We study heat conduction in one-dimensional granular gases. In particular, we consider two mechanisms of viscous dissipation during inter-grain collisions. In one, the dissipative force is proportional to the grain's velocity and dissipates not only energy but also momentum. In the other, the dissipative force is proportional to the relative velocity of the grains and therefore conserves momentum even while dissipating energy. This allows us to explore the role of momentum conservation in the heat conduction properties of this one-dimensional nonlinear system. We find normal thermal conduction whether or not momentum is conserved.
This article reports on an experimental study of the fluctuations of energy flux between a granular gas and a small driven harmonic oscillator. The DC-motor driving this system is used simultaneously as actuator and probe. The statistics of work fluctuations at controlled forcing, between the motor and the gas are examined from the viewpoint of the Fluctuation Theorem. A characteristic energy $E_c$ of the granular gas, is obtained from this relation between the probabilities of an event and its reversal.
A freely cooling granular gas with velocity dependent restitution coefficient is studied in one dimension. The restitution coefficient becomes near elastic when the relative velocity of the colliding particles is less than a velocity scale $δ$. Different statistical quantities namely density distribution, occupied and empty cluster length distributions, and spatial density and velocity correlation functions, are obtained using event driven molecular dynamic simulations. We compare these with the corresponding quantities of the sticky gas (inelastic gas with zero coefficient of restitution). We…
When smooth granular material is fluidized by vertically shaking a container, we find that the temperature in the direction of energy input always exceeds the temperature in the other directions. An analytical model is presented which shows how the anisotropy can be traced back to the inelasticity of the interparticle collisions and the collisions with the wall. The model compares very well with molecular dynamics simulations. It is concluded that any non-isotropic driving of a granular gas in a steady state necessarily causes anisotropy of the granular temperature.
The spontaneous clustering of a vibrofluidized granular gas is employed to generate directed transport in two different compartmentalized systems: a "granular fountain" in which the transport takes the form of convection rolls, and a "granular ratchet" with a spontaneous particle current perpendicular to the direction of energy input. In both instances, transport is not due to any system-intrinsic anisotropy, but arises as a spontaneous collective symmetry breaking effect of many interacting granular particles. The experimental and numerical results are quantitatively accounted for within a fl…
The rheology of a two-dimensional granular gas under a plane shear is investigated. From the comparison among the discrete element method, the simulation of a set of hydrodynamic equation, and the analytic solution of the steady equation of the hydrodynamic equations, it is confirmed that the fluid equations derived from the kinetic theory give us accurate results even in relatively high density cases.
We develop the rheology of a dilute granular gas mixture. Motivated by the interaction of charged granular particles, we assume that the grains interact via a square shoulder and well potential. Employing kinetic theory, we compute the temperature and the shear viscosity as functions of the shear rate. Numerical simulations confirm our results above the critical shear rate. At a shear rate below a critical value, clustering of the particles occurs.
A bi-disperse granular gas in a compartmentalized system is experimentally found to cluster competitively: Depending on the shaking strength, the clustering can be directed either towards the compartment initially containing mainly small particles, or to the one containing mainly large particles. The experimental observations are quantitatively explained within a flux model.
Dry, freely evolving granular materials in a dilute gaseous state coalesce into dense clusters only due to dissipative interactions. This clustering transition is important for a number of problems ranging from geophysics to cosmology. Here we show that the evolution of a dilute, freely cooling granular gas is determined in a universal way by the ratio of inertial flow and thermal velocities, that is, the Mach number. Theoretical calculations and direct numerical simulations of the granular Navier--Stokes equations show that irrespective of the coefficient of restitution, density or initial ve…
In a granular gas experiment of magnetized particles confined in a thin layer, the rate of dissipative collisions is tuned by adjusting the amplitude of an external magnetic field. The velocity statistics are analyzed using the dynamic and static structure factors of transverse velocity modes. Using the fluctuating hydrodynamics theory we measure the deviation from kinetic energy equipartition in this out-of-equilibrium system as a function of the dissipative collision rate. When the collision rate is decreased, the distance to equipartition becomes smaller meaning that the dynamical propertie…
In contrast to molecular gases, granular gases are characterized by inelastic collisions and require therefore permanent driving to maintain a constant kinetic energy. The kinetic theory of granular gases describes how the average velocity of the particles decreases after the driving is shut off. Moreover it predicts that the rescaled particle velocity distribution will approach a stationary state with overpopulated high-velocity tails as compared to the Maxwell-Boltzmann distribution. While this fundamental theoretical result was reproduced by numerical simulations, an experimental confirmati…
Using diamagnetically levitated particles we investigate the dynamics of the freely cooling granular gas. At early times we find good agreement with Haff's law, where the time scale for particle collisions can be determined from independent measurements. At late times, clustering of particles occurs. This can be included in a Haff-like description taking into account the decreasing number of free particles. With this a good description of the data is possible over the whole time range.
Motivated by recent experiments reporting non-Gaussian velocity distributions in driven dilute granular materials, we study by numerical simulation the properties of 2D inelastic gases. We find theoretically that the form of the observed velocity distribution is governed primarily by the coefficient of restitution $η$ and $q=N_H/N_C$, the ratio between the average number of heatings and the average number of collisions in the gas. The differences in distributions we find between uniform and boundary heating can then be understood as different limits of $q$, for $q \gg 1$ and $q \lesssim 1$ res…
The kinetic energy of a freely cooling granular gas decreases as a power law $t^{-θ}$ at large times $t$. Two theoretical conjectures exist for the exponent $θ$. One based on ballistic aggregation of compact spherical aggregates predicts $θ= 2d/(d+2)$ in $d$ dimensions. The other based on Burgers equation describing anisotropic, extended clusters predicts $θ=d/2$ when $2\le d \le 4$. We do extensive simulations in three dimensions to find that while $θ$ is as predicted by ballistic aggregation, the cluster statistics and velocity distribution differ from it. Thus, the freely cooling granular g…
In the frame of a well established lattice gas model for granular compaction, we investigate the high intensity tapping regime where a pile expands significantly during external excitation. We find that this model shows the same general trends as more sophisticated models based on molecular dynamic type simulations. In particular, a minimum in packing fraction as a function of tapping strength is observed in the reversible branch of an annealed tapping protocol.
In the preceding paper, linear response methods have been applied to obtain formally exact expressions for the parameters of Navier-Stokes order hydrodynamics. The analysis there is general, applying to both normal and granular fluids with a wide range of collision rules. Those results are specialized here to the case of smooth, inelastic hard spheres with constant coefficient of normal restitution, for further elaboration. Explicit expressions for the cooling rate, pressure, and the transport coefficients are given and compared with the corresponding expressions for a system of elastic hard s…
We present a new mode of transport of spherical particles in a horizontally vibrated channel with sawtooth shaped side walls. The underlying driving mechanism is based on an interplay of directional energy injection transformed by the sidewall collisions and density dependent interparticle collisions. Experiments and matching numerics show that the average particle velocity reaches a maximum at 60% of the maximal filling density. Introducing a spatial phase shift between the channel boundaries increases the transport velocity by an order of magnitude.
We evidence a Kovacs-like memory effect in a uniformly driven granular gas. A system of inelastic hard particles, in the low density limit, can reach a non-equilibrium steady state when properly forced. By following a certain protocol for the drive time dependence, we prepare the gas in a state where the granular temperature coincides with its long time value. The temperature subsequently does not remain constant, but exhibits a non-monotonic evolution with either a maximum or a minimum, depending on the dissipation, and on the protocol. We present a theoretical analysis of this memory effect,…
This study numerically and analytically investigates the dynamics of a rotor under viscous or dry friction as a non-equilibrium probe of a granular gas. In order to demonstrate the role of the rotor as a probe for a non-equilibrium bath, the molecular dynamics (MD) simulation of the rotor is performed under viscous or dry friction surrounded by a steady granular gas under gravity. A one- to-one map between the velocity distribution function (VDF) of the granular gas and the angular distribution function for the rotor is theoretically derived. The MD simulation demonstrates that the one-to-one …
In general, the total kinetic energy in a multicomponent granular gas of inelastic and rough hard spheres is unequally partitioned among the different degrees of freedom. On the other hand, partial energy equipartition can be reached, in principle, under appropriate combinations of the mechanical parameters of the system. Assuming common values of the coefficients of restitution, we use kinetic-theory tools to determine the conditions under which the components of a granular mixture in the homogeneous cooling state have the same translational and rotational temperatures as those of a one-compo…
The evolution to the steady state of a granular gas subject to simple shear flow is analyzed by means of computer simulations. It is found that, regardless of its initial preparation, the system reaches (after a transient period lasting a few collisions per particle) a non-Newtonian (unsteady) hydrodynamic regime, even at strong dissipation and for states where the time scale associated with inelastic cooling is shorter than the one associated with the irreversible fluxes. Comparison with a simplified rheological model shows a good agreement.
The decay of a small homogeneous perturbation of the temperature of a dilute granular gas in the steady uniform shear flow state is investigated. Using kinetic theory based on the inelastic Boltzmann equation, a closed equation for the decay of the perturbation is derived. The equation involves the generalized shear viscosity of the gas in the time-dependent shear flow state, and therefore it predicts relevant rheological effects beyond the quasi-elastic limit. A good agreement is found when comparing the theory with molecular dynamics simulation results. Moreover, the Onsager postulate on the…
We present in this work results from numerical solutions, obtained by means of the direct simulation Monte Carlo (DSMC) method, of the Boltzmann and Boltzmann--Lorentz equations for an impurity immersed in a granular gas under planar Couette flow. The DSMC results are compared with the exact solution of a recent kinetic model for the same problem. The results confirm that, in steady states and over a wide range of parameter values, the state of the impurity is enslaved to that of the host gas: it follows the same flow velocity profile, its concentration (relative to that of the granular gas) i…
We perform large-scale event-driven Molecular dynamics (MD) simulations for granular gases of particles interacting with the impact-velocity dependent restitution coefficient. We use the simplest first-principle collision model of viscoelastic spheres. Both cases of force-free and uniformly heated gases are studied. We formulate a simplified model of an effective constant restitution coefficient, which depends on a current granular temperature and compute the effective constant restitution coefficient, using the kinetic theory. We develop a theory of the velocity distribution function for driv…
The time evolution and stationary values of the entropy per particle of a homogeneous freely cooling granular gas, relative to the maximum entropy consistent with the instantaneous translational and rotational temperatures, is analyzed by means of a Sonine approximation involving fourth-degree cumulants. The results show a rich variety of dependencies of the relative entropy on time and on the coefficients of normal and tangential restitution, including a peculiar behavior in the quasi-smooth limit.
Experiments quantifying the rotational and translational motion of particles in a dense, driven, 2D granular gas floating on an air table reveal that kinetic energy is divided equally between the two translational and one rotational degrees of freedom. This equipartition persists when the particle properties, confining pressure, packing density, or spatial ordering are changed. While the translational velocity distributions are the same for both large and small particles, the angular velocity distributions scale with the particle radius. The probability distributions of all particle velocities…
The Navier-Stokes transport coefficients of a granular dense fluid of smooth inelastic hard disks or spheres are explicitly determined by solving the inelastic Enskog equation by means of Grad's moment method. The transport coefficients are explicitly determined as functions of the (constant) coefficient of restitution and the solid volume fraction. In addition, the cooling rate is also calculated to first order in the spatial gradients. The calculations are performed for an arbitrary number of dimensions. The results are not limited to small dissipation and are expected to apply at moderate d…
We study the equilibrium behavior of one-dimensional granular clusters and one-particle granular gases for a variety of velocity dependent coefficients of restitution $r$. We obtain equations describing of the long time behavior for the cluster's pressure, r.m.s. velocity and granular inter-spacing. We show that for extremely long times, clusters with velocity dependent coefficients of restitution are unstable and dissolve into homogeneous, quasi-elastic gases, but clusters with velocity independent $r$ are permanent. This is in accordance with hydrodynamic studies pointing to the transient na…
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 $T_i$ (measuring the mean kinetic energy of each species) are differ…
We study the response of a granular system at rest to an instantaneous input of energy in a localised region. We present scaling arguments that show that, in $d$ dimensions, the radius of the resulting disturbance increases with time $t$ as $t^α$, and the energy decreases as $t^{-αd}$, where the exponent $α=1/(d+1)$ is independent of the coefficient of restitution. We support our arguments with an exact calculation in one dimension and event driven molecular dynamic simulations of hard sphere particles in two and three dimensions.
The velocity distribution function of granular gases in the homogeneous cooling state as well as some heated granular gases decays for large velocities as $f\propto\exp(- {\rm const.} v)$. That is, its high-energy tail is overpopulated as compared with the Maxwell distribution. At the present time, there is no theory to describe the influence of the tail on the kinetic characteristics of granular gases. We develop an approach to quantify the overpopulated tail and analyze its impact on granular gas properties, in particular on the cooling coefficient. We observe and explain anomalously slow re…
A force-free granular gas is considered with impact-velocity dependent coefficient of restitution as it follows from the model of viscoelastic particles. We analyze structure formation in this system by means of three independent methods: Molecular Dynamics, numerical solution of the hydrodynamic equations and linear stability analysis of these equations. All these approaches indicate that structure formation occurs in force-free granular gases only as a transient process.
We investigate the evolution of the velocity distribution function of a granular gas composed of viscoelastic particles in the homogeneous cooling state, i.e. before clustering occurs. The deviation of the velocity distribution function from the Maxwellian distribution is quantified by a Sonine polynomials expansion. The first non-vanishing Sonine coefficient a_2(t), reveals a complex time dependence which allows to assign the granular gas the property of an age. We discuss the possibility to measure the age of a granular gas.
The coefficient of self-diffusion for a homogeneously cooling granular gas changes significantly if the impact-velocity dependence of the restitution coefficient $ε$ is taken into account. For the case of a constant $ε$ the particles spread logarithmically slow with time, whereas the 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.
A gas of particles which collide inelastically if their impact velocity exceeds a certain value is investigated. In difference to common granular gases, cluster formation occurs only as a transient phenomenon. We calculate the decay of temperature due to inelastic collisions. In spite of the drastically reduced dissipation at low temperature the temperature surprisingly converges to zero.
The self-diffusion coefficient of a granular gas in the homogeneous cooling state is analyzed near the shearing instability. Using mode-coupling theory, it is shown that the coefficient diverges logarithmically as the instability is approached, due to the coupling of the diffusion process with the shear modes. The divergent behavior, which is peculiar of granular gases and disappears in the elastic limit, does not depend on any other transport coefficient. The theoretical prediction is confirmed by molecular dynamics simulation results for two-dimensional systems.
We study the dynamics of a granular gas heated by the stochastic thermostat. From a Boltzmann description, we derive the hydrodynamic equations for small perturbations around the stationary state that is reached in the long time limit. Transport coefficients are identified as Green-Kubo formulas obtaining explicit expressions as a function of the inelasticity and the spacial dimension.
The dissipation rate due to inelastic collisions between equally charged, insulating particles in a granular gas is calculated. It is equal to the known dissipation rate for uncharged granular media multiplied by a Boltzmann-like factor, that originates from Coulomb repulsion. Particle correlations lead to an effective potential that replaces the bare Coulomb potential in the Boltzmann factor. Collisional cooling in a granular gas proceeds with the known t^-2 -law, until the kinetic energy of the grains becomes smaller than the Coulomb barrier. Then the granular temperature approaches a time d…
We study analytically and by event-driven molecular dynamics simulations the nonergodic and aging properties of force-free cooling granular gases with both constant and velocity-dependent (viscoelastic) restitution coefficient $\varepsilon$ for particle pair collisions. We compare the granular gas dynamics with an effective single particle stochastic model based on an underdamped Langevin equation with time dependent diffusivity. We find that both models share the same behavior of the ensemble mean squared displacement (MSD) and the velocity correlations in the small dissipation limit. However…
Dissipation in granular media leads to interesting phenomena as there are cluster formation and crystallization in non-equilibrium dynamical states. The freely cooling system is examined concerning the energy decay and the cluster evolution in time, both in two and three dimensions. Interesting parallels to percolation theory are obtained in three dimensions.
Motivated by recent advances in the investigation of fluctuation-driven ratchets and flows in excited granular media, we have carried out experimental and simulational studies to explore the horizontal transport of granular particles in a vertically vibrated system whose base has a sawtooth-shaped profile. The resulting material flow exhibits novel collective behavior, both as a function of the number of layers of particles and the driving frequency; in particular, under certain conditions, increasing the layer thickness leads to a reversal of the current, while the onset of transport as a fun…
The granular gas is a paradigm for understanding the effects of inelastic interactions in granular materials. Kinetic theory provides a general theoretical framework for describing the granular gas. Its central result is that the tail of the velocity distribution of a driven granular gas is a stretched exponential that, counterintuitively, decays slower than that of the corresponding elastic gas in equilibrium. However, a derivation of this result starting from a microscopic model is lacking. Here, we obtain analytical results for a microscopic model for a granular gas where particles with two…
We study a two-dimensional granular gas of inelastic spheres subject to multiplicative driving proportional to a power $|v(\vec{x})|^δ$ of the local particle velocity $v(\vec{x})$. The steady state properties of the model are examined for different values of $δ$, and compared with the homogeneous case $δ=0$. A driving linearly proportional to $v(\vec{x})$ 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
Distribution of granular temperatures in granular gas mixtures is investigated analytically and numerically. We analyze space uniform systems in a homogeneous cooling state (HCS) and under a uniform heating with a mass-dependent heating rate $Γ_k\sim m_k^γ$. We demonstrate that for steep size distributions of particles the granular temperatures obey a universal power-law distribution, $T_k \sim m_k^α$, where the exponent $α$ does not depend on a particular form of the size distribution, the number of species and inelasticity of the grains. Moreover, $α$ is a universal constant for a HCS and de…
The transport coefficients for a gas of smooth, inelastic hard spheres are obtained from the Boltzmann equation in the form of Green-Kubo relations. The associated time correlation functions are not simply those constructed from the fluxes of conserved densities. Instead, fluxes constructed from the reference local homogeneous distribution occur as well. The analysis exposes potential problems associated with a straightforward extension of linear response methods to granular systems.
We demonstrate the existence, as well as determine the conditions, of a Mpemba effect - a counterintuitive phenomenon where a hotter system equilibrates faster than a cooler system when quenched to a cold temperature - in anisotropically driven granular gases. In contrast to earlier studies of Mpemba effect in granular systems, the initial states are stationary, making it a suitable system to experimentally study the effect. Our theoretical predictions for the regular, inverse and strong Mpemba effects agree well with results of event-driven molecular dynamics simulations of hard discs.
We investigate the motion of a wedge-shaped object (a granular Brownian motor), which is restricted to move along the x-axis and cannot rotate, as gas particles collide with it. We show that its steady-state drift, resulting from inelastic gas-motor collisions, is dramatically affected by anisotropy in the velocity distribution of the gas. We identify the dimensionless parameter providing the dependence of this drift on shape, masses, inelasticity, and anisotropy: the anisotropy leads to a dramatic breaking of equipartition, which should easily be visible in experimental realizations.
Small perturbations of the homogeneous cooling state (HCS) for a low density granular gas are described by means of the linearized Boltzmann equation. The spectrum of the generator for this dynamics is shown to contain points corresponding to hydrodynamic excitations. The corresponding eigenvectors and eigenvalues are calculated to Navier-Stokes order and shown to agree with those obtained by the Chapman-Enskog method. The conditions for the hydrodynamic excitations to dominate all other excitations are discussed.
We investigate the validity of fluctuation theorems for an asymmetric rotor experiment in a granular gas. A first state, with a Gaussian distribution of the angular velocity, is found to be well described by a first order Langevin equation. We show that fluctuation theorems are valid for the injected work and for the total entropy production. In a second state the angular velocity distribution is double-peaked due to a spontaneous symmetry breaking: A convection roll develops in the granular gas, which strongly couples to the rotor. Surprisingly, in this case similar symmetry relations hold, w…
Equations of granular hydrostatics are used to compute the phase diagram of the recently discovered van der Waals-like phase separation in a driven granular gas. The model two-dimensional system consists of smooth hard disks in a rectangular box, colliding inelastically with each other and driven by a "thermal" wall at zero gravity. The spinodal line and the critical point of the phase separation are determined. Close to the critical point the spinodal and binodal (coexistence) lines are determined analytically. Effects of finite size of the confining box in the direction parallel to the therm…
The behavior of an isolated dilute granular gas near the threshold of its clustering instability is investigated by means of fluctuating hydrodynamics and the direct simulation Monte Carlo method. The theoretical predictions from the former are shown to be in good agreement with the simulation results. The total energy of the the system is renormalized by the fluctuations of the vorticity field. Moreover, the scaled second moment of the energy fluctuations exhibits a power-law divergent behavior
The steady state velocity fluctuations of a movable piston located on the top of a vibrated granular gas are studied by means of molecular dynamics simulations. From the second moment of the distribution, a temperature parameter for the piston is defined and compared with the granular temperature of the gas located just below it. Then the two temperature parameters refer to two interacting macroscopic systems. The equipartition of energy valid in usual molecular systems is strongly violated, and the temperature of the piston can be larger or smaller than the one of the gas, depending of the pa…
A hierarchy of equations was introduced recently to describe the kinetics of homogeneous cooling of the gas of inelastic hard spheres. It is argued that this hierarchy does not describe this system accurately, and a simple test for this is proposed. A stochastic interpretation of the hierarchy is presented.
An experimental study is presented, about transitions between Non-Equilibrium Steady States (NESS) in a dissipative medium. The core device is a small rotating blade that imposes cycles of increasing and decreasing forcings to a granular gas, shaken independently. The velocity of this blade is measured, subject to the transitions imposed by the periodic torque variation. The Hatano-Sasa equality, that generalises the second principle of thermodynamics to NESS, is verified with a high accuracy (a few $10^{-3}$), at different variation rates. Besides, it is observed that the fluctuating velocity…
The stochastic properties of the total internal energy of a dilute granular gas in the steady uniform shear flow state are investigated. A recent theory formulated for fluctuations about the homogeneous cooling state is extended by analogy with molecular systems. The theoretical predictions are compared with molecular dynamics simulation results. Good agreement is found in the limit of weak inelasticity, while systematic and relevant discrepancies are observed when the inelasticity increases. The origin of this behavior is discussed.
The global energy fluctuations of a low density gas granular gas in the homogeneous cooling state near its clustering instability are studied by means of molecular dynamics simulations. The relative dispersion of the fluctuations is shown to exhibit a power-law divergent behavior. Moreover, the probability distribution of the fluctuations presents data collapse as the system approaches the instability, for different values of the inelasticity. The function describing the collapse turns out to be the same as the one found in several molecular equilibrium and non-equilibrium systems, except for …
The linear stability of granular gas that reflects the contribution of self-gravitational force of mass density perturbations is investigated in order to clarify the condition of competition between clustering instability and Jeans instability. It is found that the condition depends on three parameters: the mass density $ρ_0$, the collision rate $ω_0$, and the rate of energy loss per collision $ε$. When $\sqrt{Gρ_0}\llεω_0$, clustering instability dominates, while when $εω_0\ll\sqrt{Gρ_0}$, Jeans instability dominates. These instabilities are characterized by the decrease and increase, respect…
