Residence times of branching diffusion processes.

The residence time of a branching Brownian process is the amount of time that the mother particle and all its descendants spend inside a domain. Using the Feynman-Kac formalism, we derive the residence-time equation as well as the equations for its moments for a branching diffusion process with an arbitrary number of descendants. This general approach is illustrated with simple examples in free space and in confined geometries where explicit formulas for the moments are obtained within the long time limit. In particular, we study in detail the influence of the branching mechanism on those moments. The present approach can also be applied to investigate other additive functionals of branching Brownian process.

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.

Neutron fluctuations: The importance of being delayed.

The neutron population in a nuclear reactor is subject to fluctuations in time and in space due to the competition of diffusion by scattering, births by fission events, and deaths by absorptions. As such, fission chains provide a prototype model for the study of spatial clustering phenomena. In order for the reactor to be operated in stationary conditions at the critical point, the population of prompt neutrons instantaneously emitted at fission must be in equilibrium with the much smaller population of delayed neutrons, emitted after a Poissonian time by nuclear decay of the fissioned nuclei. In this work, we will show that the delayed neutrons, although representing a tiny fraction of the total number of neutrons in the reactor, actually have a key impact on the fluctuations, and their contribution is very effective in quenching the spatial clustering.

Clustering of branching Brownian motions in confined geometries.

We study the evolution of a collection of individuals subject to Brownian diffusion, reproduction, and disappearance. In particular, we focus on the case where the individuals are initially prepared at equilibrium within a confined geometry. Such systems are widespread in physics and biology and apply for instance to the study of neutron populations in nuclear reactors and the dynamics of bacterial colonies, only to name a few. The fluctuations affecting the number of individuals in space and time may lead to a strong patchiness, with particles clustered together. We show that the analysis of this peculiar behavior can be rather easily carried out by resorting to a backward formalism based on the Green's function, which allows the key physical observables, namely, the particle concentration and the pair correlation function, to be explicitly derived.

Counting statistics: a Feynman-Kac perspective.

By building upon a Feynman-Kac formalism, we assess the distribution of the number of collisions in a given region for a broad class of discrete-time random walks in absorbing and nonabsorbing media. We derive the evolution equation for the generating function of the number of collisions, and we complete our analysis by examining the moments of the distribution and their relation to the walker equilibrium density. Some significant applications are discussed in detail: in particular, we revisit the gambler's ruin problem and generalize to random walks with absorption the arcsine law for the number of collisions on the half-line.

Residence time and collision statistics for exponential flights: the rod problem revisited.

Many random transport phenomena, such as radiation propagation, chemical-biological species migration, or electron motion, can be described in terms of particles performing exponential flights. For such processes, we sketch a general approach (based on the Feynman-Kac formalism) that is amenable to explicit expressions for the moments of the number of collisions and the residence time that the walker spends in a given volume as a function of the particle equilibrium distribution. We then illustrate the proposed method in the case of the so-called rod problem (a one-dimensional system), and discuss the relevance of the obtained results in the context of Monte Carlo estimators.

Collision-number statistics for transport processes.

Physical observables are often represented as walkers performing random displacements. When the number of collisions before leaving the explored domain is small, the diffusion approximation leads to incongruous results. In this Letter, we explicitly derive an explicit formula for the moments of the number of particle collisions in an arbitrary volume, for a broad class of transport processes. This approach is shown to generalize the celebrated Kac formula for the moments of residence times. Some applications are illustrated for bounded, unbounded and absorbing domains.

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.

Collision statistics for random flights with anisotropic scattering and absorption.

For a broad class of random walks with anisotropic scattering kernels and absorption, we derive explicit formulas that allow expressing the moments of the collision number n(V) performed in a volume V as a function of the particle equilibrium distribution. Our results apply to arbitrary domains V and boundary conditions, and allow assessing the hitting statistics for systems where the typical displacements are comparable to the domain size, so that the diffusion limit is possibly not attained. An example is discussed for one-dimensional random flights with exponential displacements, where analytical calculations can be carried out.