Exact Collision Vertex for Stress and Heat Flux

Abstract

For the hard-sphere Boltzmann equation, we derive an exact nonlinear generating vertex for collisional stress and heat-flux production. The complete two-input Hermite hierarchy for stress resums to a universal traceless tensor times one scalar radial law, while the heat-flux vertex follows from the same law by an exact contraction. Taylor differentiation yields universal bilinear collision tensors with exact stress and heat-flux gradings, without a finite moment closure. DSMC tests, including wall-driven flows and a bimodal Mach-5 shock, recover independently sampled Boltzmann productions and the predicted convergence, while the converged production is independent of the thermal Gaussian foot used for its Hermite representation.