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Symmetry and invariance properties of the Boltzmann equation on different groups. (English) Zbl 0328.35070

The study of the solutions of the Boltzmann equation is easier if we use the invariance properties of this equation on different groups. It is well known that the collision integral operator of the Boltzmann equation possesses in-variance properties under the \(O_3\) rotation group in the \(\mathcal E_{\vec v}\) space of the velocities and is accordingly reducible when it is expressed in the standard basis of this group. Yet this standard basis is a spherical one and the boundary conditions, which the Boltzmann equation must fullfill, can have cubic or cylindrical symmetries. Then we have to study the properties of collision integral operator of Klein \(V\) and \(\mathrm{SO}_2\) group and its reducibility in the standard bases of these groups. Precise selection rules have been deduced using the Wigner-Eckart theorem, first when the collision integral operator is a linear one, then when it is expressed in its general, nonlinear form. Owing to its reducibility, the operator is expressed according to the direct sum of matrices associated with the irreducible representations of the invariance groups which are basically nonequivalent. We show as a matter of fact that some of them are necessarily equal and we complete this study by indicating the symmetry properties of the matrix elements. All the above properties are those which result only from the study of the expression of the collision operator in the velocity space \(\mathcal E_{\vec v}\). One can go further and consider the phase space \((\mathcal E_{\vec r}\times\mathcal E_{\vec v})\) as a whole. Under these circumstances, it is possible to show that the Boltzmann equation possesses invariance properties which are determined according to the symmetries of the force \(F\) applied to the particles of the system. Especially, besides demonstrating the selection rules and invariant subspaces when the force \(F\) is invariant under rotation or zero, we express the Boltzmann equation eigenfunctions in the subspace \((\mathcal E_{\vec r}\times\mathcal E_{\vec v})\) which is the tensor product of the spaces corresponding to the angular parts of the variables \(\vec r\) and \(\vec v\). This brings the Boltzmann equation back to a differential equation system corresponding to the variable \(|\vec r|\) to which the Chapman-Enskog process is still applicable. Lastly, we show that the moments of the distribution function expressed in a local basis are finite linear combinations of the moments connecting to an absolute basis.
Reviewer: J. N. Massot

MSC:

35Q20 Boltzmann equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
76P05 Rarefied gas flows, Boltzmann equation in fluid mechanics
82B40 Kinetic theory of gases in equilibrium statistical mechanics
20G45 Applications of linear algebraic groups to the sciences
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