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Marielle Simon

Publications and source records attributed to Marielle Simon.

At least 19 recordsLinked to original sources
arXiv2026

The hydrodynamic limit for the energy transport in a stochastically perturbed harmonic chain of oscillators

We consider a pinned harmonic chain perturbed by random velocity flips, which conserve the total energy. Previous results have established the diffusive hydrodynamic limit for the energy profile at the level of expectations. In this article, under suitable fourth-moment bounds on the initial data, we prove for the first time a law of large numbers for the empirical energy distribution. Our approach is based on the Wigner distribution and shows that its random fluctuations vanish in the macroscopic limit. Consequently, the empirical energy profile converges in probability to the deterministic solution of the corresponding heat equation.

math.PR↗
arXiv2025

Scaling Relations For The CLG's Critical Exponents

We consider, in any dimension, the constrained lattice gas introduced by Rossi et al., which is an exclusion process on a d-dimensional lattice following the additional constraint that only particles with at least one occupied neighbour can jump. In dimension d > 2, this model features self-organized criticality at some critical density of particles. Numerical simulations predict the existence of scaling exponents close to criticality, and several relations can be derived between these exponents. The goal of this article is to give a mathematical framework for these relations, which have been numerically established in a companion article.

math-ph↗
arXiv2024

Heat flow in a periodically forced, unpinned thermostatted chain

We prove the hydrodynamic limit for a one-dimensional harmonic chain of interacting atoms with a random flip of the momentum sign. The system is open: at the left boundary it is attached to a heat bath at temperature $T_-$, while at the right endpoint it is subject to an action of a force which reads as $\bar F + \frac 1{\sqrt n} \widetilde{\mathcal F} (n^2 t)$, where $\bar F \ge0$ and $\widetilde{\mathcal F}(t)$ is a periodic function. Here $n$ is the size of the microscopic system. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities - the volume stretch and the energy - converge, as $n\to+\infty$, to the solution of a non-linear diffusive system of conservative partial differential equations with a Dirichlet type and Neumann boundary conditions on the left and the right endpoints, respectively.

math.PR↗
arXiv2024

Hydrodynamic limit for a chain with thermal and mechanical boundary forces

We prove the hydrodynamic limit for a one dimensional harmonic chain with a random flip of the momentum sign. The system is open and subject to two thermostats at the boundaries and to an external tension at one of the endpoints. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a non-linear diffusive system of conservative partial differential equations.

math.PR↗
arXiv2024

Microscopic derivation of non-local models with anomalous diffusions from stochastic particle systems

This paper considers a large class of nonlinear integro-differential scalar equations which involve an anomalous diffusion (e.g. driven by a fractional Laplacian) and a non-local singular convolution kernel. Each of those singular equations is obtained as the macroscopic limit of an interacting particle system modeled as N coupled stochastic differential equations driven by L\'evy processes. In particular we derive quantitative estimates between the microscopic empirical measure of the particle system and the solution to the limit equation in some non-homogeneous Sobolev space. Our result only requires very weak regularity on the interaction kernel, therefore it includes numerous applications, e.g.: the 2d turbulence model (including the quasi-geostrophic equation) in sub-critical regime, the 2d generalized Navier-Stokes equation, the fractional Keller-Segel equation in any dimension, and the fractal Burgers equation.

math.PR↗
arXiv2024

Hydrodynamic behavior near dynamical criticality of a facilitated conservative lattice gas

We investigate a $2d$-conservative lattice gas exhibiting a dynamical active-absorbing phase transition with critical density $\rho_c$. We derive the hydrodynamic equation for this model, showing that all critical exponents governing the large scale behavior near criticality can be obtained from two independent ones. We show that as the supercritical density approaches criticality, distinct length scales naturally appear. Remarkably, this behavior is different from the subcritical one. Numerical simulations support the critical relations and the scale separation.

cond-mat.stat-mech↗
arXiv2024

Hydrodynamic limit for an open facilitated exclusion process with slow and fast boundaries

We study the symmetric facilitated exclusion process (FEP) on the finite one-dimensional lattice $\lbrace 1,\dots ,N-1\rbrace$ when put in contact with boundary reservoirs, whose action is subject to an additional kinetic constraint in order to enforce ergodicity, and whose speed is of order $N^{-\theta}$ for some parameter $\theta$. We derive its hydrodynamic limit as $N\to\infty$, in the diffusive space-time scaling, when the initial density profile is supercritical. More precisely, the macroscopic density of particles evolves in the bulk according to a fast diffusion equation as in the periodic case, which is now subject to boundary conditions that can be of Dirichlet, Robin or Neumann type depending on the parameter $\theta$. In the Dirichlet case, the FEP exhibits a very peculiar behaviour: unlike for the classical SSEP, and due to the two-phased nature of FEP, the reservoirs impose boundary densities which do not coincide with their equilibrium densities. The proof is based on the classical entropy method, but requires significant adaptations to account for the FEP's non-product stationary states and to deal with the non-equilibrium setting.

math.PR↗
arXiv2023

On the Conversion of Work into Heat: Microscopic Models and Macroscopic Equations

We summarize and extend some of the results obtained recently for the microscopic and macroscopic behavior of a pinned harmonic chain, with random velocity flips at Poissonian times, acted on by a periodic force {at one end} and in contact with a heat bath at the other end. Here we consider the case where the system is in contact with two heat baths at different temperatures and a periodic force is applied at any position. This leads in the hydrodynamic limit to a heat equation for the temperature profile with a discontinuous slope at the position where the force acts. Higher dimensional systems, unpinned cases and anharmonic interactions are also considered.

cond-mat.stat-mech↗
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↗
arXiv2023

Weakly asymmetric facilitated exclusion process

We consider the facilitated exclusion process, an interacting particle system on the integer line where particles hop to one of their left or right neighbouring site only when the other neighbouring site is occupied by a particle. A peculiarity of this system is that, starting from the step initial condition, the density profile develops a downward jump discontinuity around the position of the first particle, unlike other exclusion processes such as the asymmetric simple exclusion process (ASEP). In the weakly asymmetric regime, we show that the field of particle positions around the jump discontinuity converges to the solution of the multiplicative noise stochastic heat equation (i.e. the exponential of a solution to the KPZ equation) on a half-line subject to Dirichlet boundary condition, with initial condition given by the derivative of a Dirac delta function. We prove this result by reformulating the problem in terms of ASEP on a half-line with a boundary reservoir, for which we extend known proofs of convergence to deal with Dirichlet boundary condition and the very singular type of initial condition that arises in our case.

math.PR↗
arXiv2022

Mapping hydrodynamics for the facilitated exclusion and zero-range processes

We derive the hydrodynamic limit for two degenerate lattice gases, the \emph{facilitated exclusion process} (FEP) and the \emph{facilitated zero-range process} (FZRP), both in the symmetric and the asymmetric case. For both processes, the hydrodynamic limit in the symmetric case takes the form of a diffusive Stefan problem, whereas the asymmetric case is characterized by a hyperbolic Stefan problem. Although the FZRP is attractive, a property that we extensively use to derive its hydrodynamic limits in both cases, the FEP is not. To derive the hydrodynamic limit for the latter, we exploit that of the zero-range process, together with a classical mapping between exclusion and zero-range processes, both at the microscopic and macroscopic level. Due to the degeneracy of both processes, the asymmetric case is a new result, but our work also provides a simpler proof than the one that was previously proposed for the FEP in the symmetric case in \cite{blondel2021stefan}.

math.PR↗
arXiv2021

Asymmetric attractive zero-range processes with particle destruction at the origin

We investigate the macroscopic behavior of asymmetric attractive zero-range processes on $\mathbb{Z}$ where particles are destroyed at the origin at a rate of order $N^\beta$, where $\beta \in \mathbb{R}$ and $N\in\mathbb{N}$ is the scaling parameter. We prove that the hydrodynamic limit of this particle system is described by the unique entropy solution of a hyperbolic conservation law, supplemented by a boundary condition depending on the range of $\beta$. Namely, if $\beta \geqslant 0$, then the boundary condition prescribes the particle current through the origin, whereas if $\beta<0$, the destruction of particles at the origin has no macroscopic effect on the system and no boundary condition is imposed at the hydrodynamic limit.

math.PR↗
arXiv2020

Convergence of a degenerate microscopic dynamics to the porous medium equation

We derive the porous medium equation from an interacting particle system which belongs to the family of exclusion processes, with nearest neighbor exchanges. The particles follow a degenerate dynamics, in the sense that the jump rates can vanish for certain configurations, and there exist blocked configurations that cannot evolve. In [Gonçalves-Landim-Toninelli '09] it was proved that the macroscopic density profile in the hydrodynamic limit is governed by the porous medium equation (PME), for initial densities uniformly bounded away from $0$ and $1$. In this paper we consider the more general case where the density can take those extreme values. In this context, the PME solutions display a richer behavior, like moving interfaces, finite speed of propagation and breaking of regularity. As a consequence, the standard techniques that are commonly used to prove this hydrodynamic limits cannot be straightforwardly applied to our case. We present here a way to generalize the \emph{relative entropy method}, by involving approximations of solutions to the hydrodynamic equation, instead of exact solutions.

math.PR↗
arXiv2020

Stefan problem for a non-ergodic facilitated exclusion process

We consider the facilitated exclusion process, which is a nonergodic, kinetically constrained exclusion process. We show that in the hydrodynamic limit, its macroscopic behavior is governed by a free boundary problem. The particles evolve on the one-dimensional lattice according to jump rates which are degenerate, since they can vanish on non-trivial configurations and create distinct phases: indeed, configurations can be totally blocked (they cannot evolve under the dynamics), ergodic (they belong to an irreducible component), or transient (after a transitive period of time they will become either blocked or ergodic). We additionally prove that the microscopic separation into blocked/ergodic phases fully coincides with the moving interface problem given by the hydrodynamic equation.

math.PR↗
arXiv2020

Hydrodynamic limit for a facilitated exclusion process

We study the hydrodynamic limit for a periodic $1$-dimensional exclusion process with a dynamical constraint, which prevents a particle at site $x$ from jumping to site $x\pm1$ unless site $x\mp1$ is occupied. This process with degenerate jump rates admits transient states, which it eventually leaves to reach an ergodic component, assuming that the initial macroscopic density is larger than $\frac{1}{2}$, or one of its absorbing states if this is not the case. It belongs to the class of conserved lattice gases (CLG) which have been introduced in the physics literature as systems with active-absorbing phase transition in the presence of a conserved field. We show that, for initial profiles smooth enough and uniformly larger than the critical density $\frac{1}{2}$, the macroscopic density profile for our dynamics evolves under the diffusive time scaling according to a fast diffusion equation (FDE). The first step in the proof is to show that the system typically reaches an ergodic component in subdiffusive time.

math.PR↗
arXiv2020

Equilibrium fluctuations for the disordered harmonic chain perturbed by an energy conserving noise

We investigate the macroscopic behavior of the disordered harmonic chain of oscillators, through energy diffusion. The Hamiltonian dynamics of the system is perturbed by a degenerate conservative noise. After rescaling space and time diffusively, we prove that energy fluctuations in equilibrium evolve according to a linear heat equation. The diffusion coefficient is obtained from the non-gradient Varadhan's approach, and is equivalently defined through the Green-Kubo formula. Since the perturbation is very degenerate and the symmetric part of the generator does not have a spectral gap, the standard non-gradient method is reviewed under new perspectives.

math.PR↗