Hydrodynamic limit of boundary driven exclusion processes with nonreversible boundary dynamics

Using duality techniques, we derive the hydrodynamic limit for one-dimensional, boundary-driven, symmetric exclusion processes with different types of non-reversible dynamics at the boundary, for which the classical entropy method fails

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.Comment: One major correction has been done. An author has been adde

Limite hydrodynamique pour un processus d'exclusion actif .

Collective dynamics can be observed among many animal species, and have given rise in the last decades to an active and interdisciplinary eld of study. Such behaviors are often modeled by active matter, in which each individual is self-driven and tends to update its velocity depending on the one of its neighbors. In a classical model introduced by Vicsek & al., as well as in numerous related active matter models, a phase transition between chaotic behavior at high temperature and global order at low temperature can be observed. Even though ample evidence of these phase transitions has been obtained for collective dynamics, from a mathematical standpoint, such active systems are not fully understood yet. Signicant progress has been achieved in the recent years under an assumption of mean-eld interactions, however to this day, few rigorous results have been obtained for models involving purely local interactions. In this paper, we describe a lattice active particle system, in which particles interact locally to align their velocities. We obtain rigorously, using the formalism developed for hydrodynamic limits of lattice gases, the scaling limit of this out-of-equilibrium system, for which numerous technical and theoretical diculties arise.L'étude des dy-namiques collectives, observables chez de nombreuses espèces animales, a motivé dans les dernières décennies un champ de recherche actif et transdisciplinaire. De tels comportements sont souvent modélisés par de la matière active, c'est-à-dire par des modèles dans lesquels chaque individu est caractérisé par une vitesse propre qui tend à s'ajuster selon celle de ses voisins. De nombreux modèles de matière active sont liés à un modèle fondateur proposé en 1995 par Vicsek & al.. Ce dernier, ainsi que de nombreux modèles proches, présentent une transition de phase entre un com-portement chaotique à haute température, et un comportement global et cohérent à faible température. De nombreuses preuves numériques de telles transitions de phase ont été obtenues dans le cadre des dynamiques collectives. D'un point de vue mathématique, toutefois, ces systèmes actifs sont encore mal compris. Plusieurs résultats ont été obtenus récemment sous une approximation de champ moyen, mais il n'y a encore à ce jour que peu d'études mathématiques de modèles actifs faisant intervenir des interactions purement microscopiques. Dans cet article, nous décrivons un système de particules actives sur réseau interagissant localement pour aligner leurs vitesses. Nous obtenons rigoureusement, à l'aide du formalisme des limites hydrodynamiques pour les gaz sur réseau, la limite macroscopique de ce système hors-équilibre, qui pose de nombreuses difficultés techniques et théoriques

Stationary fluctuations for the facilitated exclusion process

We derive the stationary fluctuations for the Facilitated Exclusion Process (FEP) in one dimension in the symmetric, weakly asymmetric and asymmetric cases. Our proof relies on the mapping between the FEP and the zero-range process, and extends the strategy in \cite{erignoux2022mapping}, where hydrodynamic limits were derived for the FEP, to its stationary fluctuations. Our results thus exploit works on the zero-range process's fluctuations \cite{gonccalves2010equilibrium,gonccalves2015stochastic}, but we also provide a direct proof in the symmetric case, for which we derive a sharp estimate on the equivalence of ensembles for the FEP's stationary states.Comment: 38page

Nematic first order phase transition for liquid crystals in the van der Waals--Kac limit

In this paper we revisit and extend some mathematical aspects of Onsager's theory of liquid crystals that have been investigated in recent years by different communities (statistical mechanics, analysis and probability). We introduce a model of anisotropic molecules with three-dimensional orientations interacting via a Kac-type interaction. We prove that, in the limit in which the range of the interaction is sent to infinity after the thermodynamic limit, the free energy tends to the infimum of an effective energy functional \`a la Onsager. We then prove that, if the spherical harmonic transform of the angular interaction has a negative minimum, this effective free energy functional displays a first order phase transition as the total density of the system increases

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, [4], 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 [2, 3] 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

Hydrodynamic limit for a facilitated exclusion process

International audienceWe 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 ± 1 unless site x 1 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 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 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.Nous étudions la limite hydrodynamique d'un système d'exclusion unidimensionnel avec une contrainte dynamique, qui empêche une particule en x de sauter en x ± 1 à moins que x ∓ 1 soit occupé. Ce processus à taux de sauts dégénérés admet des états transients, qu'il finit par quitter pour atteindre une composante ergodique dans le cas où la densité initiale macroscopique est supérieure à 1 2 , ou un de ses états absorbants dans l'autre cas. Ce processus fait partie des gaz conservatifs sur réseau, qui ont été introduits dans la litérature physique comme systèmes présentant une transition de phase active-absorbante en présence d'un champ conservé. Nous montrons que pour des profils initiaux de densité suffisamment réguliers et strictement supérieurs à 1 2 , le profil de densité macroscopique évolue à l'échelle diffusive suivant une équation de diffusion rapide (FDE). La première étape de la preuve consiste à montrer que, typiquement, le système atteint une composante ergodique en temps sous-diffusif

On the hydrodynamics of active matter models on a lattice

Active matter has been widely studied in recent years because of its rich phenomenology, whose mathematical understanding is still partial. We present some results, based on [8, 17] linking microscopic lattice gases to their macroscopic limit, and explore how the mathematical state of the art allows to derive from various types of microscopic dynamics their hydrodynamic limit. We present some of the crucial aspects of this theory when applied to weakly asymmetric active models. We comment on the specific challenges one should consider when designing an active lattice gas, and in particular underline mathematical and phenomenological differences between gradient and non-gradient models. Our purpose is to provide the physics community, as well as member of the mathematical community not specialized in the mathematical derivation of scaling limits of lattice gases, some key elements in defining microscopic models and deriving their hydrodynamic limit

Hydrodynamic limit for an active stochastic lattice gas

L'étude des dynamiques collectives, observables chez de nombreuses espèces animales, a motivé dans les dernières décennies un champ de recherche actif et transdisciplinaire. De tels comportements sont souvent modélisés par de la matière active, c'est-à-dire par des modèles dans lesquels chaque individu est caractérisé par une vitesse propre qui tend à s'aligner avec celle de ses voisins.Dans un modèle fondateur proposé par Vicsek et al., ainsi que dans de nombreux modèles de matière active liés à ce dernier, une transition de phase entre un comportement chaotique à forte température, et un comportement global et cohérent à faible température, a été observée. De nombreuses preuves numériques de telles transitions de phase ont été obtenues dans le cadre des dynamiques collectives. D'un point de vue mathématique, toutefois, ces systèmes actifs sont encore mal compris. Plusieurs résultats ont été obtenus récemment sous une approximation de champ moyen, mais il n'y a encore à ce jour que peu d'études mathématiques de modèles actifs faisant intervenir des interactions purement microscopiques.Dans ce manuscrit, nous décrivons un système de particules actives sur réseau interagissant localement pour aligner leurs vitesses. L'objet de cette thèse est l'obtention rigoureuse, à l'aide du formalisme des limites hydrodynamiques pour les gaz sur réseau, de la limite macroscopique de ce système hors-équilibre, qui pose de nombreuses difficultés techniques et théoriques.Collective dynamics can be observed among many animal species, and have given rise in the last decades to an active and interdisciplinary field of study. Such behaviors are usually modeled by active matter, in which each individual is self-driven and tends to align its velocity with that of its neighbors.In a classical model introduced by Vicsek & al., as well as in numerous related active matter models, a phase transition between chaotic behavior at high temperature and global order at low temperature can be observed. Even though ample evidence of these phase transitions has been obtained for collective dynamics, from a mathematical standpoint, such active systems are not fully understood yet. Some progress has been achieved in the recent years under an assumption of mean-field interactions, however to this day, few rigorous results have been obtained for models involving purely local interactions.In this manuscript, we describe a lattice active particle system interacting locally to align their velocities. This thesis aims at rigorously obtaining, using the formalism developed for hydrodynamic limits of lattice gases, the scaling limit of this out-of-equilibrium system, for which numerous technical and theoretical difficulties arise