Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful

We present the group classification of the one-dimensional Boltzmann equation with respect to the function $\mathcal F=\mathcal{F}(t,x,c)$ characterizing an external force field under the assumption that the physically meaningful constraints $dx=c\,dt$, $dc=\mathcal{F}\,dt$, $dt=0$, and $dx=0$ are imposed on the variables. We show that for all functions $\mathcal{F}$, the algebra is finite-dimensional, and its maximum dimension is eight, which corresponds to the equation with a zero $\mathcal{F}$.