A kinetic model for polyatomic gas with quasi-resonant collisions leading to bi-temperature relaxation processes

TL;DR

This paper develops a kinetic model for polyatomic gases incorporating quasi-resonant collisions, leading to a two-temperature relaxation process and providing a rigorous framework for multi-temperature Maxwellian equilibria.

Contribution

It introduces quasi-resonant kernels into the Boltzmann framework, enabling the study of multi-temperature relaxation in polyatomic gases with a rigorous mathematical foundation.

Findings

Established an H-theorem for the model

Demonstrated two-phase relaxation towards equilibrium

Validated predictions with numerical simulations

Abstract

In this article, we extend the Boltzmann framework for polyatomic gases by introducing quasi-resonant kernels, which relax resonant interactions, for which kinetic and internal energies are separately conserved and lead to equilibrium states with two temperatures. We establish an H-theorem and analyze the quasi-resonant model's asymptotic behaviour, demonstrating a two-phase relaxation process: an initial convergence towards a two-temperature Maxwellian state followed by gradual relaxation of the two temperatures towards each other. Numerical simulations validate our theoretical predictions. The notion of quasi-resonance provides the first rigorous framework of a Boltzmann dynamics for which the distribution is at all times close to a multi-temperature Maxwellian, relaxing towards a one-temperature Maxwellian.