The rich behavior of the Boltzmann equation for dissipative gases

TL;DR

This paper introduces a new analytic method for solving the Boltzmann equation in dissipative gases, extending previous results to a broader range of interactions and forcing mechanisms, and combining it with numerical simulations to analyze granular gases.

Contribution

A novel analytic approach for the Boltzmann equation that overcomes previous limitations and applies to diverse particle interactions and forcing mechanisms.

Findings

Established a stability criterion linked to the velocity distribution

Identified power-law behaviors in marginal stability cases

Provided a framework for interpreting granular gas experiments

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.