Stationary states and energy cascades in inelastic gases

We study the formation of high energy tails in a one-dimensional kinetic model for granular gases, the so-called Inelastic Maxwell Model. We introduce a time- discretized version of the stochastic process, and show that continuous time implies larger fluctuations of the particles energies. This is due to a statistical relation between the number of inelastic collisions undergone by a particle and its average energy. This feature is responsible for the high energy tails in the...

Analytic solutions $F(v,t)$ of the nonlinear Boltzmann equation in $d$-dimensions are studied for a new class of dissipative models, called inelastic repulsive scatterers, interacting through pseudo-power law repulsions, characterized by a strength parameter $\nu$, and embedding inelastic hard spheres ($\nu=1$) and inelastic Maxwell models ($\nu=0$). The systems are either freely cooling without energy input or driven by thermostats, e.g. white noise, and approach stable none...

We consider collisional models for granular particles and analyze the conditions under which the restitution coefficient might be a constant. We show that these conditions are not consistent with known collision laws. From the generalization of the Hertz contact law for viscoelastic particles we obtain the coefficient of normal restitution \epsilon as a function of the normal component of the impact velocity v_{imp}. Using \epsilon(v_{imp}) we describe the time evolution of t...

We conduct a molecular dynamics simulation of an inelastic gas system utilizing an event-driven algorithm combined with a thermostat mechanism. Initially, the kinetic energy of the system experiences a decay before settling into a non-equilibrium steady state. To explore the aging characteristics, we analyze the velocity autocorrelation function, denoted as \( C(t_w, t) \). Our findings indicate that \( C(t_w, t) \) exhibits a dependence on both waiting time \( t_w \) and cor...

The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation are studied in the QE-limit (Quasi-Elastic; infinitesimal dissipation) by a combination of analytical and numerical techniques. Their behavior at large velocities differs qualitatively from that for higher dimensional systems. In our generic model, a dissipative fluid is maintained in a non-equilibrium steady state by a stochastic or deterministic driving force. The velocity distribution for stochas...

We present a universal description of the velocity distribution function of granular gases, $f(v)$, valid for both, small and intermediate velocities where $v$ is close to the thermal velocity and also for large $v$ where the distribution function reveals an exponentially decaying tail. By means of large-scale Monte Carlo simulations and by kinetic theory we show that the deviation from the Maxwell distribution in the high-energy tail leads to small but detectable variation o...

We study the dynamics of a homogeneous granular gas heated by a stochastic thermostat, in the low density limit. It is found that, before reaching the stationary regime, the system quickly "forgets" the initial condition and then evolves through a universal state that does not only depend on the dimensionless velocity, but also on the instantaneous temperature, suitably renormalized by its steady state value. We find excellent agreement between the theoretical predictions at ...

We study the single particle velocity distribution for a granular fluid of inelastic hard spheres or disks, using the Enskog-Boltzmann equation, both for the homogeneous cooling of a freely evolving system and for the stationary state of a uniformly heated system, and explicitly calculate the fourth cumulant of the distribution. For the undriven case, our result agrees well with computer simulations of Brey et al. \cite{brey}. Corrections due to non-Gaussian behavior on cooli...

We present a new numerical algorithm based on a relative energy scaling for collisional kinetic equations allowing to study numerically their long time behavior, without the usual problems related to the change of scales in velocity variables. It is based on the knowledge of the hydrodynamic limit of the model considered, but is able to compute solutions for either dilute or dense regimes. Several applications are presented for Boltzmann like equations. This method is particu...

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 $t^{\alpha}$, and the energy decreases as $t^{-\alpha d}$, where the exponent $\alpha=1/(d+1)$ is independent of the coefficient of restitution. We support our arguments with an exact calculation in one dimension and event driven molecular...