discussed.

Granular fluid thermostated by a bath of elastic hard spheres

The homogeneous steady state of a fluid of inelastic hard spheres immersed in a bath of elastic hard spheres kept at equilibrium is analyzed by means of the first Sonine approximation to the (spatially homogeneous) Enskog-Boltzmann equation. The temperature of the granular fluid relative to the bath temperature and the kurtosis of the granular distribution function are obtained as functions of the coefficient of restitution, the mass ratio, and a dimensionless parameter beta measuring the cooling rate relative to the friction constant. Comparison with recent results obtained from an iterative numerical solution of the Enskog-Boltzmann equation [Biben et al., Physica A 310, 308 (2002)] shows an excellent agreement. Several limiting cases are also considered. In particular, when the granular particles are much heavier than the bath particles (but have a comparable size and number density), it is shown that the bath acts as a white noise external driving. In the general case, the Sonine approximation predicts the lack of a steady state if the control parameter beta is larger than a certain critical value beta(c) that depends on the coefficient of restitution and the mass ratio. However, this phenomenon appears outside the expected domain of applicability of the approximation.

Collision statistics of driven granular materials

We present an experimental investigation of the statistical properties of spherical granular particles on an inclined plane that are excited by an oscillating side wall. The data is obtained by high-speed imaging and particle tracking techniques. We identify all particles in the system and link their positions to form trajectories over long times. Thus, we identify particle collisions to measure the effective coefficient of restitution and find a broad distribution of values for the same impact angles. We find that the energy inelasticity can take on values greater than one, which implies that the rotational degrees of freedom play an important role in energy transfer. We also measure the distance and the time between collision events in order to directly determine the distribution of path lengths and the free times. These distributions are shown to deviate from expected theoretical forms for elastic spheres, demonstrating the inherent clustering in this system. We describe the data with a two-parameter fitting function and use it to calculate the mean free path and collision time. We find that the ratio of these values is consistent with the average velocity. The velocity distributions are observed to be strongly non-Gaussian and do not demonstrate any apparent universal behavior. We report the scaling of the second moment, which corresponds to the granular temperature, and higher order moments as a function of distance from the driving wall. Additionally, we measure long-time correlation functions in both space and in the velocities to probe diffusion in a dissipative gas.

Velocity distribution of fluidized granular gases in the presence of gravity

The velocity distribution of a fluidized dilute granular gas in the direction perpendicular to the gravitational field is investigated by means of molecular dynamics simulations. The results indicate that the velocity distribution can be exactly described neither by a Gaussian nor by a stretched exponential law. Moreover, it does not exhibit any kind of scaling. In fact, the actual shape of the distribution depends on the number of monolayers at rest, on the restitution coefficient and on the height at what it is measured. The role played by the number of particle-particle collisions as compared with the number of particle-wall collisions is discussed.