Nonequilibrium steady solutions of the Boltzmann equation

TL;DR

This paper investigates nonequilibrium steady states of the Boltzmann equation under forcing and dissipation, revealing warm cascade solutions and their relation to system parameters through theory and numerical simulations.

Contribution

It introduces a differential approximation model for the Boltzmann equation and predicts warm cascade steady solutions with a novel relation between forcing, dissipation, and temperature.

Findings

Steady solutions are warm cascades.

Temperature is independent of forcing amplitude.

Numerical simulations confirm theoretical predictions.

Abstract

We report a study of the homogeneous isotropic Boltzmann equation for an open system. We seek for nonequilibrium steady solutions in presence of forcing and dissipation. Using the language of weak turbulence theory, we analyze the possibility to observe Kolmogorov-Zakharov steady distributions. We derive a differential approximation model and we find that the expected nonequilibrium steady solutions have always the form of warm cascades. We propose an analytical prediction for relation between the forcing and dissipation and the thermodynamic quantities of the system. Specifically, we find that the temperature of the system is independent of the forcing amplitude and determined only by the forcing and dissipation scales. Finally, we perform direct numerical simulations of the Boltzmann equation finding consistent results with our theoretical predictions.