Anomalous self-diffusion in a freely evolving granular gas near the shearing instability.

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 in 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.

Power-law decay of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state.

The hydrodynamic part of the velocity autocorrelation function of a granular fluid in the homogeneous cooling state has been calculated by using mode-coupling theory for a finite system with periodic boundary conditions. The existence of the shearing instability, leading to a divergent behavior of the velocity flow fluctuations, is taken into account. A time region in which the velocity autocorrelation function exhibits a power-law decay, when time is measured by the number of collisions per particle, has been been identified. Also the explicit form of the exponential asymptotic long time decay has been obtained. The theoretical prediction for the power-law decay is compared with molecular dynamics simulation results, and a good agreement is found, after taking into account finite size corrections. The effects of approaching the shearing instability are also explored.

Nonequilibrium solutions of the Boltzmann equation under the action of an external force.

We construct a novel class of exact solutions to the Boltzmann equation, in both its classical and quantum formulation, for arbitrary collision laws. When the system is subjected to a specific external forcing, the precise form of which is worked out, nonequilibrium dampingless solutions are admissible. They do not contradict the H theorem, but are constructed from its requirements. Interestingly, these solutions hold for time-dependent confinement. We exploit them, in a reverse-engineering perspective, to work out a protocol that shortcuts any adiabatic transformation between two equilibrium states in an arbitrarily short time span, for an interacting system. Particle simulations of the direct Monte Carlo type fully corroborate the analytical predictions.

Shearing instability of a dilute granular mixture.

The shearing instability of a dilute granular mixture composed of smooth inelastic hard spheres or disks is investigated. By using the Navier-Stokes hydrodynamic equations, it is shown that the scaled transversal velocity mode exhibits a divergent behavior, similarly to what happens in one-component systems. The theoretical prediction for the critical size is compared with direct Monte Carlo simulations of the Boltzmann equations describing the system, and a good agreement is found. The total energy fluctuations in the vicinity of the transition are shown to scale with the second moment of the distribution. The scaling distribution function is the same as found in other equilibrium and nonequilibrium phase transitions, suggesting the existence of some kind of universality.

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.

Volume fluctuations and compressibility of a vibrated granular gas.

The volume fluctuations in the steady state reached by a vibrated granular gas of hard particles confined by a movable piston on the top are investigated by means of event-driven simulations. Also, a compressibility factor, measuring the response in volume of the system to a change in the mass of the piston, is introduced and measured. From the second moment of the volume fluctuations and the compressibility factor, an effective temperature is defined by using the same relation as obeyed by equilibrium molecular systems. The interpretation of this effective temperature and its relationship with the granular temperature of the gas, and also with the velocity fluctuations of the movable piston, is discussed. It is found that the ratio of the temperature based on the volume fluctuations to the temperature based on the piston kinetic energy obeys simple dependencies on the inelasticity and on the piston-particle mass ratio.

Hydrodynamic character of the nonequipartition of kinetic energy in binary granular gases.

The influence of the heating mechanism on the kinetic energy densities of the components of a vibrated granular mixture is investigated. Collisions of the particles with the vibrating wall are inelastic and characterized by two coefficients of normal restitution, one for each of the two species. By means of molecular-dynamics simulations, it is shown that the nonequipartition of kinetic energy is not affected by the differential mechanism of energy injection aside the usual boundary layer around the wall. The macroscopic state of the mixture in the bulk is defined by intensive variables that do not include the partial granular temperatures of the components.

Vibrated granular gas confined by a piston.

The steady state of a vibrated granular gas confined by a movable piston on top is discussed. Particular attention is given to the hydrodynamic boundary conditions to be used when solving the inelastic Navier-Stokes equations. The relevance of an exact general condition relating the grain fluxes approaching and moving away from each of the walls is emphasized. It is shown how it can be used to get a consistent hydrodynamic description of the boundaries. The obtained expressions for the fields do not contain any undetermined parameter. Comparison of the theoretical predictions with molecular-dynamics simulation results is carried out, and a good agreement is observed for low density and not too large inelasticity. A practical way of introducing small finite density corrections to the dilute limit theory is proposed to improve the accuracy of the theory.

Shear state of freely evolving granular gases.

Hydrodynamic equations are used to identify the final state reached by a freely evolving granular gas above but close to its shear instability. The theory predicts the formation of a two bands shear state with a steady density profile. There is a modulation between temperature and density profiles as a consequence of the energy balance, the density fluctuations remaining small, without producing clustering. Moreover, the time dependence of the velocity field can be scaled out with the squared root of the average temperature of the system. The latter follows the Haff law, but with an effective cooling rate that is smaller than that of the free homogeneous state. The theoretical predictions are compared with numerical results for inelastic hard disks obtained by using the direct Monte Carlo simulation method, and a good agreement is obtained for low inelasticity.

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.

Energy partition and segregation for an intruder in a vibrated granular system under gravity.

The difference of temperatures between an impurity and the surrounding gas in an open vibrated granular system is studied. It is shown that, in spite of the high inhomogeneity of the state, the temperature ratio remains constant in the bulk of the system. The lack of energy equipartition is associated to the change of sign of the pressure diffusion coefficient for the impurity at certain values of the parameters of the system, leading to a segregation criterium. The theoretical predictions are consistent with previous experimental results, and also in agreement with molecular dynamics simulation results reported in this Letter.

Scaling and universality of critical fluctuations in granular gases.

The total energy fluctuations of a low-density granular gas in the homogeneous cooling state near the threshold of the 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 symmetric of the one found in several molecular equilibrium and nonequilibrium systems.

Simulation study of the Green-Kubo relations for dilute granular gases.

The Green-Kubo relations for dilute granular gases are employed to compute their transport coefficients by means of the direct simulation Monte Carlo method. This requires not only to follow the dynamics of the system, but also to identify some modified fluxes appearing in the time-correlation functions. The results are compared with those obtained from the Boltzmann equation by means of the Chapman-Enskog procedure in the first Sonine approximation. A good agreement is found for the shear viscosity over a wide range of inelasticities. Nevertheless, for the two transport coefficients associated with the heat flux, significant discrepancies appear for strong inelasticity. Their origin is discussed, showing that they are partially due to the presence of velocity correlations in the homogeneous cooling state of a dilute granular fluid.

Energy fluctuations in the homogeneous cooling state of granular gases.

Starting from the hierarchy of equations for microscopic densities in phase space, a general theory for fluctuations and correlations in a dilute granular gas of hard particles is developed. Then, the particular case of the homogeneous cooling state is addressed. Explicit expressions for some distributions describing the presence of velocity correlations and their dynamics are obtained. These correlations are inherent to the dissipative dynamics of the collisions. The implications for the behavior of the total energy of the system are analyzed and the results are expressed in terms of a fluctuation-dissipation theorem. The theoretical predictions are shown to be in agreement with results obtained by molecular dynamics simulations, which also indicate that energy fluctuations are well described by a Gaussian distribution.

Steady-state representation of the homogeneous cooling state of a granular gas.

The properties of a dilute granular gas in the homogeneous cooling state are mapped to those of a stationary state by means of a change in the time scale that does not involve any internal property of the system. The new representation is closely related with a general property of the granular temperature in the long time limit. The physical and practical implications of the mapping are discussed. In particular, simulation results obtained by the direct simulation Monte Carlo method applied to the scaled dynamics are reported. This includes ensemble averages and also the velocity autocorrelation function, as well as the self-diffusion coefficient obtained from the latter by means of the Green-Kubo representation. In all cases, the obtained results are compared with theoretical predictions.

Validity of the Boltzmann equation to describe low-density granular systems.

The departure of a granular gas in the instable region of parameters from the initial homogeneous cooling state is studied. Results from molecular dynamics and from direct Monte Carlo simulation of the Boltzmann equation are compared. The results indicate that the Boltzmann equation accurately predicts the low-density limit of the system. The relevant role played by the parallelization of the velocities as time proceeds and the dependence of this effect on the density is analyzed in detail.

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.

Transversal inhomogeneities in dilute vibrofluidized granular fluids.

The spontaneous symmetry breaking taking place in the direction perpendicular to the energy flux in a dilute vibrofluidized granular system is investigated, using both a hydrodynamic description and simulation methods. The latter include molecular dynamics and direct Monte Carlo simulation of the Boltzmann equation. A marginal stability analysis of the hydrodynamic equations, carried out in the WKB approximation, is shown to be in good agreement with the simulation results. The shape of the hydrodynamic profiles beyond the bifurcation is discussed.

Hydrodynamic Maxwell demon in granular systems.

Spontaneous symmetry breaking in a vibrated system confined into two connected compartments in the absence of external fields is reported. For a small number of particles, the grains are equipartitioned, but if it is increased beyond a critical value, the number of particles in each of the compartments becomes different in the steady state, and the number of particles in one of the compartments decreases monotonically tending to a given value. This phase transition is accurately described by the hydrodynamic equations for a granular gas. The relationship with previous phenomena of phase separation in vibrofluidized granular materials is discussed.

Hydrodynamics of an open vibrated granular system.

Using the hydrodynamic description and molecular dynamics simulations, the steady state of a fluidized granular system in the presence of gravity is studied. For an open system, the density profile exhibits a maximum, while the temperature profile goes through a minimum at high altitude, beyond that the temperature increases with the height. The existence of the minimum is explained by the hydrodynamic equations if the presence of a collisionless boundary layer is taken into account. The energy dissipated by interparticle collisions is also computed. A good agreement is found between theory and simulation. The relationship with previous works is discussed.