Alain Mazzolo

On the properties of the chord length distribution, from integral geometry to reactor physics

Abstract In a couple of recent papers, Sjostrand [Ann Nucl Eng 29 (2002) 1607] and Kruijf and Kloosterman [Ann Nucl Eng 30 (2003) 549] addressed the question of the meaning of the average chord length in reactor physics. This paper offers a link between integral geometry (where chord length distribution and its properties come from) and its applications in reactor physics. In particular we specify which relations between chord length moments and simple geometric properties of the body remain valid for non-convex bodies. Monte Carlo simulations illustrates these points.

Branching exponential flights: travelled lengths and collision statistics

The evolution of several physical and biological systems, ranging from neutron transport in multiplying media to epidemics or population dynamics, can be described in terms of branching exponential flights, a stochastic process which couples a Galton–Watson birth–death mechanism with random spatial displacements. Within this context, one is often called to assess the length lV that the process travels in a given region V of the phase space, or the number of visits nV to this same region. In this paper, we address this issue by resorting to the Feynman–Kac formalism, which allows characterizing the full distribution of lV and nV and in particular deriving explicit moment formulas. Some other significant physical observables associated to lV and nV, such as the survival probability, are discussed as well, and results are illustrated by revisiting the classical example of the rod model in nuclear reactor physics.

Collision densities and mean residence times for d-dimensional exponential flights.

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 or biological species migration, and electron motion. We introduce a general framework for d-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 exact (where possible) or asymptotic results, among which are the stationary probability density for two-dimensional systems, a long-standing issue in physics, and the mean residence time in a given volume. Bounded or unbounded domains as well as scattering or absorbing domains are examined, and Monte Carlo simulations are performed so as to support our findings.

Properties of branching exponential flights in bounded domains

In a series of recent works, important results have been reported concerning the statistical properties of exponential flights evolving in bounded domains, a widely adopted model for finite-speed transport phenomena (Blanco S. and Fournier R., Europhys. Lett., 61 (2003) 168; Mazzolo A., Europhys. Lett., 68 (2004) 350; Benichou O. et al., Europhys. Lett., 70 (2005) 42). Motivated by physical and biological systems where random spatial displacements are coupled with Galton-Watson birth-death mechanisms, such as neutron multiplication, diffusion of reproducing bacteria or spread of epidemics, in this letter we extend those results in two directions, via a Feynman-Kac formalism. First, we characterize the occupation statistics of exponential flights in the presence of absorption and branching, and give explicit moment formulas for the total length travelled by the walker and the number of performed collisions in a given domain. Then, we show that the survival and escape probability can be derived as well by resorting to a similar approach.

Finite-size effects and percolation properties of Poisson geometries

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.

Benchmark solutions for transport in d-dimensional Markov binary mixtures

Abstract 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 [1] (recently revisited by Brantley [2] ) 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 1 d slab, 2 d extruded, and full 3 d 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 two- and 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.

Monte Carlo particle transport in random media: The effects of mixing statistics

Abstract Particle transport in random media obeying a given mixing statistics is key in several applications in nuclear reactor physics and more generally in diffusion phenomena emerging in optics and life sciences. Exact solutions for the ensemble-averaged physical observables are hardly available, and several approximate models have been thus developed, providing a compromise between the accurate treatment of the disorder-induced spatial correlations and the computational time. In order to validate these models, it is mandatory to use reference solutions in benchmark configurations, typically obtained by explicitly generating by Monte Carlo methods several realizations of random media, simulating particle transport in each realization, and finally taking the ensemble averages for the quantities of interest. In this context, intense research efforts have been devoted to Poisson (Markov) mixing statistics, where benchmark solutions have been derived for transport in one-dimensional geometries. In a recent work, we have generalized these solutions to two and three-dimensional configurations, and shown how dimension affects the simulation results. In this paper we will examine the impact of mixing statistics: to this aim, we will compare the reflection and transmission probabilities, as well as the particle flux, for three-dimensional random media obtained by using Poisson, Voronoi and Box stochastic tessellations. For each tessellation, we will furthermore discuss the effects of varying the fragmentation of the stochastic geometry, the material compositions, and the cross sections of the background materials.

Discrete Feynman-Kac formulas for branching random walks

Branching random walks are key to the description of several physical and biological systems, such as neutron multiplication, genetics and population dynamics. For a broad class of such processes, in this Letter we derive the discrete Feynman-Kac equations for the probability and the moments of the number of visits

Collision-number statistics for transport processes.

n_V

Counting statistics: a Feynman-Kac perspective.

of the walker to a given region