09 et 10 septembre 2013 - Université de Rouen

## Lundi 09 septembre

10h45 : Ouverture officielle
11h00 - 12h30 : Anna De Masi - Stochastic particle systems and hydrodynamic limit with free boundaries (Part I).
12h30 - 14h00 : Déjeuner
14h00 - 14h45 : Kirone Mallick - Large deviations of the current in the exclusion process with open boundaries.
14h45 - 15h30 : Julien Reygner - Nonequilibrium steady states of the aerogel dynamics.
15h30 - 15h45 : Pause café
15h45 - 17h15 : Anna De Masi - Stochastic particle systems and hydrodynamic limit with free boundaries (Part II).
17h15 - 18h00 : Peter Nejjar - Anomalous shock fluctuations in TASEP and last passage percolation models.

## Mardi 10 septembre

9h00 - 10h30 : Anna De Masi - Stochastic particle systems and hydrodynamic limit with free boundaries (Part III).
10h30 - 10h45 : Pause café
10h45 - 11h30 : Marielle Simon - Hydrodynamic limits for the velocity-flip model.
11h30 - 12h15 : Kevin Kuoch - A multitype contact process: phase transition and hydrodynamic limit.
12h15 - 13h45 : Déjeuner
13h45 - 14h30 : Oriane Blondel - Diffusion coefficient in low temperature kinetically constrained models.
14h30 - 15h15 : Christophe Poquet - Noise induced escape problem and phase reduction.
15h15 - 15h30 : Pause café
15h30 - 16h15 : Giuseppe Genovese - Random Walk with Obstacle.

## Mini-cours

Anna De Masi (L'Aquila, Italy) Stochastic particle systems and hydrodynamic limit with free boundaries. Hydrodynamics describes the collective behavior of particle systems. The validity of the hydrodynamic picture is based on a local equilibrium property, namely that locally the system is close to one of its equilibrium measures which is specified by an order parameter, usually the particle’s density. Rigorous proofs for stochastic particle systems are by now well established in fair generality after the works of Varadhan and collaborators, [1].
The hydrodynamic equations do not take into account boundary effects which are instead determined by the forces acting to keep the system confined in a bounded region. In this way hydrodynamic PDE’s have to be complemented with the correct boundary conditions. The most studied case is when the boundary forces are due to reservoirs which fix the densities at the boundaries. If the boundary densities are non homogeneous, then the density gradients produce currents that flow through the system according to Fick’s law. I will briefly discuss this phenomenon in the case of the symmetric simple exclusion process where the analysis is very elementary.
The aim of these lectures is instead to study the case when the region confining the system is determined by the state of the system itself. In continuum mechanics such situations are called free boundary problems and the prototipe is the Stefan problem were the system evolve according to the heat equation in a domain $\Omega_t$ with Dirichlet boundary conditions and with the local speed of the boundary determined by the normal gradient of the solution.
I will present some one dimensional simple stochastic particle systems were some of these issues can be analyzed in details, [2], [3].

References
[1] C. Kipnis, C. Landim. Scaling limits of interacting particle systems Springer-Verlag 1999.
[2] A.De Masi, P.A.Ferrari, E.Presutti. (2013) Symmetric simple exclusion process with free boundaries, preprint. http://arxiv.org/abs/1304.0701.
[3] G. Carinci, C. Giardinà, A. De Masi, E. Presutti. (2013) Hydrodynamic and super-hydrodynamic limits in a particle system with topological interactions, in preparation.

## Conférences

Oriane Blondel (LPMA, Paris 7) Diffusion coefficient in low temperature kinetically constrained models. Kinetically constrained spin models have been introduced in the physics literature to model glassy dynamics. They are interacting particle systems with a dynamics of creation/destruction of particle, with the specificity that a particular local constraint has to be satisfied to allow an update. We inject in the system a particle performing an independent random walk, constrained to jump only between empty sites. We analyze both non-cooperative models and the East model. We prove that the probe particle diffuses in a non-degenerate way when the particle density is smaller than 1 and we analyze the asymptotics of the diffusion coefficient when the density goes to 1. For the non cooperative models we prove a power law scaling conjectured by physicists. Instead for the East model we show that the diffusion coefficient is comparable to the spectral gap, disproving the physicists' conjecture.

Giuseppe Genovese (Paris Descartes) Random Walk with Obstacle. We will look at the correction to pure diffusive behavior in the asymptotic of the probability of a random walk on ${\mathbb Z}^d$ with a point obstacle in every dimensions $d$. If there will be time, we will discuss some issues about the limiting continuos process.

Kirone Mallick (CEA, Saclay) Large deviations of the current in the exclusion process with open boundaries. The asymmetric simple exclusion process is used as a template to study various aspects of non-equilibrium statistical physics. It appears in many models of low-dimensional transport with constraints. In the steady state, a non-vanishing current is carried through the system. We shall explain how to derive the statistics of the current for an ASEP with open boundaries and shall give exact combinatorial formulas valid for systems of all sizes and for all values of the parameters. Our results are obtained by using an extension of the Matrix Product Representation method.

Kevin Kuoch (Paris Descartes) A multitype contact process: phase transition and hydrodynamic limit. We introduce the "multitype contact process with competitive immigration" (MCPci) as a generalized contact process. Here, individuals encounter a hostile population, randomly dropped on each site, that either blocks or slows its natural growth.
First, we exhibit a phase transition according to the immigration parameter, that is, there exists a unique critical value in terms of survival of the process. In particular, the critical MCPci dies out. Considering a random environment in the unidimensional case, we show new critieria for survival. Then, by adding a rapid-stirring, we obtain an hydrodynamic limit which is a reaction-diffusion equation showing up the mean-field equations of the MCPci.

Peter Nejjar (Bonn, Allemagne) Anomalous shock fluctuations in TASEP and last passage percolation models. We consider the totally asymmetric simple exclusion process with initial conditions and/or jump rates such that shocks are generated. If the initial condition is deterministic, then the shock at time t will have a width of order $t^{1/3}$. We determine the law of particle positions in the large time limit around the shock in a few models. In particular, we cover the case where at both sides of the shock the process of the particle positions is asymptotically described by the Airy_1 process. The limiting distribution is a product of two distribution functions, which is a consequence of the fact that at the shock two characteristics merge and of the slow decorrelation along the characteristics. We show that the result generalizes to generic last passage percolation models.

Christophe Poquet (LPMA, Paris 7) Noise induced escape problem and phase reduction. The phase reduction of dynamical systems of large dimensions is widely used in applied sciences. We will see that this kind of reduction is valid for the study of the noise induced escape problem in the case of systems obtained by perturbation of a gradient flow : if the perturbed flow admits a hyperbolic stable curve (obtained by perturbation of a stable stationary curve of the gradient flow), then the noise induced escape problem from a stable fixed point of this curve can be well approximated by a one-dimensional problem.

Julien Reygner (École des Ponts ParisTech et Université Pierre et Marie Curie) Nonequilibrium steady states of the aerogel dynamics. An aerogel is a porous material derived from a gel, in which the liquid component is replaced with gas. Experimentally, it exhibits very low thermal conductivity. At the microscopic scale, it is described by a periodic lattice, in each cell of which a gas molecule is confined. Between neighbouring cells, the molecules interact through hard sphere potentials.
We shall explain how the study of this model reduces to a high-dimensional stochastic billard with non classical reflection rules. Standard techniques in the analysis of chains of oscillators are not very appropriate to address the long time behaviour of such a model. In the simple case of a system of two molecules put in contact with thermal baths, we introduce a general representation of the joint process of the positions and momenta of the particles, which allows us to describe the set of its nonequilibrium steady states. An key point in the proof is the use of the renewal theorem to quantify the marginal action of the thermal baths on each molecule.

Marielle Simon (ENS Lyon) Hydrodynamic limits for the velocity-flip model. We will be interested in microscopic models of atoms whose time evolution is governed by a hybrid dynamics, namely a combination of deterministic and stochastic dynamics. We add a stochastic noise to the classical Newton’s equations of motion, such that the main features of the underlying Hamiltonian system are not destroyed. More precisely, we will study the diffusive scaling limit for a chain of N coupled oscillators for which the Hamiltonian dynamics is perturbed with random flips of velocities. As a result, the total energy of the system is still conserved along the evolution, but momentum conservation is no longer valid. This stochastic noise provides good ergodic properties, and allows to derive the so-called hydrodynamic equations, which describe the macroscopic behavior of the system.