Inelastic Gases Research Articles

Hydrodynamic Singularities and Clustering in a Freely Cooling Inelastic Gas

We employ hydrodynamic equations to follow the clustering instability of a freely cooling dilute gas of inelastically colliding spheres into a well-developed nonlinear regime. We simplify the problem by dealing with a one-dimensional coarse-grained flow. We observe that at a late stage of the instability the shear stress becomes negligibly small, and the gas flows solely by inertia. As a result the flow formally develops a finite-time singularity, as the velocity gradient and the gas density diverge at some location. We argue that flow by inertia represents a generic intermediate asymptotic of unstable free cooling of dilute inelastic gases.

Application of Simple Models to the Study of Nonequilibrium Behaviour of Inelastic Gases

The dynamics of cooling inelastic gases and the evolution of their velocity fields is addressed by studying a series of simplified models. We first discuss a model of inelastic hard particles, which shows an uncorrelated transient phase (homogeneous cooling state or Haff regime) followed by the emergence of structures in the velocity and density field. Motivated by the linear stability analysis of the Haff regime, which predicts the appearance of density clusters only after the formation of structures in the velocity field, we focus our attention on the velocities of the gas particles. We study the so-called inelastic Maxwell model (IMM), first in a version with infinite connectivity which is the analog of mean-field spin systems. Secondly, we embed the model onto a lattice (in one dimension and two dimension), in order to observe spatial correlations. The mean-field IMM has the advantage that it lends itself to analytical treatment: in one dimension we find an exact asymptotic scaling solution for the probability density function (p.d.f.) of velocities. On the other hand, the lattice version displays typical spatial features of an inelastic gas, e.g., the transient Haff regime followed by the coarsening of structures in the velocity field (shocks in one dimension, vortices and shocks in two dimension), the so-called “return to the Gaussian” phenomenon of the velocity p.d.f. observed in MD simulations, etc. We show that the growth of structures in the lattice model is similar to that of domains in a diffusive field, but presents a short-scale disorder (“internal noise”) which is induced by the randomizing effect of collisions. In the lattice model, we can also establish the presence or absence of a mesoscopic scale which is required for a hydrodynamics description of the field evolution.

Velocity distributions in dissipative granular gases.

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 eta 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>>1 and q less, similar 1, respectively.

Inelastically scattering particles and wealth distribution in an open economy.

Using the analogy with inelastic granular gases we introduce a model for wealth exchange in society. The dynamics is governed by a kinetic equation, which allows for self-similar solutions. The scaling function has a power-law tail, the exponent being given by a transcendental equation. In the limit of continuous trading, a closed form of the wealth distribution is calculated analytically.

Velocity fluctuations in electrostatically driven granular media.

We study experimentally the particle velocity fluctuations in an electrostatically driven dilute granular gas. The velocity distributions have strong deviations from a Maxwellian form over a wide range of parameters. We have found that the tails of the distribution functions are consistent with a stretched exponential law with typical exponents of the order 3/2. Molecular dynamic simulations shows qualitative agreement with experimental data. Our results suggest that this non-Gaussian behavior is typical of most inelastic gases with both short- and long-range interactions.

Molecular dynamics simulations of vibrated granular gases.

We present molecular dynamics simulations of monodisperse or bidisperse inelastic granular gases driven by vibrating walls, in two dimensions (without gravity). Because of the energy injection at the boundaries, a situation often met experimentally, density and temperature fields display heterogeneous profiles in the direction perpendicular to the walls. A general equation of state for an arbitrary mixture of fluidized inelastic hard spheres is derived and successfully tested against numerical data. Single-particle velocity distribution functions with non-Gaussian features are also obtained, and the influence of various parameters (inelasticity coefficients, density, etc.) are analyzed. The validity of a recently proposed random restitution coefficient model is assessed through the study of projected collisions onto the direction perpendicular to that of energy injection. For the binary mixture, the nonequipartition of translational kinetic energy is studied and compared both to experimental data and to the case of homogeneous energy injection ("stochastic thermostat"). The rescaled velocity distribution functions are found to be very similar for both species.

Scaling, multiscaling, and nontrivial exponents in inelastic collision processes.

We investigate velocity statistics of homogeneous inelastic gases using the Boltzmann equation. Employing an approximate uniform collision rate, we obtain analytic results valid in arbitrary dimension. In the freely evolving case, the velocity distribution is characterized by an algebraic large-velocity tail, P(v,t) approximately v(-sigma). The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension d and the dissipation coefficient epsilon. Although the velocity distribution follows a scaling form, its moments exhibit multiscaling asymptotic behavior. Furthermore, the velocity autocorrelation function decays algebraically with time, A(t)=<v(0).v(t)> approximately t(-alpha), with a nonuniversal dissipation-dependent exponent alpha=1/epsilon. In the forced case, the steady state Fourier transform is obtained via a cumulant expansion. Even in this case, velocity correlations develop and the velocity distribution is non-Maxwellian.

Nontrivial velocity distributions in inelastic gases

We study freely evolving and forced inelastic gases using the Boltzmann equation. We consider uniform collision rates and obtain analytical results valid for arbitrary spatial dimension d and arbitrary dissipation coefficient epsilon. In the freely evolving case, we find that the velocity distribution decays algebraically, P(v,t) ~ v^{-sigma} for sufficiently large velocities. We derive the exponent sigma(d,epsilon), which exhibits nontrivial dependence on both d and epsilon, exactly. In the forced case, the velocity distribution approaches a steady-state with a Gaussian large velocity tail.

Shocklike Dynamics of Inelastic Gases

We provide a simple physical picture which suggests that the asymptotic dynamics of inelastic gases in one dimension is independent of the degree of inelasticity. Statistical characteristics, including velocity fluctuations and the velocity distribution, are identical to those of a perfectly inelastic sticky gas, which in turn is described by the inviscid Burgers equation. Asymptotic predictions of this continuum theory, including the ${t}^{\ensuremath{-}2/3}$ temperature decay and the development of discontinuities in the velocity profile, are verified numerically for inelastic gases.

Inelastic granular gas: Computer simulations and kinetic theory of the cooling state

We developed computer simulations of an inelastic granular gas which show that the energy decay proposed by Goldhirsch and Zanetti [Phys. Rev. Lett. 70, 1619 (1993)] has a limited validity. Moreover, we give an exact solution of the Liouville equation for the moments of the two-body homogeneous cooling distribution. The latter includes velocity correlations which raises questions about the derivation of kinetic equations for inelastic gases.

Towards granular hydrodynamics in two dimensions

We study steady-state properties of inelastic gases in two-dimensions in the presence of an energy source. We generalize previous hydrodynamic treatments to situations where high and low density regions coexist. The theoretical predictions compare well with numerical simulations in the nearly elastic limit. It is also seen that the system can achieve a nonequilibrium steady-state with asymmetric velocity distributions, and we discuss the conditions under which such situations occur.