Discrete Symmetries of the Linear Boltzmann equation

Abstract

Group theoretical considerations originating from Wigner are applied to the few group diffusion equation (DE) and to the linear Boltzmann transport equation (TE). The irreducible subspaces invariant under the symmetries of the TE or DE determine specific incoming current patterns. If these patterns are used in the response matrix formulation, the response matrix will be diagonal. A method for obtaining a solution that belongs to a given irreducible subspace is provided. These considerations are utilized in a coarse-mesh program. It is shown that the solution to a boundary condition problem is determined by the irreducible decomposition of the boundary condition and by the solutions belonging to the irreducible boundary conditions. Finally, the solution of the TE in an infinite lattice is investigated and the results are utilized to solve the TE in a finite lattice.