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.

中文翻译：

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玻尔兹曼方程和弗拉索夫方程的群分析

我们展示了多维 Boltzmann 和 Vlasov 方程组分析的结果。对于玻尔兹曼方程，我们得到对称群的等价群和分类关系，并在没有外力的情况下研究这些关系。我们发现了一个尺度悖论：我们表明，对于任何碰撞积分，对于移位群不变的方程不允许均匀膨胀，因为方程的左侧和右侧的尺度不同。特别是，这适用于经典的 Boltzmann 方程。对于 Vlasov 方程，我们还得到了对称群的等价群和分类关系，并在没有外力的情况下，对 Vlasov 方程允许包含伽利略群的群的粒子间相互作用进行分类。