The Boltzmann equation for driven systems of inelastic soft spheres

Abstract

We study a generic class of inelastic soft sphere models with a binary collision rate $g^{\nu }$ that depends on the relative velocity $g$. This includes previously studied inelastic hard spheres ($\nu =1$) and inelastic Maxwell molecules ($\nu =0$). We develop a new asymptotic method for analyzing large deviations from Gaussian behavior for the velocity distribution function $f(c)$. The framework is that of the spatially uniform nonlinear Boltzmann equation and special emphasis is put on the situation where the system is driven by white noise. Depending on the value of exponent $\nu$, three different situations are reported. For $\nu <-2$, the non-equilibrium steady state is a repelling fixed point of the dynamics. For $\nu >-2$, it becomes an attractive fixed point, with velocity distributions $f(c)$ having stretched exponential behavior at large $c$. The corresponding dominant behavior of $f(c)$ is computed together with sub-leading corrections. In the marginally stable case $\nu =-2$, the high energy tail of $f(c)$ is of power law type and the associated exponents are calculated. Our analytical predictions are confronted with Monte Carlo simulations, with a remarkably good agreement.

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• Theory of Transport Processes and the Method of the Nonequilibrium Statistical Operator
• Granular Brownian Motor
• Memory effect in uniformly heated granular gases
• Fluctuating hydrodynamics for driven granular gases
• Third and fourth degree collisional moments for inelastic Maxwell models
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• Hydrodynamics of inelastic Maxwell models
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• Unsteady non-Newtonian hydrodynamics in granular gases
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• Inelastic Maxwell models for monodisperse gas-solid flows
• Multicomponent Modified Boltzmann Equation and Thermalization