Particle and radiation leakage importance: definition, analysis, and interpretation

The concept of leakage importance function has been introduced and analyzed for physical systems governed by the Boltzmann transport equation. This leakage importance function represents a measure of the relative importance of source particles located at every point in phase space in contributing to the leakage and provides insight regarding the specific physical process that leads to leakage. The equation satisfied by the leakage importance function has been derived by using adjoint operators. It has been shown that procedures that are customarily used to derive an equation obeyed by an importance function suitable for an integral parameter such as a detector response or an eigenvalue lead to difficulties when directly applied to derive an equation obeyed by the leakage importance function. This is because, although leakage is also an integral parameter (i.e., a functional of the forward flux density), leakage is expressed in terms of a surface integral rather than in terms of volume integrals such as those appearing in expressions of detector responses or eigenvalues. Therefore, a procedure that departs from the customary course has been devised to derive the equation satisfied by the leakage importance function