Wigner-Eckart Factorization of the Spectral Boltzmann Collision Operator

We reduce the eight-dimensional weak form of the bilinear Boltzmann collision operator to a five-dimensional kinematic core by rigidly rotating the laboratory frame to align with the colliding pair and integrating over the $\mathrm{S}\mathrm{O}(3)$ rotation group. This reduction yields an exact Wigner–Eckart factorization within a spectral Galerkin framework of associated Laguerre polynomials and spherical harmonics. The decomposition decouples the angular geometry from the scattering physics. The former, represented by Clebsch–Gordan coefficients, is evaluated exactly, while the latter is evaluated to machine precision by a spectrally convergent singular quadrature strategy. By explicitly zeroing specific entries, the macroscopic collision invariants are embedded without approximation. Cache-optimized contractions deliver up to a 37-fold single-core speedup and a 1000-fold memory reduction over standard dense Cartesian formulations. The approach is validated against analytical solutions for Maxwell molecules and infinite-order Chapman–Enskog viscosity coefficients for hard spheres.