 Expanding Gaussian solution of the Boltzmann equation

This spatially nonconstant local Maxwellian is constant along free characteristic curves. Its collision term vanishes by the collision invariants, so it solves the full Boltzmann equation. It is positive, rapidly decaying and integrable in phase space. Completing its velocity square gives local mean $\alpha tx/(\beta +\alpha t^2)$ and local variance proportional to $(\beta +\alpha t^2{)}^{-1}$, while its spatial profile broadens with time.