Mustapha Mourragui

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.

Hydrodynamic limit of mean zero asymmetric zero range processes in infinite volume

Abstract We prove the hydrodynamic behaviour of mean zero, asymmetric zero range processes evolving on the infinite lattice Z d. The proof relies on a bound, uniform in the volume, for the entropy production of processes in large finite volume. Such an entropy production bound, uniform in the volume, was first proved by Fritz in [6] to extend to infinite volume the proof of Guo Papanicolaou and Varadhan of hydrodynamic behaviour of interacting particle systems. Our approach follows a method introduced by Yau in [12].

Phase segregation dynamics for the Blume{Capel model with Kac interaction

We consider the Glauber and Kawasaki dynamics for the Blume-Capel spin model with weak long-range interaction on the infinite lattice: a ferromagnetic d-dimensional lattice system with the spin variable [sigma] taking values in {-1,0,1} and pair Kac potential . The Kawasaki dynamics conserves the empirical averages of [sigma] and [sigma]2 corresponding to local magnetization and local concentration. We study the behaviour of the system under the Kawasaki dynamics on the spatial scale [gamma]-1 and time scale [gamma]-2. We prove that the empirical averages converge in the limit [gamma]-->0 to the solutions of two coupled equations, which are in the form of the flux gradient for the energy functional. In the case of the Glauber dynamics we still scale the space as [gamma]-1 but look at finite time and prove in the limit of vanishing [gamma] the law of large number for the empirical fields. The limiting fields are solutions of two coupled nonlocal equations. Finally, we consider a nongradient dynamics which conserves only the magnetization and get a hydrodynamic equation for it in the diffusive limit which is again in the form of the flux gradient for a suitable energy functional.

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≥3, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a quasilinear 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.

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 ??0 and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasilinear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, when ? is 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.

Macroscopic evolution of particle systems with random field Kac interactions

We consider a lattice gas interacting via a Kac interaction Jγ(|x−y|) of range γ−1, γ>0, x,yd and under the influence of an external random field given by independent bounded random variables with a translation invariant distribution. We study the evolution of the system through a conservative dynamics, i.e. particles jump to nearest neighbour empty sites, with rates satisfying a detailed balance condition with respect to the equilibrium measure. We prove that rescaling space as γ−1 and time as γ−2, in the limit γ→0, for dimension d≥3, the macroscopic density profile ρ satisfies, a.s. with respect to the random field, a nonlinear integral differential equation, with a diffusion matrix determined by the statistical properties of the external random field. The result holds for all values of the density, also in the presence of phase segregation, and the equation is in the form of the flux gradient for the energy functional.