= Expanding Gaussian solution of the Boltzmann equation
{title2=$f(t,x,v)=A\exp(-\alpha|x-tv|^2-\beta|v|^2),\quad A,\alpha,\beta>0$}
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.
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