arXiv subjects
Publications and source records attributed to Patrícia Gonçalves.
In this article we derive the fractional porous medium equation for any power of the fractional Laplacian as the hydrodynamic limit of a microscopic dynamics of random particles with long range interactions, but the jump rate highly depends on the occupancy near the sites where the interactions take place.
We analyze the generalized symmetric exclusion process, which allows at most $α$ particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter $θ\in\mathbb R$. We prove that the hydrodynamic behavior is given by the heat equation and depending on the value of $θ$, the equation is supplemented with different boundary conditions. Setting $α= 1$ we find the results known in [1] for the symmetric simple exclusion process.
In this article we analyse the hydrodynamical behavior of the symmetric exclusion process with long jumps and in the presence of a slow barrier. The jump rates for fast bonds are given by a transition probability $p(\cdot)$ which is symmetric and has finite variance, while for slow bonds the jump rates are given $p(\cdot)αn^{-β}$ (with $α>0$ and $β\geq 0$), and correspond to jumps from $\mathbb{Z}_{-}^{*}$ to $\mathbb N$. We prove that: if there is a fast bond from $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, then the hydrodynamic limit is given by the heat equation with no boundary conditions; otherwise, it is given by the previous equation if $0\leq β<1$, but for $β\geq 1$ boundary conditions appear, namely, we get Robin (linear) boundary conditions if $β=1$ and Neumann boundary conditions if $β>1$.
We construct a non reversible exclusion process with Bernoulli product invariant measure and having, in the diffusive hydrodynamic scaling, a non symmetric diffusion matrix, that can be explicitly computed. The antisymmetric part does not affect the evolution of the density but it is relevant for the evolution of the current. Switching on a weak external field we obtain a symmetric mobility matrix that is related just to the symmetric part of the diffusion matrix by an Einstein relation. We argue that this fact is typical within a class of generalized gradient models. We consider for simplicity the model in dimension $d=2$, but a similar behavior can be also obtained in higher dimensions.
We analyse the hydrodynamical behavior of the long jumps symmetric exclusion process in the presence of a slow barrier. The jump rates are given by a symmetric transition probability $p(\cdot)$ with infinite variance. When jumps occur from $\mathbb{Z}_{-}^{*}$ to $\mathbb N$ the rates are slowed down by a factor $αn^{-β}$ (with $α>0$ and $β\geq 0$). We obtain several partial differential equations given in terms of the regional fractional Laplacian on $\mathbb R^*$ and with different boundary conditions. Surprisingly, in opposition to the diffusive regime, we get different regimes depending on whether $α=1$ (all bonds with the same rate) or $α\neq 1$.
The purpose of this article is to study the hydrodynamic limit of the symmetric exclusion process with long jumps and in contact with infinitely extended reservoirs for a particular critical regime. The jumps are given in terms of a transition probability that can have finite or infinite variance and the hydrodynamic equation is a diffusive equation, in the former case, or a fractional equation, in the latter case. In this work we treat the critical case, that is, when the variance is infinite and grows logarithmically with the size of the system. This is the case in which there is a transition from diffusive to super-diffusive behavior.
We consider the symmetric simple exclusion process evolving on the interval of length $n-1$ in contact with reservoirs of density $ρ\in (0,1)$ at the boundary. We use Yau's relative entropy method to show that if the initial measure is associated with a profile $u_0:[0,1] \to (0,1)$, then at explicit times $t^n(b)$ that depend on $u_0$, the distance to equilibrium, in total variation distance, converges, as $n \to \infty$, to a profile $\mathcal G(γe^{-b})$. The parameter $γ$ also depends on the initial profile $u_0$ and $\mathcal G(m)$ stands for the total variation distance $\|\mathcal N (m,1) - \mathcal N(0,1)\|_{\mathrm{TV}}$.
We prove a large deviations principle for the empirical measure of the one dimensional symmetric simple exclusion process in contact with reservoirs. The dynamics of the reservoirs is slowed down with respect to the dynamics of the system, that is, the rate at which the system exchanges particles with the boundary reservoirs is of order $n^{-θ}$, where $n$ is number of sites in the system, $θ$ is a non negative parameter, and the system is taken in the diffusive time scaling. Two regimes are studied here, the subcritical $θ\in(0,1)$ whose hydrodynamic equation is the heat equation with Dirichlet boundary conditions and the supercritical $θ\in(1,+\infty)$ whose hydrodynamic equation is the heat equation with Neumann boundary conditions. In the subcritical case $θ\in(0,1)$, the rate function that we obtain matches the rate function corresponding to the case $θ=0$ which was derived on previous works (see \cite{blm,flm}), but the challenges we faced here are much trickier. In the supercritical case $θ\in(1,+\infty)$, the rate function is equal to infinity outside the set of trajectories which preserve the total mass, meaning that, despite the discrete system exchanges particles with the reservoirs, this phenomena has super-exponentially small probability in the diffusive scaling limit.
We study the hydrodynamic and hydrostatic limits of the one-dimensional open symmetric inclusion process with slow boundary. Depending on the value of the parameter tuning the interaction rate of the bulk of the system with the boundary, we obtain a linear heat equation with either Dirichlet, Robin or Neumann boundary conditions as hydrodynamic equation. In our approach, we combine duality and first-second class particle techniques to reduce the scaling limit of the inclusion process to the limiting behavior of a single, non-interacting, particle.
We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.
We study the convergence of the weak solution of the porous medium equation with a type of Robin boundary conditions, by tuning a parameter either to zero or to infinity. The convergence is in the strong sense, with respect to the $L^2$-norm, and the limiting function solves the same equation with Neumann (resp. Dirichlet) boundary conditions when the parameter is taken to zero (resp. infinity). Our approach is to consider an underlying microscopic dynamics whose space-time evolution of the density is ruled by the solution of those equations and from this, we derive sufficiently strong energy estimates which are the keystone to the proof of our convergence result.
By a probabilistic method we provide an explicit fundamental solution of the Cauchy problem associated to the heat equation on the half-line with constant drift and Dirichlet boundary condition at zero.
We analyze the hydrodynamic behavior of the porous medium model in a discrete space $\{0,\ldots, n\}$, where the sites $0$ and $n$ stand for reservoirs. Our strategy relies on the entropy method of Guo, Papanicolau and Varadhan. However, this method cannot be straightforwardly applied, since there are configurations that do not evolve according to the dynamics (blocked configurations). In order to avoid this problem, we slightly perturbed the dynamics in such a way that the macroscopic behavior of the system keeps following the porous medium equation, but with boundary conditions which depend on the reservoirs strength's.
In [AAV] Amir, Angel and Valk{ó} studied a multi-type version of the totally asymmetric simple exclusion process (TASEP) and introduced the TASEP speed process, which allowed them to answer delicate questions about the joint distribution of the speed of several second-class particles in the TASEP rarefaction fan. In this paper we introduce the analogue of the TASEP speed process for the totally asymmetric zero-range process (TAZRP), and use it to obtain new results on the joint distribution of the speed of several second-class particles in the TAZRP with a reservoir. There is a close link from the speed process to questions about stationary distributions of multi-type versions of the TAZRP; for example we are able to give a precise description of the contents of a single site in equilibrium for a multi-type TAZRP with continuous labels.
We derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with slowed stochastic reservoirs which are regulated by a factor $n^{-θ}$. Depending on the range of $θ$ we obtain processes with various boundary conditions. Moreover, as a consequence of the previous result we deduce the non-equilibrium stationary fluctuations by using the matrix ansatz method which gives us information on the stationary measure for the model. The main ingredient to prove these results is the derivation of precise bounds on the two-point space-time correlation function, which are a consequence of precise bounds on the transition probability of some underlying random walks.
We prove the non-equilibrium fluctuations for the one-dimensional symmetric simple exclusion process with a slow bond. This generalizes a result of T. Franco, A. Neumann and P. Gonçalves (2013), which dealt with the equilibrium fluctuations. The foundation stone of our proof is a precise estimate on the correlations of the system, and that is by itself one of the main novelties of this paper. To obtain these estimates, we first deduce a spatially discrete PDE for the covariance function and we relate it to the local times of a random walk in a non-homogeneous environment via Duhamel's principle. Projection techniques and coupling arguments reduce the analysis to the problem of studying the local times of the classical random walk. We think that the method developed here can be applied to a variety of models, and we provide a discussion on this matter.
Detailed radiation analysis of instruments flown in space is critical to ensure mission safety, often requiring the use of state-of-the-art particle transport simulation tools. Geant4 is one of the most powerful toolkits to simulate the interaction and the passage of particles through matter. This framework however, is not prepared to receive Standard for The Exchange of Product data (STEP) files, the most versatile Computer-Aided Design (CAD) format, as inputs requiring previous conversion to other formats. This, especially when the instruments have complex shapes, and/or a large number of volumes may lead to loss of detail and under or overestimation of the quantities under study. Though several solutions have been proposed to import complex geometries into Geant4, so far, only commercial options are available. In this paper we present a new tool, GUIMesh, that embeds FreeCAD libraries, an open-source CAD editor, to tessellate volumes, and convert them to Geometry Description Markup Language (GDML), a Geant4 readable format, in a straightforward way. Several degrees of freedom are given to the user regarding the mesh and choice of material. Different geometries were tested for p material definition, geometry and navigation errors, successfully validating the method used.
We consider the symmetric exclusion process with jumps given by a symmetric, translation invariant, transition probability $p(\cdot)$. The process is put in contact with stochastic reservoirs whose strength is tuned by a parameter $θ\in\mathbb R$. Depending on the value of the parameter $θ$ and the range of the transition probability $p(\cdot)$ we obtain the hydrodynamical behavior of the system. The type of hydrodynamic equation depends on whether the underlying probability $p(\cdot)$ has finite or infinite variance and the type of boundary condition depends on the strength of the stochastic reservoirs, that is, it depends on the value of $θ$. More precisely, when $p(\cdot)$ has finite variance we obtain either a reaction or reaction-diffusion equation with Dirichlet boundary conditions or the heat equation with different types of boundary conditions (of Dirichlet, Robin or Neumann type). When $p(\cdot)$ has infinite variance we obtain a fractional reaction-diffusion equation given by the regional fractional Laplacian with Dirichlet boundary conditions but for a particular strength of the reservoirs.