 Nonlinear Boltzmann collision operator

For equal-mass elastic collisions, one reflection parametrization is $v^{\prime }=v-((v-v_{\ast })\cdot n)n$ and $v_{\ast }^{\prime }=v_{\ast }+((v-v_{\ast })\cdot n)n$. The fixed-$n$ transformation preserves pairwise momentum, kinetic energy and velocity-pair measure. A nonnegative kernel invariant under particle interchange and collision reversal yields a gain-loss operator. Its weak Boltzmann collision identity proves conservation and its entropy identity proves the Boltzmann H theorem. Unlike the Maxwell molecule collision operator, this general kernel may depend on relative speed.