arXiv subjects

Otávio Menezes

Publications and source records attributed to Otávio Menezes.

7 recordsLinked to original sources

arXiv2021

Sharp Convergence to Equilibrium for the SSEP with Reservoirs

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 $\|\mathcal N (m,1) - \mathcal N(0,1)\|_{\mathrm{TV}}$.

math.PR↗

arXiv2021

Variable speed symmetric random walk driven by symmetric exclusion

We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.

math.PR↗

arXiv2018

Non-equilibrium Fluctuations of Interacting Particle Systems

We obtain the large scale limit of the fluctuations around its hydrodynamic limit of the density of particles of a weakly asymmetric exclusion process in dimension up to three. The proof is based upon a sharp estimate on the relative entropy of the law of the process with respect to product reference measures associated to the hydrodynamic limit profile, which holds in any dimension and is of independent interest. As a corollary of this entropy estimate, we obtain some quantitative bounds on the speed of convergence of the aforementioned hydrodynamic limit.

math.PR↗

arXiv2018

Non-equilibrium and stationary fluctuations for the SSEP with slow boundary

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.

math.PR↗

arXiv2018

Non-equilibrium fluctuations for a reaction-diffusion model via relative entropy

We look at a superposition of symmetric simple exclusion and Glauber dynamics in the discrete torus in dimension 1. For this model, we prove that the fluctuations around the hydrodynamic limit are described, in the diffusive scale, by an infinite-dimensional Ornstein-Uhlenbeck process. Our proof technique is an adaptation of Yau's Relative Entropy Method that is robust enough to be adapted to other exclusion models. To cut the technical details to a minimum, we assume that the process starts from a product measure with a custom-chosen density, for which the solution of the hydrodynamic equation is stationary. Although we prove fluctuations only in dimension 1, we provide an estimate on the entropy production that holds for any dimension and a proof of the Boltzmann-Gibbs principle that applies in dimension smaller than 3.

math.PR↗

arXiv2018

Symmetric exclusion as a random environment: invariance principle

We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.

math.PR↗

arXiv2017

Exclusion process with slow boundary

We study the hydrodynamic and the hydrostatic behavior of the Simple Symmetric Exclusion Process with \emph{slow boundary}. The term \emph{slow boundary} means that particles can be born or die at the boundary sites, at a rate proportional to $N^{-θ}$, where $θ> 0$ and $N$ is the scaling parameter. In the bulk, the particles exchange rate is equal to $1$. In the hydrostatic scenario, we obtain three different linear profiles, depending on the value of the parameter $θ$; in the hydrodynamic scenario, we obtain that the time evolution of the spatial density of particles, in the diffusive scaling, is given by the weak solution of the heat equation, with boundary conditions that depend on $ θ$. If $θ\in(0,1)$, we get Dirichlet boundary conditions, (which is the same behavior if $θ=0$, see \cite{f}); if $θ=1$, we get Robin boundary conditions; and, if $θ\in(1,\infty)$, we get Neumann boundary conditions.

math.PR↗