Scaling Solutions of Inelastic Boltzmann Equations with Over-populated High Energy Tails

This paper deals with solutions of the nonlinear Boltzmann equation for spatially uniform freely cooling inelastic Maxwell models for large times and for large velocities, and the nonuniform convergence to these limits. We demonstrate how the velocity distribution approaches in the scaling limit to a similarity solution with a power law tail for general classes of initial conditions and derive a transcendental equation from which the exponents in the tails can be calculated. Moreover on the basis of the available analytic and numerical results for inelastic hard spheres and inelastic Maxwell models we formulate a conjecture on the approach of the velocity distribution function to a scaling form.Comment: 15 pages, 4 figures. Accepted in J. Statistical Physic

Extension of Haff's cooling law in granular flows

The total energy E(t) in a fluid of inelastic particles is dissipated through inelastic collisions. When such systems are prepared in a homogeneous initial state and evolve undriven, E(t) decays initially as t^{-2} \aprox exp[ - 2\epsilon \tau] (known as Haff's law), where \tau is the average number of collisions suffered by a particle within time t, and \epsilon=1-\alpha^2 measures the degree of inelasticity, with \alpha the coefficient of normal restitution. This decay law is extended for large times to E(t) \aprox \tau^{-d/2} in d-dimensions, far into the nonlinear clustering regime. The theoretical predictions are quantitatively confirmed by computer simulations, and holds for small to moderate inelasticities with 0.6< \alpha< 1.Comment: 7 pages, 4 PostScript figures. To be published in Europhysics Letter

Asymptotic solutions of the nonlinear Boltzmann equation for dissipative systems

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 nonequilibrium steady states, or marginally stable homogeneous cooling states, where the data, $v^d_0(t) F(v,t)$ plotted versus $c=v/v_0(t)$, collapse on a scaling or similarity solution $f(c)$, where $v_0(t)$ is the r.m.s. velocity. The dissipative interactions generate overpopulated high energy tails, described generically by stretched Gaussians, $f(c) \sim \exp[-\beta c^b]$ with $0 < b < 2$, where $b=\nu$ with $\nu>0$ in free cooling, and $b=1+{1/2} \nu$ with $\nu \geq 0$ when driven by white noise. Power law tails, $f(c) \sim 1/c^{a+d}$, are only found in marginal cases, where the exponent $a$ is the root of a transcendental equation. The stability threshold depend on the type of thermostat, and is for the case of free cooling located at $\nu=0$. Moreover we analyze an inelastic BGK-type kinetic equation with an energy dependent collision frequency coupled to a thermostat, that captures all qualitative properties of the velocity distribution function in Maxwell models, as predicted by the full nonlinear Boltzmann equation, but fails for harder interactions with $\nu>0$.Comment: Submitted to: "Granular Gas Dynamics", T. Poeschel, N. Brilliantov (eds.), Lecture Notes in Physics, Vol. LNP 624, Springer-Verlag, Berlin-Heidelberg-New York, 200

Towards a Landau-Ginzburg-type Theory for Granular Fluids

In this paper we show how, under certain restrictions, the hydrodynamic equations for the freely evolving granular fluid fit within the framework of the time dependent Landau-Ginzburg (LG) models for critical and unstable fluids (e.g. spinodal decomposition). The granular fluid, which is usually modeled as a fluid of inelastic hard spheres (IHS), exhibits two instabilities: the spontaneous formation of vortices and of high density clusters. We suppress the clustering instability by imposing constraints on the system sizes, in order to illustrate how LG-equations can be derived for the order parameter, being the rate of deformation or shear rate tensor, which controls the formation of vortex patterns. From the shape of the energy functional we obtain the stationary patterns in the flow field. Quantitative predictions of this theory for the stationary states agree well with molecular dynamics simulations of a fluid of inelastic hard disks.Comment: 19 pages, LaTeX, 8 figure

Heating without heat: thermodynamics of passive energy filters between finite systems

Passive filters allowing the exchange of particles in a narrow band of energy are currently used in micro-refrigerators and energy transducers. In this letter, we analyze their thermal properties using linear irreversible thermodynamics and kinetic theory, and discuss a striking phenomenon: the possibility of increasing or decreasing simultaneously the temperatures of two systems without any supply of energy. This occurs when the filter induces a flow of particles whose energy is between the average energies of the two systems. Here we show that this selective transfer of particles does not need the action of any sort of Maxwell demon and can be carried out by passive filters without compromising the second law of thermodynamics. The phenomenon allows us to design cycles between two reservoirs at temperatures $T_1<T_2$ that are able to reach temperatures below $T_1$ or above $T_2$.Comment: 5 pages, 3 figure