Group analysis of the Boltzmann and Vlasov equations

We present results of a group analysis of the multidimensional Boltzmann and Vlasov equations. For the Boltzmann equation, we obtain the equivalence group and classifying relations for the symmetry group and study these relations in the case where external forces are absent. We discover a scale paradox: we show that for any collision integral, an equation that is invariant with respect to the shift group does not admit uniform dilations, because the left- and right-hand sides of the equation scale differently. In particular, this holds for the classical Boltzmann equation. For the Vlasov equation, we also obtain the equivalence group and classifying relations for the symmetry group and classify the interparticle interactions for which the Vlasov equation admits groups containing the Galilean group in the case where external forces are absent.