A continuous time random walk model of transport in variably saturated heterogeneous porous media

We propose a unified physical framework for transport in variably saturated porous media. This approach allows fluid flow and solute migration to be treated as ensemble averages of fluid and solute particles, respectively. We consider the cases of homogeneous and heterogeneous porous materials. Within a fractal mobile-immobile (MIM) continuous time random walk framework, the heterogeneity will be characterized by algebraically decaying particle retention-times. We derive the corresponding (nonlinear) continuum limit partial differential equations and we compare their solutions to Monte Carlo simulation results. The proposed methodology is fairly general and can be used to track fluid and solutes particles trajectories, for a variety of initial and boundary conditions.Comment: 12 pages, 9 figure

Counting statistics: a Feynman-Kac perspective

By building upon a Feynman-Kac formalism, we assess the distribution of the number of hits in a given region for a broad class of discrete-time random walks with scattering and absorption. We derive the evolution equation for the generating function of the number of hits, and 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 hits on the half-line.Comment: 10 pages, 6 figure

Universal properties of branching random walks in confined geometries

Characterizing the occupation statistics of a radiation flow through confined geometries is key to such technological issues as nuclear reactor design and medical diagnosis. This amounts to assessing the distribution of the travelled length $\ell$ and the number of collisions $n$ performed by the underlying stochastic transport process, for which remarkably simple Cauchy-like formulas were established in the case of branching Pearson random walks with exponentially distributed jumps. In this Letter, we show that such formulas strikingly carry over to the much broader class of branching processes with arbitrary jumps, provided that scattering is isotropic and the average jump size is finite.Comment: 5 pages, 3 figure

The critical catastrophe revisited

The neutron population in a prototype model of nuclear reactor can be described in terms of a collection of particles confined in a box and undergoing three key random mechanisms: diffusion, reproduction due to fissions, and death due to absorption events. When the reactor is operated at the critical point, and fissions are exactly compensated by absorptions, the whole neutron population might in principle go to extinction because of the wild fluctuations induced by births and deaths. This phenomenon, which has been named critical catastrophe, is nonetheless never observed in practice: feedback mechanisms acting on the total population, such as human intervention, have a stabilizing effect. In this work, we revisit the critical catastrophe by investigating the spatial behaviour of the fluctuations in a confined geometry. When the system is free to evolve, the neutrons may display a wild patchiness (clustering). On the contrary, imposing a population control on the total population acts also against the local fluctuations, and may thus inhibit the spatial clustering. The effectiveness of population control in quenching spatial fluctuations will be shown to depend on the competition between the mixing time of the neutrons (i.e., the average time taken for a particle to explore the finite viable space) and the extinction time.Comment: 16 pages, 6 figure

A model of dispersive transport across sharp interfaces between porous materials

Recent laboratory experiments on solute migration in composite porous columns have shown an asymmetry in the solute arrival time upon reversal of the flow direction, which is not explained by current paradigms of transport. In this work, we propose a definition for the solute flux across sharp interfaces and explore the underlying microscopic particle dynamics by applying Monte Carlo simulation. Our results are consistent with previous experimental findings and explain the observed transport asymmetry. An interpretation of the proposed physical mechanism in terms of a flux rectification is also provided. The approach is quite general and can be extended to other situations involving transport across sharp interfaces.Comment: 4 pages, 4 figure

Properties of branching exponential flights in bounded domains

Branching random flights are key to describing the evolution of many physical and biological systems, ranging from neutron multiplication to gene mutations. When their paths evolve in bounded regions, we establish a relation between the properties of trajectories starting on the boundary and those starting inside the domain. Within this context, we show that the total length travelled by the walker and the number of performed collisions in bounded volumes can be assessed by resorting to the Feynman-Kac formalism. Other physical observables related to the branching trajectories, such as the survival and escape probability, are derived as well.Comment: 5 pages, 2 figure