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Abstract

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.

Hydrodynamic Burnett equations for inelastic Maxwell models of granular gases

“…the flow velocity 4) and the granular temperature 5) where V = v − u is the peculiar velocity. By taking moments in the Boltzmann equation with respect to 1, v, and v 2 , the balance equations are obtained:…”

“…(3.9) is not known, its velocity moments can be in principle obtained in a recursive way [5,6]. In particular, the fourth-degree cumulant is [43] …”

“…Two of them are relevant here, the inelastic hard-sphere model (IHSM) and the inelastic Maxwell model (IMM) [3][4][5][6][7]. The minimal version of the IHSM corresponds to a collection of smooth hard spheres or disks that undergo inelastic collisions, with a velocity-independent coefficient of normal restitution α [8,9].…”

Derivation of Tsallis statistics from dynamical equations for a granular gas

“…A large amount of research has recently been devoted to the study of spatially uniform systems with dissipative interactions ("granular matter") (see, e.g., [1][2][3][4][5][6] and references therein). Among the issues addressed, much attention is deserved to transport properties and therefore to the structure of the asymptotic velocity statistics.…”

The Inelastic Maxwell Model

“…Historically, it played an important role in the development of kinetic theory [32][33][34][35][36], and it remains the subject of current research [37][38][39]. Recently, it has been realized that the Maxwell model is analytically tractable even for inelastic collisions [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. In this review, we detail dynamics of uniform inelastic gases, isolated impurities in uniform gases, and binary mixtures.…”