Stationary states and energy cascades in inelastic gases

We report a general class of steady and transient states of granular gases. We find that the kinetic theory of inelastic gases admits stationary solutions with a power-law velocity distribution, f(v) ~ v^(-sigma). The exponent sigma is found analytically and depends on the spatial dimension, the degree of inelasticity, and the homogeneity degree of the collision rate. Driven steady-states, with the same power-law tail and a cut-off can be maintained by injecting energy at a l...

We study inelastic gases in two dimensions using event-driven molecular dynamics simulations. Our focus is the nature of the stationary state attained by rare injection of large amounts of energy to balance the dissipation due to collisions. We find that under such extreme driving, with the injection rate much smaller than the collision rate, the velocity distribution has a power-law high energy tail. The numerically measured exponent characterizing this tail is in excellent ...

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) ~ v^{-sigma}. The exponent sigma(d,epsilon), a nontrivial root of an integral equation, varies continuously with the spatial dimension, d, and the dissipation coe...

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,...

This review is a kinetic theory study investigating the effects of inelasticity on the structure of the non-equilibrium states, in particular on the behavior of the velocity distribution in the high energy tails. Starting point is the nonlinear Boltzmann equation for spatially homogeneous systems, which supposedly describes the behavior of the velocity distribution function in dissipative systems as long as the system remains in the homogeneous cooling state, i.e. on relative...

Within the framework of the homogeneous non-linear Boltzmann equation, we present a new analytic method, without the intrinsic limitations of existing methods, for obtaining asymptotic solutions. This method permits extension of existing results for Maxwell molecules and hard spheres to large classes of particle interactions, from very hard spheres to softer than Maxwell molecules, as well as to more general forcing mechanisms, beyond free cooling and white noise driving. By ...

We use the framework of sample space reducing processes (SSR) as an alternative to Boltzmann equation based approaches, to derive the energy and velocity distribution functions of an inelastic gas in a box as an example for a dissipative, driven system. SSR processes do not assume molecular chaos and are characterized by a specific type of eigenvalue equation whose solutions represent stationary distribution functions. The equations incorporate the geometry of inelastic colli...

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...

The velocity distribution in a homogeneously cooling granular gas has been studied in the viscoelastic regime when the restitution coefficient of colliding particles depends on the impact velocity. We show that for viscoelastic particles the simple scaling hypothesis is violated, i.e., that the time dependence of the velocity distribution does not scale with the mean square velocity as in the case of particles interacting via a constant restitution coefficient. The deviation ...

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 ...