Planning
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.
Supports de conférences
Anna De Masi :
Stochastic particle systems and hydrodynamic limit with free boundaries.
Kevin Kuoch :
A multitype contact process: phase transition and
hydrodynamic limit.
Kirone Mallick :
Large deviations of the current in the exclusion process with open
boundaries.
Christophe Poquet :
Noise induced escape problem and phase reduction.
Julien Reygner :
Nonequilibrium steady states of the aerogel dynamics.
Marielle Simon :
Hydrodynamic limits for the velocity-flip model.
Résumés des exposés
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.