Quasi-elastic solutions to the nonlinear Boltzmann equation for dissipative gases

TL;DR

This paper investigates the behavior of solutions to the one-dimensional nonlinear Boltzmann equation in the quasi-elastic limit, revealing unique velocity distribution features and singularities under different driving forces.

Contribution

It provides a detailed analysis of the quasi-elastic limit for 1D dissipative gases, highlighting differences from higher-dimensional systems and classifying velocity distribution singularities.

Findings

Stretched exponential tails with exponent 2b for stochastic driving.

Singular multi-peaked velocity distributions for deterministic driving.

Qualitative differences in large velocity behavior compared to higher dimensions.

Abstract

The solutions of the one-dimensional homogeneous nonlinear Boltzmann equation are studied in the QE-limit (Quasi-Elastic; infinitesimal dissipation) by a combination of analytical and numerical techniques. Their behavior at large velocities differs qualitatively from that for higher dimensional systems. In our generic model, a dissipative fluid is maintained in a non-equilibrium steady state by a stochastic or deterministic driving force. The velocity distribution for stochastic driving is regular and for infinitesimal dissipation, has a stretched exponential tail, with an unusual stretching exponent $b_{QE} = 2b$, twice as large as the standard one for the corresponding $d$-dimensional system at finite dissipation. For deterministic driving the behavior is more subtle and displays singularities, such as multi-peaked velocity distribution functions. We classify the corresponding velocity distributions according to the nature and scaling behavior of such singularities.