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Patricia Gonçalves

Publications and source records attributed to Patricia Gonçalves.

28 records · Page 2Linked to original sources
arXiv2016

Stochastic Burgers equation from long range exclusion interactions

We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form $p_n(\cdot)=s(\cdot)+γ_na(\cdot)$, such that its symmetric part $s(\cdot)$ is irreducible with finite variance and its antisymmetric part is absolutely bounded by $s(\cdot).$ We prove that under diffusive time scaling and strength of asymmetry $\sqrt n γ_n \to_{n\to\infty} b\neq 0$, the equilibrium density fluctuations are given by the unique energy solution of the stochastic Burgers equation.

math.PR↗
arXiv2014

From normal diffusion to superdiffusion of energy in the evanescent flip noise limit

We consider a harmonic chain perturbed by an energy conserving noise depending on a parameter $γ$. When $γ$ is of order one, the energy diffuses according to the standard heat equation after a space-time diffusive scaling. On the other hand, when $γ=0$, the energy superdiffuses according to a 3/4 fractional heat equation after a subdiffusive space-time scaling. In this paper, we study the existence of a crossover between these two regimes as a function of $γ$.

math.PR↗
arXiv2014

The Einstein relation for the KPZ equation

We compute the non-universal constants in the KPZ equation in one dimension, in terms of the thermodynamical quantities associated to the underlying microscopic dynamics. In particular, we derive the second-order Einstein relation $λ= \frac{1}{2}\frac{d^2}{dρ^2} χ(ρ) D(ρ)$ for the transport coefficient $λ$ of the KPZ equation, in terms of the conserved quantity $ρ$, the diffusion coefficient $D$ and the static compressibility of the system $χ$.

math.PR↗
arXiv2013

On the asymmetric zero-range in the rarefaction fan

We consider the one-dimensional asymmetric zero-range process starting from a step decreasing profile. In the hydrodynamic limit this initial condition leads to the rarefaction fan of the associated hydrodynamic equation. Under this initial condition and for totally asymmetric jumps, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For partially asymmetric jumps we derive the Law of Large Numbers for the position of a second class particle under the initial configuration in which all the positive sites are empty, all the negative sites are occupied with infinitely many first class particles and with a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle, this particle chooses randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation through some sort of renormalization function. By coupling the zero-range with the exclusion process we derive some limiting laws for more general initial conditions.

math.PR↗
arXiv2013

Phase transition in equilibrium fluctuations of symmetric slowed exclusion

We analyze the equilibrium fluctuations of the density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is $αn^-β$, where $α,β\geq{0}$ and $n$ is the scaling parameter. Depending on the regime of $β$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value $β=1$, starting a tagged particle near the slow bond, we obtain a family of gaussian processes indexed in $α$, interpolating a fractional brownian motion of Hurst exponent 1/4 and the degenerate process equal to zero.

math.PR↗
arXiv2013

Nonlinear fluctuations of weakly asymmetric interacting particle systems

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.

math.PR↗
arXiv2013

Phase transition of a Heat equation with Robin's boundary conditions and exclusion process

For a heat equation with Robin's boundary conditions which depends on a parameter $α>0$, we prove that its unique weak solution $ρ^α$ converges, when $α$ goes to zero or to infinity, to the unique weak solution of the heat equation with Neumann's boundary conditions or the heat equation with periodic boundary conditions, respectively. To this end, we use uniform bounds on a Sobolev norm of $ρ^α$ obtained from the hydrodynamic limit of the symmetric slowed exclusion process, plus a careful analysis of boundary terms.

math.PR↗
arXiv2012

Scaling limits of additive functionals of interacting particle systems

Using the renormalization method introduced in \cite{GJ}, we prove what we call the {\em local} Boltzmann-Gibbs principle for conservative, stationary interacting particle systems in dimension $d=1$. As applications of this result, we obtain various scaling limits of additive functionals of particle systems, like the occupation time of a given site or extensive additive fields of the dynamics. As a by-product of these results, we also construct a novel process, related to the stationary solution of the stochastic Burgers equation.

math.PR↗
arXiv2011

Hydrodynamical behavior of symmetric exclusion with slow bonds

We consider the exclusion process in the one-dimensional discrete torus with $N$ points, where all the bonds have conductance one, except a finite number of slow bonds, with conductance $N^{-β}$, with $β\in[0,\infty)$. We prove that the time evolution of the empirical density of particles, in the diffusive scaling, has a distinct behavior according to the range of the parameter $β$. If $β\in [0,1)$, the hydrodynamic limit is given by the usual heat equation. If $β=1$, it is given by a parabolic equation involving an operator $\frac{d}{dx}\frac{d}{dW}$, where $W$ is the Lebesgue measure on the torus plus the sum of the Dirac measure supported on each macroscopic point related to the slow bond. If $β\in(1,\infty)$, it is given by the heat equation with Neumann's boundary conditions, meaning no passage through the slow bonds in the continuum.

math.PR↗