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We complete the investigation of the feasibility in principle to close the moment system for the kinetic equation via invariant relations obtained by group analysis methods. For the one-dimensional equation, we obtain an affirmative answer, find an invariant closure of the moment system, and reveal several new phenomena. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional Boltzmann equation f t + c f x + (F f) c = 0 with a function F depending on (t,x,c,f) and obtain the complete group classification of such equations in the class of point changes of whole set of variables (t,x,c,f). for this, we impose additional conditions on the transformations for the invariance of (a) the relations dx = c dt and d c = F d t , (b) the lines dt = dx = 0, and (c) the form f dx dc, which fix the physical meaning of the used variables and the relations between them. [ABSTRACT FROM AUTHOR]