Kinetics of inhomogeneous cooling in granular fluids

Intruder dynamics in a frictional granular fluid: A molecular dynamics study

We study the dynamics of an intruder moving through a fluidized granular medium in three dimensions (d=3). The intruder and grains have both translational and rotational degrees of freedom. The energy-dissipation mechanism is solid friction between all pairs of particles. We keep the granular system fluidized even at rather high densities by randomly perturbing the linear and angular velocities of the grains. We apply a constant external force of magnitude F to the intruder and obtain its steady-state velocity V_{s} in the center-of-mass frame of the grains. The F-V_{s} relation is of great interest in the industrial processing of granular matter and has been the subject of most experiments on this problem. We also obtain the mobility, which is proportional to the inverse viscosity, as a function of the volume fraction ϕ. This is shown to diverge at the jamming volume fraction. For ϕ below the jamming fraction, we find that V_{s}∼F for small F and V_{s}∼F^{1/2} for large F. The intruder shows diffusive motion in the plane perpendicular to the direction of the external force.

Size-polydisperse dust in molecular gas: Energy equipartition versus nonequipartition

We investigate numerically and analytically size-polydisperse granular mixtures immersed into a molecular gas. We show that the equipartition of granular temperatures of particles of different sizes is established; however, the granular temperatures significantly differ from the temperature of the molecular gas. This result is surprising since, generally, the energy equipartition is strongly violated in driven granular mixtures. Qualitatively, the obtained results do not depend on the collision model, being valid for a constant restitution coefficient ɛ, as well as for the ɛ for viscoelastic particles. Our findings may be important for astrophysical applications, such as protoplanetary disks, interstellar dust clouds, and comets.

Coarsening dynamics in the Vicsek model of active matter

We study the flocking model introduced by Vicsek et al. (Phys. Rev. Lett. 75, 1226 (1995)) in the "coarsening" regime. At standard self-propulsion speeds, we find two distinct growth laws for the coupled density and velocity fields. The characteristic length scale of the density domains grows as [Formula: see text] (with [Formula: see text] , while the velocity length scale grows much faster, viz., [Formula: see text] (with [Formula: see text] . The spatial fluctuations in the density and velocity fields are studied by calculating the two-point correlation function and the structure factor, which show deviations from the well-known Porod's law. This is a natural consequence of scattering from irregular morphologies that dynamically arise in the system. At large values of the scaled wave vector, the scaled structure factors for the density and velocity fields decay with powers -2.6 and -1.52 , respectively.

Temperature distribution in driven granular mixtures does not depend on mechanism of energy dissipation

We study analytically and numerically the distribution of granular temperatures in granular mixtures for different dissipation mechanisms of inelastic inter-particle collisions. Both driven and force-free systems are analyzed. We demonstrate that the simplified model of a constant restitution coefficient fails to predict even qualitatively a granular temperature distribution in a homogeneous cooling state. At the same time we reveal for driven systems a stunning result - the distribution of temperatures in granular mixtures is universal. That is, it does not depend on a particular dissipation mechanism of inter-particles collisions, provided the size distributions of particles is steep enough. The results of the analytic theory are compared with simulation results obtained by the direct simulation Monte Carlo (DSMC). The agreement between the theory and simulations is perfect. The reported results may have important consequences for fundamental science as well as for numerous application, e.g. for the experimental modelling in a lab of natural processes.

Mixing rate in Classical Many Body Systems

Mixing in many body systems is intuitively understood as the change in time of the set of neighbors surrounding each particle. Its rate and its development over time hold important clues to the behavior of many body systems. For example, gas particles constantly change their position and surrounding particles, while in solids one expects the motion of the atoms to be limited by a fixed set of neighboring atoms. In other systems the situation is less clear. For example, agitated granular systems may behave like a fluid, a solid or glass, depending on various parameter such as density and friction. Thus, we introduce a parameter which describes the mixing rate in many body systems in terms of changes of a properly chosen adjacency matrix. The parameter is easily measurable in simulations but not in experiment. To demonstrate an application of the concept, we simulate a many body system, with particles interacting via a two-body potential and calculate the mixing rate as a function of time and volume fraction. The time dependence of the mixing rate clearly indicates the onset of crystallization.

Pattern, growth, and aging in aggregation kinetics of a Vicsek-like active matter model

Via molecular dynamics simulations, we study kinetics in a Vicsek-like phase-separating active matter model. Quantitative results, for isotropic bicontinuous pattern, are presented on the structure, growth, and aging. These are obtained via the two-point equal-time density-density correlation function, the average domain length, and the two-time density autocorrelation function. Both the correlation functions exhibit basic scaling properties, implying self-similarity in the pattern dynamics, for which the average domain size exhibits a power-law growth in time. The equal-time correlation has a short distance behavior that provides reasonable agreement between the corresponding structure factor tail and the Porod law. The autocorrelation decay is a power-law in the average domain size. Apart from these basic similarities, the overall quantitative behavior of the above-mentioned observables is found to be vastly different from those of the corresponding passive limit of the model which also undergoes phase separation. The functional forms of these have been quantified. An exceptionally rapid growth in the active system occurs due to fast coherent motion of the particles, mean-squared-displacements of which exhibit multiple scaling regimes, including a long time ballistic one.

Dimension dependence of clustering dynamics in models of ballistic aggregation and freely cooling granular gas

Via event-driven molecular dynamics simulations we study kinetics of clustering in assemblies of inelastic particles in various space dimensions. We consider two models, viz., the ballistic aggregation model (BAM) and the freely cooling granular gas model (GGM), for each of which we quantify the time dependence of kinetic energy and average mass of clusters (that form due to inelastic collisions). These quantities, for both the models, exhibit power-law behavior, at least in the long time limit. For the BAM, corresponding exponents exhibit strong dimension dependence and follow a hyperscaling relation. In addition, in the high packing fraction limit the behavior of these quantities become consistent with a scaling theory that predicts an inverse relation between energy and mass. On the other hand, in the case of the GGM we do not find any evidence for such a picture. In this case, even though the energy decay, irrespective of packing fraction, matches quantitatively with that for the high packing fraction picture of the BAM, it is inversely proportional to the growth of mass only in one dimension, and the growth appears to be rather insensitive to the choice of the dimension, unlike the BAM.

Ballistic aggregation in systems of inelastic particles: Cluster growth, structure, and aging

We study far-from-equilibrium dynamics in models of freely cooling granular gas and ballistically aggregating compact clusters. For both the cases, from event-driven molecular dynamics simulations, we have presented detailed results on structure and dynamics in space dimensions d=1 and 2. Via appropriate analyses it has been confirmed that the ballistic aggregation mechanism applies in d=1 granular gases as well. Aging phenomena for this mechanism, in both the dimensions, have been studied via the two-time density autocorrelation function. This quantity is demonstrated to exhibit scaling property similar to that in the standard phase transition kinetics. The corresponding functional forms have been quantified and the outcomes have been discussed in connection with the structural properties. Our results on aging establish a more complete equivalence between the granular gas and the ballistic aggregation models in d=1.

Clustering and velocity distributions in granular gases cooling by solid friction

We present large-scale molecular dynamics simulations to study the free evolution of granular gases. Initially, the density of particles is homogeneous and the velocity follows a Maxwell-Boltzmann (MB) distribution. The system cools down due to solid friction between the granular particles. The density remains homogeneous, and the velocity distribution remains MB at early times, while the kinetic energy of the system decays with time. However, fluctuations in the density and velocity fields grow, and the system evolves via formation of clusters in the density field and the local ordering of velocity field, consistent with the onset of plug flow. This is accompanied by a transition of the velocity distribution function from MB to non-MB behavior. We used equal-time correlation functions and structure factors of the density and velocity fields to study the morphology of clustering. From the correlation functions, we obtain the cluster size, L, as a function of time, t. We show that it exhibits power law growth with L(t)∼t^{1/3}.

Energy decay in three-dimensional freely cooling granular gas

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≤d≤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 gas fits to neither the ballistic aggregation or a Burgers equation description.

Velocity distribution function and effective restitution coefficient for a granular gas of viscoelastic particles

We perform large-scale event-driven molecular dynamics (MD) simulations for granular gases of particles interacting with the impact-velocity-dependent restitution coefficient ε(v(imp)). We use ε(v(imp)) as it follows from the simplest first-principles 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 ε(eff), which depends on a current granular temperature, and we compute ε(eff) using the kinetic theory. We develop a theory of the velocity distribution function for driven gases of viscoelastic particles and analyze the evolution of granular temperature and of the Sonine coefficients, which characterize the form of the velocity distribution function. We observe that for not large dissipation the simulation results are in an excellent agreement with the theory for both the homogeneous cooling state and uniformly heated gases. At the same time, a noticeable discrepancy between the theory and MD results for the Sonine coefficients is detected for large dissipation. We analyze the accuracy of the simplified model based on the effective restitution coefficient ε(eff), and we conclude that this model can accurately describe granular temperature. It provides also an acceptable accuracy for the velocity distribution function for small dissipation, but it fails when dissipation is large.

Intermediate regimes in granular Brownian motion: superdiffusion and subdiffusion

Brownian motion in a granular gas in a homogeneous cooling state is studied theoretically and by means of molecular dynamics. We use the simplest first-principles model for the impact-velocity dependent restitution coefficient, as it follows for the model of viscoelastic spheres. We reveal that for a wide range of initial conditions the ratio of granular temperatures of Brownian and bath particles demonstrates complicated nonmonotonic behavior, which results in a transition between different regimes of Brownian dynamics: It starts from the ballistic motion, switches later to a superballistic one, and turns at still later times into subdiffusion; eventually normal diffusion is achieved. Our theory agrees very well with the molecular dynamics results, although extreme computational costs prevented us from detecting the final diffusion regime. Qualitatively, the reported intermediate diffusion regimes are generic for granular gases with any realistic dependence of the restitution coefficient on the impact velocity.

Coarse-grained dynamics of the freely cooling granular gas in one dimension

We study the dynamics and structure of clusters in the inhomogeneous clustered regime of a freely cooling granular gas of point particles in one dimension. The coefficient of restitution is modeled as r(0)<1 or 1, depending on whether the relative speed is greater or smaller than a velocity scale δ. The effective fragmentation rate of a cluster is shown to rise sharply beyond a δ-dependent time scale. This crossover is coincident with the velocity fluctuations within a cluster becoming order δ. Beyond this crossover time, the cluster-size distribution develops a nontrivial power-law distribution, whose scaling properties are related to those of the velocity fluctuations. We argue that these underlying features are responsible for the recently observed nontrivial coarsening behavior in the one-dimensional freely cooling granular gas.

da Vinci fluids, catch-up dynamics and dense granular flow

We introduce and study a da Vinci fluid, a fluid whose dissipation is dominated by solid friction. We analyse the flow rheology of a discrete model and then coarse-grain it to the continuum. We find that the model gives rise to behaviour that is characteristic of dense granular fluids. In particular, it leads to plug flow. We analyse the nucleation mechanism of plugs and their development. We find that plug boundaries generically expand and we calculate the growth rate of plug regions. In systems whose internal effective dynamic and static friction coefficients are relatively uniform we find that the linear size of plug regions grows as (time)¹(/)³ . The suitability of the model to granular materials is discussed.

Critical scaling and aging in cooling systems near the jamming transition

We conduct athermal simulations of freely cooling, viscous soft spheres around the jamming transition density varphi(J) and find evidence for a growing length xi(t) that governs relaxation to mechanical equilibrium. xi(t) is manifest in both the velocity correlation function and the spatial correlations in a scalar measure of local force balance which we define. Data for different densities varphi can be collapsed onto two master curves by scaling xi(t) and t by powers of |varphi-varphi(J)|, indicative of critical scaling. Furthermore, particle transport for varphi>varphi(J) exhibits aging and superdiffusion similar to a range of soft matter experiments, suggesting a common origin. Finally, we explain how xi(t) at late times maps onto known behavior away from varphi(J).

Equivalence of the freely cooling granular gas to the sticky gas

A freely cooling granular gas with a 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 delta . 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 find that in the inhomogeneous cooling regime, for times smaller than a crossover time t{1} , where t{1} approximately delta;{-1} , the behavior of the granular gas is equivalent to that of the sticky gas. When delta-->0 , then t{1}-->infinity and, hence, the results support an earlier claim that the freely cooling inelastic gas is described by the inviscid Burgers equation. For a real granular gas with finite delta , the existence of the time scale t{1} shows that, for large times, the granular gas is not described by the inviscid Burgers equation.

Violation of the Porod law in a freely cooling granular gas in one dimension

We study a model of freely cooling inelastic granular gas in one dimension, with a restitution coefficient which approaches the elastic limit below a relative velocity scale delta. While at early times (t<<delta;{-1}) the gas behaves as a completely inelastic sticky gas conforming to predictions of earlier studies, at late times (t>>delta;{-1}) it exhibits a new fluctuation-dominated phase ordering state. We find distinct scaling behavior for the (i) density distribution function, (ii) occupied and empty gap distribution functions, (iii) the density structure function, and (iv) the velocity structure function, as compared to the completely inelastic sticky gas. The spatial structure functions (iii) and (iv) violate the Porod law. Within a mean-field approximation, the exponents describing the structure functions are related to those describing the spatial gap distribution functions.

Velocity distributions and aging in a cooling granular gas

We use large-scale molecular dynamics simulations to study freely evolving granular gases with dimensionality d=2,3 . The system dissipates kinetic energy (or cools) due to inelastic collisions between granular particles. The density and velocity fields are approximately homogeneous at early times, and the system is said to be in a homogeneous cooling state (HCS). However, fluctuations in the density and velocity fields grow, and the system evolves into an inhomogeneous cooling state (ICS). We study the nature of velocity distributions in both the HCS and ICS. We also investigate the aging property of the velocity autocorrelation function.

Active nematics are intrinsically phase separated

Two-dimensional nonequilibrium nematic steady states, as found in agitated granular-rod monolayers or films of orientable amoeboid cells, were predicted [Europhys. Lett. 62, 196 (2003)10.1209/epl/i2003-00346-7] to have giant number fluctuations, with the standard deviation proportional to the mean. We show numerically that the steady state of such systems is macroscopically phase separated, yet dominated by fluctuations, as in the Das-Barma model [Phys. Rev. Lett. 85, 1602 (2000)10.1103/PhysRevLett.85.1602]. We suggest experimental tests of our findings in granular and living-cell systems.