Duality in transport theory

Abstract

The relationship between the forward and the adjoint or backward linear particle transport equations is treated from a different viewpoint. Both deterministic and stochastic transport equations are considered. The starting point is that at the level of the Green's function, the forward and backward transport equations represent a dual alternative for calculating the same quantity, and a close analogy with the similar properties of the forward and backward Kolmogorov or master equations of discrete Markovian stochastic processes is stressed. This analogy is exploited in that derivation methods of such master equations are transferred to transport equations. It is shown that forward and backward (adjoint) transport equations can be derived directly from the Markovian property of the transport process, without explicitly resorting to adjoint techniques. This principle is referred to as duality in this paper. The first application is a new derivation of the conventional forward and adjoint transport equations from one common balance equation, expressing the Markovian property of the transport process. Then forward- and backward-type transport master equations are derived, and it is shown that the corresponding transport equations are their first moments. Reasons for the rare occurrence of the forward master transport equation are discussed. From the master equations, second-order forward and backward moment equations are also derived. It is seen that for the derivation of forward-type equations for the doublet density, the forward two-point master equations are an especially convenient tool when external sources are involved.