Boltzmann equation for dissipative gases in homogeneous states with nonlinear friction

TL;DR

This paper analyzes the velocity distribution in dissipative gases with nonlinear friction, deriving analytical forms and validating them with numerical simulations, revealing different behaviors in stable and marginally stable states.

Contribution

It extends previous methods to include nonlinear friction, deriving the high energy tail and sub-leading corrections for dissipative gases in steady states.

Findings

Stretched exponential tails in stable steady states.

Power-law tails in marginal stability cases.

Analytical predictions validated by numerical simulations.

Abstract

Combining analytical and numerical methods, we study within the framework of the homogeneous non-linear Boltzmann equation, a broad class of models relevant for the dynamics of dissipative fluids, including granular gases. We use the new method presented in a previous paper [J. Stat. Phys. 124, 549 (2006)] and extend our results to a different heating mechanism, namely a deterministic non-linear friction force. We derive analytically the high energy tail of the velocity distribution and compare the theoretical predictions with high precision numerical simulations. Stretched exponential forms are obtained when the non-equilibrium steady state is stable. We derive sub-leading corrections and emphasize their relevance. In marginal stability cases, power-law behaviors arise, with exponents obtained as the roots of transcendental equations. We also consider some simple BGK (Bhatnagar, Gross, Krook) models, driven by similar heating devices, to test the robustness of our predictions.