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Mustapha Mourragui

Publications and source records attributed to Mustapha Mourragui.

12 recordsLinked to original sources
arXiv2026

Boundary driven weakly asymmetric Blume-Capel model: Large deviations for mixed Dirichlet-Neumann boundary conditions

We consider the Blume-Capel spin model on a finite cylinder with reservoirs at the boundary. A model with spin variable $\sigma$ taking values in {-1, 0, 1}, with the superposition of two dynamics: in the bulk, the spins evolve according to a weakly asymmetric dynamics; and the boundary dynamics follows a mechanism of creation, annihilation and spin flip, its action is accelerated differently on the left and on the right in a way to produce mixed boundary conditions. For the dynamics in the bulk, two quantities are conserved, the magnetization which corresponds to the sum of the spin values, and the concentration which corresponds to the sum of the squared spin values. We first establish, in the diffusive scaling, the hydrodynamic limit for this model which states that the couple of empirical measures (magnetization, concentration) converges to the solution of a system of coupled equations with mixed boundary conditions. Then we prove the associated dynamical large deviations principle.

math.DS↗
arXiv2024

Bridging bulk and surface: An interacting particle system towards the field-road diffusion model

We recover the so-called field-road diffusion model as the hydrodynamic limit of an interacting particle system. The former consists of two parabolic PDEs posed on two sets of different dimensions (a "field" and a "road" in a population dynamics context), and coupled through exchange terms between the field's boundary and the road. The latter stands as a Symmetric Simple Exclusion Process (SSEP): particles evolve on two microscopic lattices following a Markov jump process, with the constraint that each site cannot host more than one particle at the same time. The system is in contact with reservoirs that allow to create or remove particles at the boundary sites. The dynamics of these reservoirs are slowed down compared to the diffusive dynamics, to reach the reactions and the boundary conditions awaited at the macroscopic scale. This issue of bridging two spaces of different dimensions is, as far as we know, new in the hydrodynamic limit context, and raises perspectives towards future related works.

math.AP↗
arXiv2023

Hydrodynamic and hydrostatic limit for a generalized contact process with mixed boundary conditions

We consider an interacting particle system which models the sterile insect technique. It is the superposition of a generalized contact process with exchanges of particles on a finite cylinder with open boundaries (see Kuoch et al., 2017). We show that when the system is in contact with reservoirs at different slowdown rates, the hydrodynamic limit is a set of coupled non linear reaction-diffusion equations with mixed boundary conditions. We also prove the hydrostatic limit when the macroscopic equations exhibit a unique attractor.

math.PR↗
arXiv2018

Metastability of the two-dimentional Blume-Capel model with zero chemical potential and small magnetic field on a large torus

We consider the Blume-Capel model with zero chemical potential and small magnetic field in a two-dimensional torus whose length increaseswith the inverse of the temeprature. We prove the mestastable behavior and that starting from a configuration with only negative spins, the process visits the configuration with only 0-spins on its way to the ground state which is the configuration with all spins equal to +1.

math.PR↗
arXiv2018

Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions

We consider continuous-time Markov chains which display a family of wells at the same depth. We provide sufficient conditions which entail the convergence of the finite-dimensional distributions of the order parameter to the ones of a finite state Markov chain. We also show that the state of the process can be represented as a time-dependent convex combination of metastable states, each of which is supported on one well.

math.PR↗
arXiv2016

A boundary driven generalised contact process with exchange of particles: Hydrodynamics in infinite volume

We consider a two species process which evolves in a finite or infinite domain in contact with particles reservoirs at different densities, according to the superposition of a generalised contact process and a rapid-stirring dynamics in the bulk of the domain, and a creation/annihilation mechanism at its boundaries. For this process, we study the law of large numbers for densities and current. The limiting equations are given by a system of non-linear reaction-diffusion equations with Dirichlet boundary conditions.

math.PR↗
arXiv2014

Large deviations of the empirical current for the boundary driven Kawasaki process with long range interaction

We consider a lattice gas evolving in a bounded cylinder of length 2N + 1 and interacting via a Neuman Kac interaction of range N, in contact with particles reservoirs at different densities. We investigate the associated law of large numbers and large deviations of the empirical current and of the density. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by a nonlocal, nonlinear evolution equation with Dirichlet boundary conditions.

math.PR↗
arXiv2012

Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states

A particle system with a single locally-conserved field (density) in a bounded interval with different densities maintained at the two endpoints of the interval is under study here. The particles interact in the bulk through a long range potential parametrized by $β\ge 0$ and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasi-linear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, for $β$ small enough, it is also demonstrated that the empirical particle density obeys a law of large numbers with respect to the stationary measures (hydrostatic). The macroscopic particle density solves a non local, stationary, transport equation.

math.PR↗
arXiv2012

Quenched large deviations for Glauber evolution with Kac interaction and Random Field

We study a spin-flip model with Kac type interaction, in the presence of a random field given by i.i.d. bounded random variables. The system, spatially inhomogeneous, evolves according to a non conservative (Glauber) dynamics. We show an almost sure (with respect to the random field) large deviations principle for the empirical magnetizations of this process. The rate functional depends on the statistical properties of the external random field, it is lower semicontinuous with compact level sets.

math.PR↗
arXiv2009

Dynamical large deviations for the boundary driven weakly asymmetric exclusion process

We consider the weakly asymmetric exclusion process on a bounded interval with particle reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. We prove the associated dynamical large deviations principle.

math.PR↗
arXiv2008

Lattice gas model in random medium and open boundaries: hydrodynamic and relaxation to the steady state

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions $d \ge 3$, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a non linear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation.

math.PR↗