Random tessellations of the space represent a class of prototype models of heterogeneous media, which are central in several applications in physics, engineering, and life sciences. In this work, we investigate the statistical properties of d-dimensional isotropic Poisson geometries by resorting to Monte Carlo simulation, with special emphasis on the case d=3. We first analyze the behavior of the key features of these stochastic geometries as a function of the dimension d and the linear size L of the domain. Then, we consider the case of Poisson binary mixtures, where the polyhedra are assigned two labels with complementary probabilities. For this latter class of random geometries, we numerically characterize the percolation threshold, the strength of the percolating cluster, and the average cluster size.

Eigenvalue problems for neutron transport in random geometries are key for many applications, ranging from reactor design to criticality safety. In this work we examine the behaviour of the reactivity and of the kinetics parameters (the effective delayed neutron fraction and the effective neutron generation time) for three-dimensional UOX and MOX assembly configurations where a portion of the fuel pins has been randomly fragmented by using various mixing statistics. For this purpose, we have selected stochastic tessellations of the Poisson, Voronoi and Box type, which provide convenient models for the random partitioning of space, and we have generated an ensemble of assembly realizations; for each geometry realization, criticality calculations have been performed by using the Monte Carlo code TRIPOLI-4 R , developed at CEA. We have then examined the evolution of the ensemble-averaged observables of interest as a function of the average chord length of the random geometries, which is roughly proportional to the correlation length of the fuel fragmentation. The methodology proposed in this work is fairly general and could be applied, e.g., to the assessment of re-criticality probability following severe accidents.

In the context of linear transport in homogeneous non-stochastic media, the well-known Cauchy formulas express a remarkably elegant relation between the contributions to the angular flux within an arbitrary body due to particles starting from the body surface and those due to particles starting from the interior of the body. In this letter, we will extend these results to binary stochastic mixtures, a prototype class of random media that emerges in a broad spectrum of applications in physical and life sciences, ranging from hydrodynamical instabilities to tracer diffusion in biological tissues. In particular, for both quenched disorder and annealed disorder models we will establish generalized Cauchy-like formulas that appear to have a universal character and bear a close similarity to those previously derived for non-stochastic media.

In a recent article, Krapivsky and Redner (J. Stat. Mech. 093208 (2018)) established that the distribution of the first hitting times for a diffusing particle subject to hitting an absorber is independent of the direction of the external flow field. In the present paper, we build upon this observation and investigate when the conditioning on the diffusion leads to a process that is totally independent of the flow field. For this purpose, we adopt the Langevin approach, or more formally the theory of conditioned stochastic differential equations. This technique allows us to derive a large variety of stochastic processes: in particular, we introduce a new kind of Brownian bridge ending at two different final points and calculate its fundamental probabilities. This method is also very well suited for generating statistically independent paths. Numerical simulations illustrate our findings.

Linear particle transport in stochastic media is key to such relevant applications as neutron diffusion in randomly mixed immiscible materials, light propagation through engineered optical materials, and inertial confinement fusion, only to name a few. We extend the pioneering work by Adams, Larsen and Pomraning Adams et al. (1989) (recently revisited by Brantley Brantley (2011)) by considering a series of benchmark configurations for mono-energetic and isotropic transport through Markov binary mixtures in dimension d. The stochastic media are generated by resorting to Poisson random tessellations in 1d slab, 2d extruded, and full 3d geometry. For each realization, particle transport is performed by resorting to the Monte Carlo simulation. The distributions of the transmission and reflection coefficients on the free surfaces of the geometry are subsequently estimated, and the average values over the ensemble of realizations are computed. Reference solutions for the benchmark have never been provided before for twoand three-dimensional Poisson tessellations, and the results presented in this paper might thus be useful in order to validate fast but approximated models for particle transport in Markov stochastic media, such as the celebrated Chord Length Sampling algorithm.

Particle transport in Markov mixtures can be addressed by the so-called Chord Length Sampling (CLS) methods, a family of Monte Carlo algorithms taking into account the effects of stochastic media on particle propagation by generating on-the-fly the material interfaces crossed by the random walkers during their trajectories. Such methods enable a significant reduction of computational resources as opposed to reference solutions obtained by solving the Boltzmann equation for a large number of realizations of random media. CLS solutions, which neglect correlations induced by the spatial disorder, are faster albeit approximate, and might thus show discrepancies with respect to reference solutions. In this work we propose a new family of algorithms (called 'Poisson Box Sampling', PBS) aimed at improving the accuracy of the CLS approach for transport in d-dimensional binary Markov mixtures. In order to probe the features of PBS methods, we will focus on three-dimensional Markov media and revisit the benchmark problem originally proposed by Adams, Pomraning Adams et al. (1989) and extended by Brantley Brantley (2011): for these configurations we will compare reference solutions, standard CLS solutions and the new PBS solutions for scalar particle flux, transmission and reflection coefficients. PBS will be shown to perform better than CLS at the expense of a reasonable increase in computational time.

The Chord Length Sampling (CLS) algorithm is a powerful Monte Carlo method that models the effects of stochastic media on particle transport by generating on-the-fly the material interfaces seen by the random walkers during their trajectories. This annealed disorder approach, which formally consists of solving the approximate Levermore-Pomraning equations for linear particle transport, enables a considerable speed-up with respect to transport in quenched disorder, where ensemble-averaging of the Boltzmann equation with respect to all possible realizations is needed. However, CLS intrinsically neglects the correlations induced by the spatial disorder, so that the accuracy of the solutions obtained by using this algorithm must be carefully verified with respect to reference solutions based on quenched disorder realizations. When the disorder is described by Markov mixing statistics, such comparisons have been attempted so far only for one-dimensional geometries, of the rod or slab type. In this work we extend these results to Markov media in two-dimensional (extruded) and three-dimensional geometries, by revisiting the classical set of benchmark configurations originally proposed by Adams, Pomraning Adams et al. (1989) and extended by Brantley Brantley (2011). In particular, we examine the discrepancies between CLS and reference solutions for scalar particle flux and transmission/reflection coefficients as a function of the material properties of the benchmark specifications and of the system dimensionality.

Assessing the impact of random media for eigenvalue problems plays a central role in nuclear reactor physics and criticality safety. In a recent work (Larmier et al., 2018a), we have applied a probabilistic model based on stochastic tessellations in order to describe fuel degradation following severe accidents with partial melting and rearrangement of the resulting debris. The distribution of the multiplication factor and of the kinetics parameters as a function of the mixing statistics model and of the typical correlation length of the tessellation were examined in detail for a benchmark configuration consisting in a fuel assembly with UOX or MOX fuel pins. In this paper, we extend our previous findings by including in the stochastic tessellation model the effects of anisotropy that might result from gravity and material stratification: for this purpose, we adopt the broad class of anisotropic Poisson geometries. We examine the evolution of the statistical properties of the tessellations, including the volume, surface and chord length of the cells for various anisotropy laws, and compare them to the case of isotropic Poisson geometries. Then, we discuss the behaviour of the key observables of interest for eigenvalue problems in anisotropic tessellations by revisiting the fuel assembly benchmark calculations proposed in (Larmier et al., 2018a). The effects of anisotropic random media on the multiplication factor and on the kinetics parameters will be carefully examined.