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.
Cauchy's formula was originally established for random straight paths crossing a body B⊂Rn and basically relates the average chord length through B to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in B. Similar results are also obtained for the average number of collisions performed by the walker in B. Invariance properties of random walks in bounded domains are a topic of growing interest since they contribute to improving our understanding of diffusion in confined geometries. Recently, limited to Pearson random walks with exponentially distributed straight paths, it has been shown that under isotropic uniform incidence, the average length of the trajectories through the domain is independent of the random walk characteristic and depends only on the ratio of the volume's domain over its surface. In this paper, thanks to arguments of integral geometry, we generalize this property to any isotropic bounded stochastic process and we give the conditions of its validity for isotropic unbounded stochastic processes. The analytical form for the traveled distance from the boundary to the first scattering event that ensures the validity of the Cauchy formula is also derived. The generalization of the Cauchy formula is an analytical constraint that thus concerns a very wide range of stochastic processes, from the original Pearson random walk to a Rayleigh distribution of the displacements, covering many situations of physical importance. 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. Abstract The classic Milne problem of radiative transfer is considered within the context of transport in a statistical medium. It is pointed out that the statistical nature of the medium leads to two different definitions for this Milne problem. Explicit analytic results are given for each definition, under the assumption of binary homogeneous Markov statistics. The two solutions are qualitatively as well as quantitatively quite different. The microstructure of a medium strongly influences how light propagates through it. The amount of disorder it contains determines whether the medium is transparent or opaque. Theory predicts that exciting such a medium homogeneously and isotropically makes some of its optical properties depend only on the medium’s outer geometry. Here, we report an optical experiment demonstrating that the mean path length of light is invariant with respect to the microstructure of the medium it scatters through. Using colloidal solutions with varying concentration and particle size, the invariance of the mean path length is observed over nearly two orders of magnitude in scattering strength. Our results can be extended to a wide range of systems—however ordered, correlated, or disordered—and apply to all wave-scattering problems. Starting from a simple animal-biology example, a general, somewhat counter-intuitive property of diffusion random walks is presented. It is shown that for any (non-homogeneous) purely diffusing system, under any isotropic uniform incidence, the average length of trajectories through the system (the average length of the random walk trajectories from entry point to first exit point) is independent of the characteristics of the diffusion process and therefore depends only on the geometry of the system. This exact invariance property may be seen as a generalization to diffusion of the well-known mean-chord-length property (Case K. M. and Zweifel P. F., Linear Transport Theory (Addison-Wesley) 1967), leading to broad physics and biology applications. For connected regions of arbitrary shape containing a distribution of scattering centers that need not be homogeneous, the average path length for particles that traverse the medium is shown to be unaffected by the presence of the scattering centers, provided that the incidence of the particles is uniform and isotropic, that no absorption or particle reproduction occurs in the medium, and that the scattering probabilities satisfy detailed balance. A statistical correlation function of basic importance in the study of two-phase random media (such as suspensions, porous media, and composites) is the chord-length distribution function p(z). We show that p(z) is related to another fundamentally important morphological descriptor studied by us previously, namely, the lineal-path function L(z), which gives the probability of finding a line segment of length z wholly in one of the phases when randomly thrown into the sample. We derive exact series representations of the chord-length distribution function for media comprised of spheres with a polydispersivity in size for arbitrary space dimension D. For the special case of spatially uncorrelated spheres (i.e., fully penetrable spheres), we determine exactly p(z) and the mean chord length @math , the first moment of p(z). We also obtain corresponding formulas for the case of impenetrable (i.e., spatially correlated) polydispersed spheres.