When lifting the assumption of spatially-independent scattering centers in classical linear transport theory, collision rate is no longer proportional to vector flux radiance because the macroscopic cross-section @math depends on the distance @math to the previous collision or boundary. This creates a nonlocal relationship between collision rate and flux and requires revising a number of familiar deterministic and Monte Carlo methods. We generalize collision and track-length estimators to support unbiased estimation of either flux integrals or collision rates in generalized radiative transfer (GRT). To provide benchmark solutions for the Monte Carlo estimators, we derive the four Green's functions for the isotropic point source in infinite media with isotropic scattering. Additionally, new moment-preserving diffusion approximations for these Green's functions are derived and reduce to algebraic expressions involving the first four moments of the free-path lengths between collisions.
 Abstract This paper extends a recently introduced theory describing particle transport for random statistically homogeneous systems in which the distribution function p ( s ) for chord lengths between scattering centers is non-exponential. Here, we relax the previous assumption that p ( s ) does not depend on the direction of flight Ω ; this leads to a new generalized linear Boltzmann equation that includes angular-dependent cross sections, and to a new generalized diffusion equation that accounts for anisotropic behavior resulting from the statistics of the system. The kinetic theory of gases was proposed by J. Clerk Maxwell [34, 35] and L. Boltzmann [5] in the second half of the XIXth century. Because the existence of atoms, on which kinetic theory rested, remained controversial for some time, it was not until many years later, in the XXth century, that the tools of kinetic theory became of common use in various branches of physics such as neutron transport, radiative transfer, plasma and semiconductor physics, etc. In classical kinetic models a particle free path distribution is exponential, but this is more likely to be an exception than a rule. In this paper we derive a generalized linear Boltzmann equation (GLBE) for a general free path distribution in the framework of Alt’s model. In the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the GLBE for small mean free paths. In the special case of the one-speed transport problem the asymptotic solution results in a diffusion approximation to the GLBE. (The subsequent events and the solution are discussed in Sections 3 and 4.) This problem, which is basically concerned with the probability density of a sum of two-dimensional random vectors, was soon found to include, or was extended to include, a significant part of the theory of mathematical probability and to have numerous applications. In a form sufficiently general for the present purpose, the random walk problem for a A>dimensional Euclidean space may be put as follows :   Abstract This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications. We are interested in non-standard transport equations where the description of the scattering events involves an additional "memory variable''. We establish the well posedness and investigate the diffusion asymptotics of such models. While the questions we address are quite classical the analysis is original since the usual dissipative properties of collisional transport equations is broken by the introduction of the memory terms.   Abstract New radiative transfer theory is developed for stochastically inhomogeneous scattering media. The three-dimensional shapes and large scale (compared to the mean free path) structures of the media are modeled by stochastic interfaces separating regions of different scattering properties. The small scale fluctuations are characterized by a pair-correlation function. The radiative transfer equation is extended to include individual scattering and propagation probabilities of a ray for each subregion as well as the probability for a ray to cross the interface between two subregions. The propagation probability is found to depend on the entire preceding path of the ray; the present formulation accounts for the two previous scatterings. A new adding doubling algorithm is developed to solve this problem numerically. Transmission through a cloud layer and backward scattering seem to be particularly sensitive to inhomogeneities. The unanswered mathematical questions of reactor theory can be divided roughly into two classes. The first class concerns the mathematical theory of the basic transport equations and of the approximations thereto. Thus, the multiplication and criticality equations are characteristic value equations but their operators do not belong to the class for which the characteristic value theory is well established: they are not normal. Much progress was made recently concerning the character and properties of the highest characteristic value and the corresponding characteristic vector but the properties of the lower characteristic values and vectors are not known. In particular, the extent of a continuous spectrum and the completeness of the whole set of characteristic vectors are not established in general. A simple example is given in which the transport operator has a continuous spectrum which has not been recognized to date. The need for a generalization of the characteristic value problem is pointed out and such a generalization is proposed.