The rich behavior of the Boltzmann equation for dissipative gases

Abstract

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 combining this method with numerical solutions, obtained from the Direct Simulation Monte Carlo (DSMC) method, we study a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We establish a criterion connecting the stability of the non-equilibrium steady state to an exponentially bound form for the velocity distribution $F$, which varies depending on the forcing mechanism. Power laws arise in marginal stability cases, of which several new cases are reported. Our results provide a minimal framework for interpreting large classes of experiments on driven granular gases.

References in corpus (7)

• The Inelastic Maxwell Model
• Velocity Distributions of Granular Gases with Drag and with Long-Range Interactions
• Self-Similarity in Random Collision Processes
• Local Equation of State and Velocity Distributions of a Driven Granular Gas
• Stationary states and energy cascades in inelastic gases
• An exactly solvable model for driven dissipative systems
• Power-law velocity distributions in granular gases

Cited by in corpus (6)

• Quasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases
• Shear-rate dependent transport coefficients for inelastic Maxwell models
• Aging to non-Newtonian hydrodynamics in a granular gas
• Granular gases under extreme driving
• Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction
• Anomalous transport of impurities in inelastic Maxwell gases