Finite-size effects and percolation properties of Poisson geometries

Abstract

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.

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

“…In this work, we have considered the effects of varying the stochastic tessellation model on the statistical properties of the resulting random media and on the transport-related physical observables, such as the reflection and the transmission probabilities. As such, this paper is a generalization of our previous findings (Larmier et al, 2017(Larmier et al, , 2016, and might be helpful for researchers interested in developing effective kernels for particle transport in disordered media. In order to single out the sensitivity of the simulation results to the various model parameters, we have proposed two benchmark configurations that are simple enough and yet retain the key physical ingredients.…”

“…For binary mixtures, percolation statistics plays an important role Larmier et al (2016); Lepage et al (2011). To fix the ideas, we will consider the percolation properties of the clusters of composition α.…”

“…The site percolation properties of three-dimensional Poisson binary mixtures have been previously addressed in Larmier et al (2016). The percolation threshold of three-dimensional Voronoi tessellations has also been estimated Jerauld et al (1984).…”

“…Table 2: Estimation of the percolation threshold, as a function of the mixing statistics m. Estimates are taken fromLarmier et al (2016);Jerauld et al (1984);Deng and Blöte (2005). The average percolation probability P c (p|L) as a function of p and of Λ ∞ , for a domain of linear size L = 10.…”

Poisson-Box Sampling algorithms for three-dimensional Markov binary mixtures

“…Exact solutions for ϕ , or more generally for some ensemble-averaged functional F[ϕ] of the particle flux, can be obtained using a so-called quenched disorder approach: an ensemble of medium realizations are first sampled from the underlying mixing statistics; then, the linear transport equation is solved for each realization by either deterministic or Monte Carlo methods, and the physical observables of interest F[ϕ] are determined; ensemble averages are finally computed. In a series of recent papers, we have provided reference solutions for particle transport in d-dimensional random media with Markov statistics Larmier et al (2017a,b), where the spatial disorder has been generated by means of homogeneous and isotropic ddimensional Poisson tessellations Larmier et al (2016).…”