arXiv subjects

Andrea Zoia

Publications and source records attributed to Andrea Zoia.

At least 37 records · Page 2Linked to original sources

arXiv2014

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.

cond-mat.stat-mech↗

arXiv2014

Cauchy's formulas for random walks in bounded domains

Cauchy's formula was originally established for random straight paths crossing a body $B \subset \mathbb{R}^{n}$ and basically relates the average chord length through $B$ to the ratio between the volume and the surface of the body itself. The original statement was later extended in the context of transport theory so as to cover the stochastic paths of Pearson random walks with exponentially distributed flight lengths traversing a bounded domain. Some heuristic arguments suggest that Cauchy's formula may also hold true for Pearson random walks with arbitrarily distributed flight lengths. For such a broad class of stochastic processes, we rigorously derive a generalized Cauchy's formula for the average length travelled by the walkers in the body, and show that this quantity depends indeed only on the ratio between the volume and the surface, provided that some constraints are imposed on the entrance step of the walker in $B$. Similar results are obtained also for the average number of collisions performed by the walker in $B$, and an extension to absorbing media is discussed.

cond-mat.stat-mech↗

arXiv2013

Spatial extent of an outbreak in animal epidemics

Characterizing the spatial extent of epidemics at the outbreak stage is key to controlling the evolution of the disease. At the outbreak, the number of infected individuals is typically small, so that fluctuations around their average are important: then, it is commonly assumed that the susceptible-infected-recovered (SIR) mechanism can be described by a stochastic birth-death process of Galton-Watson type. The displacements of the infected individuals can be modelled by resorting to Brownian motion, which is applicable when long-range movements and complex network interactions can be safely neglected, as in case of animal epidemics. In this context, the spatial extent of an epidemic can be assessed by computing the convex hull enclosing the infected individuals at a given time. We derive the exact evolution equations for the mean perimeter and the mean area of the convex hull, and compare them with Monte Carlo simulations.

q-bio.PE↗

arXiv2012

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.

cond-mat.stat-mech↗

arXiv2012

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 $\ell_V$ that the process travels in a given region $V$ of the phase space, or the number of visits $n_V$ 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 $\ell_V$ and $n_V$ and in particular deriving explicit moment formulas. Some other significant physical observables associated to $\ell_V $ and $n_V$, 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.

cond-mat.stat-mech↗

arXiv2012

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 $n_V$ of the walker to a given region $V$ in the phase space. Feynman-Kac formulas for the residence times of Markovian processes are recovered in the diffusion limit.

cond-mat.stat-mech↗

arXiv2011

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.

cond-mat.stat-mech↗

arXiv2011

Collision statistics for random flights with anisotropic scattering and absorption

For a broad class of random walks with anisotropic scattering kernel 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 (1d) random flights with exponential displacements, where analytical calculations can be carried out.

cond-mat.stat-mech↗

arXiv2011

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 {\em 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 {\em rod problem} (a 1d system), and discuss the relevance of the obtained results in the context of Monte Carlo estimators.

cond-mat.stat-mech↗

arXiv2011

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

In this paper we analyze some aspects of {\em exponential flights}, a stochastic process that governs the evolution of many random transport phenomena, such as neutron propagation, chemical/biological species migration, or 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 novel exact (where possible) or asymptotic results, among which the stationary probability density for 2d systems, a long-standing issue in Physics, and the mean residence time in a given volume. Bounded or unbounded, as well as scattering or absorbing domains are examined, and Monte Carlo simulations are performed so as to support our findings.

cond-mat.stat-mech↗

arXiv2011

Collision number statistics for transport processes

Many physical observables can be represented as a particle spending some random time within a given domain. For a broad class of transport-dominated processes, we detail how it is possible to express the moments of the number of particle collisions in an arbitrary volume in terms of repeated convolutions of the ensemble equilibrium distribution. This approach is shown to generalize the celebrated Kac formula for the moments of residence times, which is recovered in the diffusion limit. Some practical applications are illustrated for bounded, unbounded and absorbing domains.

cond-mat.stat-mech↗

arXiv2009

Hitting probability for anomalous diffusion processes

We present the universal features of the hitting probability $Q(x,L)$, the probability that a generic stochastic process starting at $x$ and evolving in a box $[0,L]$ hits the upper boundary $L$ before hitting the lower boundary at 0. For a generic self-affine process (describing, for instance, the polymer translocation through a nanopore) we show that $Q(x,L)=Q(x/L)$ and the scaling function $Q(z)\sim z^ϕ$ as $z\to 0$ with $ϕ=θ/H$ where $H$ and $θ$ are respectively the Hurst exponent and the persistence exponent of the process. This result is verified in several exact calculations including when the process represents the position of a particle diffusing in a disordered potential. We also provide numerical supports for our analytical results.

cond-mat.stat-mech↗

arXiv2009

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.

cond-mat.stat-mech↗

arXiv2009

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.

cond-mat.stat-mech↗

arXiv2009

Mass transport subject to time-dependent flow with non-uniform sorption in porous media

We address the description of solutes flow with trapping processes in porous media. Starting from a small-scale model for tracer particles trajectories, we derive the corresponding governing equations for the concentration of the mobile and immobile phases. We show that this formulation is fairly general and can easily take into account non-constant coefficients and in particular space-dependent sorption rates. The transport equations are solved numerically and a comparison with Monte Carlo particle-tracking simulations of spatial contaminant profiles and breakthrough curves is proposed, so to illustrate the obtained results.

cond-mat.stat-mech↗

arXiv2009

Spatial and temporal features of dense contaminant transport: experimental investigation and numerical modeling

We investigate the spatial and temporal features of dense contaminant plumes dynamics in porous materials. Our analysis is supported by novel experimental results concerning pollutant concentration profiles inside a vertical column setup. We describe the experimental methods and elucidate the salient outcomes of the measurements, with focus on miscible fluids in homogeneous saturated media. By resorting to a finite elements approach, we numerically solve the equations that rule the pollutants migration and compare the simulation results with the experimental data. Finally, we qualitatively explore the interfacial dynamics behavior between the dense contaminant plume and the lighter resident fluid that saturates the column.

physics.flu-dyn↗

arXiv2009

A nonlinear random walk approach to concentration-dependent contaminant transport in porous media

We propose a nonlinear random walk model to describe the dynamics of dense contaminant plumes in porous media. A coupling between concentration and velocity fields is found, so that transport displays non-Fickian features. The qualitative behavior of the pollutant spatial profiles and moments is explored with the help of Monte Carlo simulation, within a Continuous Time Random Walk approach. Model outcomes are then compared with experimental measurements of variable-density contaminant transport in homogeneous and saturated vertical columns.

cond-mat.stat-mech↗

arXiv2009

Asymptotic behavior of self-affine processes in semi-infinite domains

We propose to model the stochastic dynamics of a polymer passing through a pore (translocation) by means of a fractional Brownian motion, and study its behavior in presence of an absorbing boundary. Based on scaling arguments and numerical simulations, we present a conjecture that provides a link between the persistence exponent $θ$ and the Hurst exponent $H$ of the process, thus sheding light on the spatial and temporal features of translocation. Furthermore, we show that this conjecture applies more generally to a broad class of self affine processes undergoing anomalous diffusion in bounded domains, and we discuss some significant examples.

cond-mat.stat-mech↗