A variational principle for transport theory

Abstract

A maximal variational principle is used to construct an infinite medium Green’s function for treating the boundary value problems of the linear transport theory (neutron and radiative). For the neutrons we consider the one-speed case and correspondingly for the radiative transfer the monochromatic case. The scattering properties of the medium are presumed to be dependent on the relaxation length. Thus, for the neutrons the secondary production function depends on the neutron’s relaxation length and for the radiative transfer the albedo for single scattering is dependent on the optical depth. These two functions are kept arbitrary so that a large class of problems can be covered. The basic principle involves a functional which is an absolute maximum when the trial function is an exact solution of an integral equation of the Fredholm type. The kernel of the integral equation is required to satisfy certain symmetry and boundedness properties. We also exhibit an interesting relation between the absolute maximal and Schwinger’s stationary variational principles, which in general is neither a maximum nor a minimum.