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By studying scattering Lie groups and their associated Lie algebras, weintroduce a new method for the characterisation of collision invariants forphysical scattering families associated to smooth, convex hard particles in theparticular case that the collision invariant is of class $\mathscr{C}^{1}$.This work extends that of Saint-Raymond and Wilkinson (Communications on Pureand Applied Mathematics (2018), 71(8), pp. 1494-1534), in which the authorscharacterise collision invariants only in the case of the so-called canonicalphysical scattering family. Indeed, our method extends to the case ofnon-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 Wilkinsonas we place no symmetry hypotheses on the underlying non-spherical particleswhich make up the gas under consideration. The techniques established in thispaper also yield a new proof of the result of Boltzmann for collisioninvariants of class $\mathscr{C}^{1}$ in the classical case of hard spheres.

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