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.
GRT is an efficient and approximate form of scalar stochastic radiative transfer formed by GRT is agnostic to how @math is determined. It may be estimated using Monte Carlo sampling within specific realizations of quenched disorder @cite_1 @cite_32 @cite_39 , by deterministic analysis of a periodic structure @cite_33 , or from observed statistics of the motion of organic material @cite_44 @cite_128 . Alternatively, @math follows from specifying any one of @math , by relationships summarized in Table . The mean attenuation law from a deterministic origin @math may be measured in the laboratory, or calculated from a given stationary random function for @math @cite_50 @cite_142 @cite_6 @cite_14 @cite_106 @cite_69 @cite_70 @cite_13 .
@cite_69 @cite_14 @cite_128 @cite_33 @cite_70 @cite_1 @cite_32 @cite_6 @cite_39 @cite_44 @cite_106 @cite_50 @cite_13 @cite_142
2732070634 2154575087 2036974279 1764825637 2909892328 2026202150 2132908133 2504202545 2081489139 1970613379 2044667456 2028690468 2095576999
Abstract Turbulent clustering of particles causes an increase in the radiation transmission through gas-particle mixtures. Attempts to capture the ensemble-averaged transmission lead to a closure problem called the turbulence-radiation interaction. A simple closure model based on the particle radial distribution function is proposed to capture the effect of turbulent fluctuations in the concentration on radiation intensity. The model is validated against a set of particle-resolved ray tracing experiments through particle fields from direct numerical simulations of particle-laden turbulence. The form of the closure model is generalizable to arbitrary stochastic media with known two-point correlation functions. We survey research on radiation propagation or ballistic particle motion through media with randomly variable material density, and we investigate the topic with an emphasis on very high spatial frequencies. Our new results are based on a specific variability model consisting of a zero-mean Gaussian scaling noise riding on a constant value that is large enough with respect to the amplitude of the noise to yield overwhelmingly non-negative density. We first generalize known results about sub-exponential transmission from regular functions, which are almost everywhere continuous, to merely “measurable” ones, which are almost everywhere discontinuous (akin to statistically stationary noises), with positively correlated fluctuations. We then use the generalized measure-theoretic formulation to address negatively correlated stochastic media without leaving the framework of conventional (continuum-limit) transport theory. We thus resolve a controversy about recent claims that only discrete-point process approaches can accommodate negative correlations, i.e., anti-clustering of the material particles. We obtain in this case the predicted super-exponential behavior, but it is rather weak. Physically, and much like the alternative discrete-point process approach, the new model applies most naturally to scales commensurate with the inter-particle distance in the material, i.e., when the notion of particle density breaks down due to Poissonian—or maybe not-so-Poissonian—number-count fluctuations occur in the sample volume. At the same time, the noisy structure must prevail up to scales commensurate with the mean-free-path to be of practical significance. Possible applications are discussed. This paper develops a procedure for quantifying movement sequences in terms of move length and turning angle probability distributions. By assuming that movement is a correlated random walk, we derive a formula that relates expected square displacements to the number of consecutive moves. We show this displacement formula can be used to highlight the consequences of different searching behaviors (i.e. different probability distributions of turning angles or move lengths). Observations of Pieris rapae (cabbage white butterfly) flight and Battus philenor (pipe-vine swallowtail) crawling are analyzed as a correlated random walk. The formula that we derive aptly predicts that net displacements of ovipositing cabbage white butterflies. In other circumstances, however, net displacements are not well-described by our correlated random walk formula; in these examples movement must represent a more complicated process than a simple correlated random walk. We suggest that progress might be made by analyzing these more complicated cases in terms of higher order markov processes. 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 Direct radiation transmission in turbulent flows laden with heavy particles plays a fundamental role in systems such as clouds and particle solar receivers. Owing to their inertia, particles are preferentially concentrated by the turbulence, and the resulting voids and clusters lead to deviations in mean transmission from the classical Beer-Lambert law for exponential extinction. Additionally, the transmission fluctuations can exceed those of Poissonian media by an order of magnitude, which implies a gross misprediction in transmission statistics if the correlations in particle positions are neglected. On the other hand, tracking millions of particles in a turbulence simulation can be prohibitively expensive. We propose stochastic processes as computationally inexpensive reduced order models for the instantaneous particle number density field and radiation transmission therein. Results from the stochastic processes are compared to Monte Carlo Ray Tracing (MCRT) simulations using the particle positions obtained from the point-particle direct numerical simulation (DNS) of isotropic turbulence at a Taylor Reynolds number of 150. Accurate transmission statistics are predicted with respect to MCRT by matching the mean, variance, and correlation length of the DNS number density fields. Formulation of the stochastic processes in terms of stochastic differential equations allows an exact solution for arbitrary moments of radiation transmission to be derived from the Kolmogorov backwards equations. Higher order statistics such as the transmission variance, and correlations between particle number density, transmission, and absorption are compared to MCRT and discussed. Abstract This paper considers monochromatic radiative transfer in a diffusive three dimensional random binary mixture. The absorption coefficient, along any line-of-sight is a homogeneous Markov process, which is described by a three-dimensional Kubo-Anderson process. The transfer equation is solved numerically by Monte-Carlo simulations on a massively parallel computer (a Connection Machine) by attaching one or several photons to each processor. The implementation of the simulations on the machine is discussed in detail, in particular the association between photons and processors and the storage of the data concerning the photons and the realizations of the statistics. With a CM-2 having 8000 processors, it is possible, with an adequate strategy, to follow simultaneously millions of photons in hundreds of realizations and to reach optical thicknesses up to 100 with dispersions of order 10 -2 for the reflection and transmission coeficients. The simultations are validated, in the case of the one-dimensional rod geometry, by comparison with the exact analytical solution, constructed by averaging the solution of the non-stochastic problem (diffusion in a rod of given optical thickness) over the probability density of the optical thickness. The latter obeys a stochastic Liouville equation which is solved by a Green's function method. The influence of the dimension of the Kubo-Anderson process is studied for the case of a slab and it is shown that a slab consisting in a pile of layers (1D process) is more transparent than one which consists in a stack of lumps (3D process). A strategy for improving the efficiency of Monte-Carlo simulations, based on the distribution of the lengths of the individuals steps of the photons, is presented and discussed. This paper addresses light transport through a discrete random medium, which we define as a volume filled with macroscopic scattering geometry generated by a random process. This formulation is more general than standard radiative transport, because it can be applied to media that are made up of closely packed scatterers. A new approach to rendering these media is introduced, based on precomputed solutions to a local multiple scattering problem, including a new algorithm for generating paths through random media that moves through the interior of the medium in large strides without considering individual scattering events. A method for rendering homogeneous isotropic random media is described that generates paths using precomputed scattering solutions compressed and randomly sampled using Nonnegative Matrix Factorization. It can efficiently render discrete media, such as a large pile of glass objects, in which the individual scatterers are visible. The method is demonstrated on scenes containing tens of thousands of transparent, specular objects that are nearly impossible to render with standard global illumination techniques. 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. Stochastic models of biased random walk are discussed, which describe the behavior of chemosensitive cells like bacteria or leukocytes in the gradient of a chemotactic factor. In particular the turning frequency and turn angle distributions are derived from certain biological hypotheses on the background of related experimental observations. Under suitable assumptions it is shown that solutions of the underlying differential-integral equation approximately satisfy the well-known Patlak-Keller-Segel diffusion equation, whose coefficients can be expressed in terms of the microscopic parameters. By an appropriate energy functional a precise error estimation of the diffusion approximation is given within the framework of singular perturbation theory. Abstract Ballistic photon models of radiative transfer in discrete absorbing random media have demonstrated deviations from the Beer–Lambert–Bouguer law of exponential attenuation. A number of theoretical constructs to quantify the deviation from the Beer–Lambert–Bouguer law have appeared in the literature, several of which rely principally on a statistical measure related to the statistics of the absorber spatial positions alone. Here, we utilize a simple computational model to explore the interplay between the geometric size of the absorbing obstacles and the statistics governing the placement of the absorbers in the volume. We find that a description of the volume that depends on particle size and the spatial statistics of absorbers is not sufficient to fully characterize deviations from the Beer–Lambert–Bouguer law. Implications for future further theoretical and computational explorations of the problem are explored. Renewal theory is used to analyze linear particle transport without scattering in a random mixture of two immiscible fluids, with the statistics described by arbitrary (non‐Markovian) fluid chord length distributions. One conclusion (for unimodal distributions) that is drawn is that the mean and variance of the chord length distributions through each fluid is sufficient knowledge of the statistics to give a reasonably accurate description of the ensemble averaged intensity. Expressions for effective cross sections and an effective source to be used in the usual deterministic transport equation are also obtained. The use of these effective quantities allows statistical information to be introduced very simply into a standard transport equation. An analysis is given which shows how the transport description, including scattering, in a Markovian mixture can be modified to yield an approximate description of transport in a non‐Markovian mixture. Numerical results are given to assess the accuracy of this model... Abstract The transport of radiation through a random array of purely absorbing slabs is considered assuming that the cross section of the slabs fluctuates according to binomial statistics. An exact solution is constructed for the transmission factor through N slabs and the mean value and variance calculated. It is also shown how the complete probability distribution may be recovered. The results are extended to the case of two different types of slab material and thickness and also to the case where the slab thickness is random subject to a constant total thickness of the system. Simple results are obtained and these are compared with the solution of an analogous problem using the dichotomic Markov process method favored by Pomraning and his co-workers. No particular conclusions are drawn regarding the relative merits of these two statistics and the matter is left open for futher discussion.