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 The aim of this note is to deepen the knowledge of isotropic random flights moving in mixed Poisson media; that is we analyze the random evolution of a particle with random scattering rate Λ . By assuming Λ distributed according to some standard probability laws, we are able to explicit the probability measure of the random flights. Furthermore, we show that there exists a link between our processes and a inhomogeneous wave equation. In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak--Keller--Segel--Alt... (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 :  It is shown that statistics of records for time series generated by random walks are independent of the details of the jump distribution, as long as the latter is continuous and symmetric. In @math steps, the mean of the record distribution grows as the @math while the standard deviation grows as @math , so the distribution is non-self-averaging. The mean shortest and longest duration records grow as @math and @math , respectively. The case of a discrete random walker is also studied, and similar asymptotic behavior is found.  We consider isotropic random motion where the direction alternations occur according to the renewal epochs of a Gamma distribution with shape parameter (n−2) 2, n=4,5,6,…, in higher dimensions. We formulate a general renewal-type equation for the characteristic function and we solve the renewal equation in an iterative manner. Random flights in Rd,d≥2, with Dirichlet-distributed displacements and uniformly distributed orientation are analyzed. The explicit characteristic functions of the position X¯d(t),t>0, when the number of changes of direction is fixed are obtained. The probability distributions are derived by inverting the characteristic functions for all dimensions d of Rd and many properties of the probabilistic structure of X¯d(t),t>0 are examined. An n-step Pearson-Gamma random walk in ℝ d starts at the origin and consists of n independent steps with gamma distributed lengths and uniform orientations. The gamma distribution of each step length has a shape parameter q>0. Constrained random walks of n steps in ℝ d are obtained from the latter walks by imposing that the sum of the step lengths is equal to a fixed value. Simple closed-form expressions were obtained in particular for the distribution of the endpoint of such constrained walks for any d≥d 0 and any n≥2 when q is either (q = d 2 - 1 ) (d 0=3) or q=d−1 (d 0=2) (Le Caer in J. Stat. Phys. 140:728–751, 2010). When the total walk length is chosen, without loss of generality, to be equal to 1, then the constrained step lengths have a Dirichlet distribution whose parameters are all equal to q and the associated walk is thus named a Pearson-Dirichlet random walk. The density of the endpoint position of a n-step planar walk of this type (n≥2), with q=d=2, was shown recently to be a weighted mixture of 1+floor(n 2) endpoint densities of planar Pearson-Dirichlet walks with q=1 (Beghin and Orsingher in Stochastics 82:201–229, 2010). The previous result is generalized to any walk space dimension and any number of steps n≥2 when the parameter of the Pearson-Dirichlet random walk is q=d>1. We rely on the connection between an unconstrained random walk and a constrained one, which have both the same n and the same q=d, to obtain a closed-form expression of the endpoint density. The latter is a weighted mixture of 1+floor(n 2) densities with simple forms, equivalently expressed as a product of a power and a Gauss hypergeometric function. The weights are products of factors which depends both on d and n and Bessel numbers independent of d. Removal of oxygen from regenerative alkaline hydrogen sulfide absorption solutions before contacting the solution with a hydrogen sulfide containing process gas stream eliminates the formation of thiosulfate impurities. In this paper we analyze some aspects of exponential flights , a stochastic process that governs the evolution of many random transport phenomena, such as neutron propagation, chemical biological species migration, or electron motion. We introduce a general framework for @math -dimensional setups, and emphasize that exponential flights represent a deceivingly simple system, where in most cases closed-form formulas can hardly be obtained. We derive a number of novel exact (where possible) or asymptotic results, among which the stationary probability density for 2d systems, a long-standing issue in Physics, and the mean residence time in a given volume. Bounded or unbounded, as well as scattering or absorbing domains are examined, and Monte Carlo simulations are performed so as to support our findings. This paper deals with a new class of random flights in ℝ d , d≥2, characterized by non-uniform probability distributions on the multidimensional sphere. These random motions differ from similar models appeared in literature where the directions are taken according to the uniform law. The family of angular probability distributions introduced in this paper depends on a parameter ν≥0, which gives the anisotropy of the motion. Furthermore, we assume that the number of changes of direction performed by the random flight is fixed. The time lengths between two consecutive changes of orientation have joint probability distribution given by a Dirichlet density function. A random-medium model being the correlated distribution of points positioned in E-dimensional Euclidean space is considered. The construction of the medium starts from noncorrelated (Poisson) uniform distribution of parent particles; each of them initiates the finite Marcov chain of its descendants. The complete collection of correlation functions of all orders within the scope of the model has been obtained. The use of three-dimensional stable laws (Levy laws) as transition probability density allows one to express the correlation functions of all orders in terms of two point ones only. Some numerical results are presented and discussed in connection with fractal structure simulation.