arXiv subjects

Patricia Gonçalves

Publications and source records attributed to Patricia Gonçalves.

At least 19 recordsLinked to original sources

arXiv2026

Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics

We investigate a boundary-driven Ginzburg-Landau dynamics with long-range interactions. In the hydrodynamic limit, the macroscopic evolution is governed by a fractional heat equation with Dirichlet boundary conditions, while the corresponding stationary profile is characterized by a fractional Laplace equation. We establish a dynamical large deviations principle for the empirical measure and derive the associated stationary large deviations principle for the non-equilibrium steady state, which can be computed semi-explicitly. We further show that the stationary rate function coincides with the quasi-potential associated with the dynamical large deviations functional.

math.PR↗

arXiv2025

Derivation of stochastic Burgers on the line with a Dirichlet boundary condition at the origin

We analyze the \emph{equilibrium fluctuations} of a Hamiltonian chain of oscillators on $\mathbb{Z}$ with an exponential potential, perturbed by a conservative, symmetric noise. Under the canonical \emph{diffusive scaling} $t \mapsto t n^2$ and an interaction strength tuned by $n^{-1/2}$, the fluctuation field is known to converge to the \emph{energy solution} of the stochastic Burgers equation (SBE) on the torus~\cite{ABGS22}. We introduce a \emph{coupled moving heat bath} of strength $n^{-δ}$ acting on the particle system. We prove that for $δ\leq 1$ (the \emph{strong-coupling regime}), the equilibrium fluctuation field converges to the \emph{energy solution of the SBE with a Dirichlet boundary condition at zero}. We provide two distinct analytical characterizations of these boundary solutions, corresponding to different spaces of test functions. Conversely, for $δ> 1$ (the \emph{weak-coupling regime}), the heat bath becomes irrelevant in the scaling limit: the fluctuations converge to the standard SBE on the full line without any boundary condition, reproducing the full-line result of~\cite{GJ14}. Our analysis thus reveals a sharp \emph{critical scaling} in the coupling strength $δ$, which dictates the emergence -- or absence -- of a macroscopic boundary condition from the microscopic perturbation.

math.PR↗

arXiv2025

Advancements in Monte Carlo simulations with gMicroMC: reactive species build-up promotes radical-radical reactions at Flash dose rates

Ultra-high dose rate irradiations to water indicate an enhancement of radical-radical reactions, which could potentially correlate with the Flash effect. The purpose of this work was to extend gMicroMC to support multiple pulse simulations and Flash dose rates, and to investigate, in a pure water model, the mechanisms underlying the enhancement of radical-radical reactions under Flash conditions. gMicroMC, a GPU-based Monte Carlo track-structure algorithm, was extended to simulate multiple pulses. Pure water was exposed to multiple 70 MeV protons pulses delivering up to 20 Gy. The pulse dose rate was set to 2*10^5 and 10^6 Gy/s, while the average dose rate ranged from 0.01 to 100000 Gy/s. The G-values of H2O2 were used to monitor the influence of dose rate on radical-radical reactions. The multiple pulse extension of gMicroMC was validated against Kinetiscope. Multiple pulse simulations indicated an average dose rate threshold. Below it, complete radical depletion occurred within the pulses, leading to constant G-values. Above it, reactive species accumulated throughout the irradiation, resulting in an increase of radical-radical reactions and thus the G-values of H2O2. The average dose rate thresholds were in the order of 10 and 100 Gy/s for pulse dose rates of 2*10^5 and 10^6 Gy/s, respectively. At ultra-high dose rates, the brief intervals between pulses led to a reactive species build-up, which enhanced radical-radical reactions. This build-up is more likely to promote radical-radical reactions than the inter-track mechanism. The advancements in gMicroMC provide a sophisticated tool to study chemical dose rate dependencies.

physics.med-ph↗

arXiv2024

From ABC to KPZ

We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with $N\in\mathbb N$ points, denoted by $\mathbb T_N$, and with three species of particles that we name $A,B$ and $C$, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit $N\to\infty$, to a system of stochastic partial differential equations, that can either be the Ornstein-Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann-Lebesgue lemma which is of independent interest.

math.PR↗

arXiv2023

From Exclusion to Slow and Fast Diffusion

We construct a nearest-neighbour interacting particle system of exclusion type, which illustrates a transition from slow to fast diffusion. More precisely, the hydrodynamic limit of this microscopic system in the diffusive space-time scaling is the parabolic equation $\partial_tρ=\nabla (D(ρ)\nabla ρ)$, with diffusion coefficient $ D(ρ)=mρ^{m-1} $ where $ m\in(0,2] $, including therefore the fast diffusion regime in the range $ m\in(0,1) $, and the porous {medium} equation for $ m\in(1,2) $. The construction of the model is based on the generalized binomial theorem, and interpolates continuously in $ m $ the already known microscopic porous medium model with parameter $ m=2 $, the symmetric simple exclusion process with $ m=1 $, going down to a fast diffusion model up to any $ m>0$. The derivation of the hydrodynamic limit for the local density of particles on the one-dimensional torus is achieved via the entropy method -- with additional technical difficulties depending on the regime (slow or fast diffusion) and where new properties of the porous medium model need to be derived.

math.PR↗

arXiv2022

Hydrodynamic limit for a boundary driven super-diffusive symmetric exclusion

We study the hydrodynamic limit for a model of symmetric exclusion processes with heavy-tailed long jumps and in contact with infinitely extended reservoirs. We show how the corresponding hydrodynamic equations are affected by the parameters defining the model. The hydrodynamic equations are characterized by a class of super-diffusive operators that are given by the regional fractional Laplacian with some additional reaction terms and various boundary conditions. This work solves some questions left open in \cite{BJGO2} about the same model.

math.PR↗

arXiv2022

Non-equilibrium stationary properties of the boundary driven zero-range process with long jumps

We consider the zero-range process with long jumps and in contact with infinitely many reservoirs in its non-equilibrium stationary state. We derive the hydrostatic limit and the Fick's law, which are a consequence of a relationship between the exclusion process and the zero-range process. We also obtain the large deviation principle for the empirical density, i.e. we compute the non-equilibrium free energy.

math.PR↗

arXiv2022

Hydrodynamics for the ABC model with slow/fast boundary

In this article, we consider the ABC model in contact with slow/fast reservoirs. In this model, there is at most one particle per site, which can be of type $α\in\{A,B,C\}$ and particles exchange positions in the discrete set of points $\{1,\cdots, N-1\}$ with a weakly asymmetric rate that depends on the type of particles involved in the exchange mechanism. At the boundary points $x=1, N-1$ particles can be injected or removed with a rate that depends on the type of particles involved. We prove that the hydrodynamic limit, in the diffusive time scale, is given by a system of non-linear coupled equation with several boundary conditions, that depend on the strength of the reservoir's action.

math.PR↗

arXiv2020

On energy solutions to stochastic Burgers equation

In this review we discuss the weak KPZ universality conjecture for a class of 1-d systems whose dynamics conserves one or more quantities. As a prototype example for the former case, we will focus on weakly asymmetric simple exclusion processes, for which the density is preserved and the equilibrium fluctuations are shown to cross from the Edwards-Wilkinson universality class to the KPZ universality class. The crossover depends on the strength of the asymmetry. For the latter case, we will present an exclusion process with three species of particles, known as the ABC model, for which we aim to prove the convergence to a system of coupled stochastic Burgers' equations, i.e. gradient versions of coupled KPZ equations. We will review the recent results on this matter around the notion of energy solutions introduced in [26] for which uniqueness has been proved in [35].

math.PR↗

arXiv2020

Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part I, the critical regime and beyond

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with slowed reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size K placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The hydrodynamic limit is given by the heat equation with non-linear Robin boundary conditions or Neumann boundary conditions, the latter being in the case when the reservoirs are too slow. The proof goes through the entropy method of Guo, Papanicolaou and Varadhan. We also derive the hydrostatic limit for this model, whose proof is based on the method developed by Landim and Tsunoda. We observe that we do not make use of correlation estimates in none of our results.

math-ph↗

arXiv2020

Hydrodynamics for SSEP with non-reversible slow boundary dynamics: Part II, below the critical regime

The purpose of this article is to provide a simple proof of the hydrodynamic and hydrostatic behavior of the SSEP in contact with reservoirs which inject and remove particles in a finite size windows at the extremities of the bulk. More precisely, the reservoirs inject/remove particles at/from any point of a window of size $K$ placed at each extremity of the bulk and particles are injected/removed to the first open/occupied position in that window. The reservoirs have slow dynamics, in the sense that they intervene at speed $N^{-θ}$ w.r.t. the bulk dynamics. In the first part of this article, we treated the case $θ>1$ for which the entropy method can be adapted. We treat here the case where the boundary dynamics is too fast for the Entropy Method to apply. We prove, using duality estimates inspired by previous work by Erignoux, Landim and Xu, that the hydrodynamic limit is given by the heat equation with Dirichlet boundary conditions, where the density at the boundaries is fixed by the parameters of the model.

math.PR↗

arXiv2018

A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions

We prove the hydrodynamic limit for the symmetric exclusion process with long jumps given by a mean zero probability transition rate with infinite variance and in contact with infinitely many reservoirs with density $α$ at the left of the system and $β$ at the right of the system. The strength of the reservoirs is ruled by $κ$N --$θ$ > 0. Here N is the size of the system, $κ$ > 0 and $θ$ $\in$. Our results are valid for $θ$ $\le$ 0. For $θ$ = 0, we obtain a collection of fractional reaction-diffusion equations indexed by the parameter $κ$ and with Dirichlet boundary conditions. Their solutions also depend on $κ$. For $θ$ < 0, the hydrodynamic equation corresponds to a reaction equation with Dirichlet boundary conditions. The case $θ$ > 0 is still open. For that reason we also analyze the convergence of the unique weak solution of the equation in the case $θ$ = 0 when we send the parameter $κ$ to zero. Indeed, we conjecture that the limiting profile when $κ$ $\rightarrow$ 0 is the one that we should obtain when taking small values of $θ$ > 0.

math-ph↗

arXiv2017

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

We consider the weakly asymmetric simple exclusion process on the discrete space $\{1,...,n-1\}$, in contact with stochastic reservoirs, both with density $ρ\in{(0,1)}$ at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ$. Under time diffusive scaling $tn^2$ and for $ρ=\frac12$, when the asymmetry parameter is taken of order $1/ \sqrt n$, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and different from the Cole-Hopf solution.

math.PR↗

arXiv2017

Second order Boltzmann-Gibbs principle for polynomial functions and applications

In this paper we give a new proof of the second order Boltzmann-Gibbs principle. The proof does not impose the knowledge on the spectral gap inequality for the underlying model and it relies on a proper decomposition of the antisymmetric part of the current of the system in terms of polynomial functions. In addition, we fully derive the convergence of the equilibrium fluctuations towards 1) a trivial process in case of supper-diffusive systems, 2) an Ornstein-Uhlenbeck process or the unique energy solution of the stochastic Burgers equation, in case of weakly asymmetric diffusive systems. Examples and applications are presented for weakly and partial asymmetric exclusion processes, weakly asymmetric speed change exclusion processes and hamiltonian systems with exponential interactions.

math.PR↗

arXiv2017

Convergence to the Stochastic Burgers Equation from a degenerate microscopic dynamics

In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [Gonçalves, Jara 2014]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.

math.PR↗

arXiv2017

Crossover to the stochastic Burgers equation for the WASEP with a slow bond

We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^β}2}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\frac1{\vphantom{n^β}2}-\frac{a}{2n^{\vphantomβγ}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\fracα{2n^β}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\fracα{2n^β}-\frac{a}{2n^{\vphantomβγ}}$). Above, $α>0$, $γ\geq β\geq 0$, $a\geq 0$. For $β<1$, we show that the limit density fluctuation field is an Ornstein-Uhlenbeck process defined on the Schwartz space if $γ>\frac12$, while for $γ= \frac12$ it is an energy solution of the stochastic Burgers equation. For $γ\geqβ=1$, it is an Ornstein-Uhlenbeck process associated to the heat equation with Robin's boundary conditions. For $γ\geqβ> 1$, the limit density fluctuation field is an Ornstein-Uhlenbeck process associated to the heat equation with Neumann's boundary conditions.

math.PR↗

arXiv2017

Nonlinear Perturbation of a Noisy Hamiltonian Lattice Field Model: Universality Persistence

In [2] it has been proved that a linear Hamiltonian lattice field perturbed by a conservative stochastic noise belongs to the 3/2-Lévy/Diffusive universality class in the nonlinear fluctuating theory terminology [15], i.e. energy superdiffuses like an asymmetric stable 3/2-Lévy process and volume like a Brownian motion. According to this theory this should remain valid at zero tension if the harmonic potential is replaced by an even potential. In this work we consider a quartic anharmonicity and show that the result obtained in the harmonic case persists up to some small critical value of the anharmonicity.

math-ph↗

arXiv2017

Interpolation process between standard diffusion and fractional diffusion

We consider a Hamiltonian lattice field model with two conserved quantities, energy and volume, perturbed by stochastic noise preserving the two previous quantities. It is known that this model displays anomalous diffusion of energy of fractional type due to the conservation of the volume [5, 3]. We superpose to this system a second stochastic noise conserving energy but not volume. If the intensity of this noise is of order one, normal diffusion of energy is restored while it is without effect if intensity is sufficiently small. In this paper we investigate the nature of the energy fluctuations for a critical value of the intensity. We show that the latter are described by an Ornstein-Uhlenbeck process driven by a Lévy process which interpolates between Brownian motion and the maximally asymmetric 3/2-stable Lévy process. This result extends and solves a problem left open in [4].

math.PR↗