A Lie Algebra-Theoretic Approach to Characterisation of Collision Invariants of the Boltzmann Equation for General Convex Particles

TL;DR

This paper introduces a Lie algebra-based method to characterize collision invariants for convex particles in the Boltzmann equation, extending previous work to non-canonical and non-spherical cases without symmetry assumptions.

Contribution

It extends the characterization of collision invariants to non-canonical physical scattering and non-spherical particles, removing symmetry constraints and providing a new proof for classical hard spheres.

Findings

Extended characterization to non-canonical scattering

Removed symmetry assumptions for non-spherical particles

Provided a new proof for classical hard sphere collision invariants

Abstract

By studying scattering Lie groups and their associated Lie algebras, we introduce a new method for the characterisation of collision invariants for physical scattering families associated to smooth, convex hard particles in the particular case that the collision invariant is of class $\mathscr{C}^{1}$. This work extends that of Saint-Raymond and Wilkinson (Communications on Pure and Applied Mathematics (2018), 71(8), pp. 1494-1534), in which the authors characterise collision invariants only in the case of the so-called canonical physical scattering family. Indeed, our method extends to the case of non-canonical physical scattering, whose existence was reported in Wilkinson (Archive for Rational Mechanics and Analysis (2020), 235(3), pp. 2055-2083). Moreover, our new method improves upon the work in Saint-Raymond and Wilkinson as we place no symmetry hypotheses on the underlying non-spherical particles which make up the gas under consideration. The techniques established in this paper also yield a new proof of the result of Boltzmann for collision invariants of class $\mathscr{C}^{1}$ in the classical case of hard spheres.