Symmetry and invariance properties of the Boltzmann equation on different groups

Abstract

The introduction of the group theory in the treatment of the Boltzmann equation shows the reducibility of the collision integral operator on the invariant subspaces of Klein V or SO2 group. Especially we prove the equality of matrices representing the collision integral operator between inequivalent subspaces first in its linear form and then in its general form. These results are finally expanded to the full Boltzmann equation when we consider its properties as a whole in the phase space (ℰr×ℰv). This brings back Boltzmann equation following the Chapmann–Enskog process to the differential equation system depending solely on the variable ‖r‖. The examination of the Boltzmann equation symmetries allows us to obtain the selection rules which lead to an important simplification in theoretical as well as numerical calculations of the distribution function.