Collision invariant ($\varphi (v^{\prime })+\varphi (v_{\ast }^{\prime })=\varphi (v)+\varphi (v_{\ast })$)

A collision invariant has an unchanged pairwise sum in every allowed elastic collision. The functions $1$, each velocity coordinate, and $|v{|}^2$ encode mass, momentum and energy. Symmetrized weak collision identities then give zero moments of the collision operator against these functions. Under standard regularity assumptions, the invariants form their five-dimensional linear span in three velocity dimensions.