The relationship between the popular so-called backward or Lindhard-type transport equations for linear energetic cascades and the direct or forward Boltzmann equation description is rigorously examined for an arbitrary atomic species mix. A variational principle is systematically derived that characterizes the forward model with generalized boundary conditions (internal reflection at a free surface) and is extremized to yield self-consistently the adjoint equations and boundary conditions as components of the corresponding Euler-Lagrange system. The adjoint function is treated purely as a mathematical artifact, which follows naturally from the variational principle. Dubious physical arguments to assign adjoint boundary conditions are thereby avoided. A truly backward description is derived from the adjoint formalism, which under the assumption of space and time homogeneity, reduces to the familiar Lindhard form. The Lindhard-type equations are seen to be neither backward nor forward equations but assume a hybrid form. In contrast, the forward and truly backward (or adjoint) models are exact and of general validity. They are complementary approaches and thus describe a duality that is mediated by the variational principle.

• Received 2 September 1988

DOI:https://doi.org/10.1103/PhysRevB.39.8858