Abstracts

Leonardo De Carlo - Divergence free flows and interacting particle systems - slides

Once the rules for a dynamics of interacting particles system arechosen, we determine non reversible transition rates corresponding to a fixedinvariant measure using the equivalence of this problem with the constructionof divergence free flows on the transition graph. Since divergence free flowsare characterized by cyclic decompositions we can generate families of models from elementary cycles on the configuration space.

Chiara Franceschini - Duality under transformation and thermalization: the asymmetric Kipnis–Marchioro–Presutti (KMP) model - slides

In this talk I will present the asymmetric version of the KMP model for heat conduction. To do this I first introduce a diffusion process known as Brownian energy process (BEP) which conserve the total energy of the system and it is dual to a symmetric inclusion process of particle systems. Acting with a global change of coordinates one can map this process into its asymmetric version, with the duality relation changing accordingly. Moreover, via an instantaneous thermalization limit one can recover the asymmetric KMP process, which is currently object of investigation and for which the Fourier law with transport term that depends on the asymmetry should arise from stationary hydrodynamics.

Work in progress with Leonardo De Carlo.

Milton Jara - Sharp mixing of the Curie-Weiss model - slides

(with Freddy Hernández)

Using the stochastic Curie -Weiss model as an example, we explain a new methodology to study the convergence to equilibrium of Markov chains. Using Varadhan's formulation of large deviations principles, we develop the notion of sharp mixing, which is a generalization of the concept of mixing times of Markov chains. We show that the stochastic Curie-Weiss has sharp mixing in at least two different time scales and we identify the crossover region between these two regimes.

Rodrigo Marinho - Cutoffs for exclusion processes on finite graphs with open boundaries - slides

(with Joe P. Chen and Milton Jara)

We prove a general theorem on cutoffs for the symmetric exclusion processes on graphs with open boundaries, under the natural assumption that the graphs converge geometrically and spectrally to a compact metric measure space with Dirichlet boundary condition. Our theorem is valid on a variety of settings including, but not limited to, the discrete hypercube in $\mathbb{R}^d$ in every dimension $d$; self-similar fractal spaces and products thereof; and others.

The proof is by construction of a "cutoff martingale," the analysis thereof using $L^2$-spectral methods, and a coupling between dual exclusion processes.

Oslenne Nogueira de Araujo - Hydrodynamic limit for a boundary driven stochastic lattice gas model with many conserved quantities - slides

In this talk, we are interested in studying the hydrodynamic limit for a particle system where, in this case, the dynamics consists of a weakly asymmetric simple exclusion process in contact with slowed reservoirs with collision among particles having different velocities.

Alessandra Occelli - From ABC to KPZ - slides

We consider a simple exclusion process on the discrete ring with three species of particles, labelled A, B anc C. The mass of each species is conserved. The particles exchange position under the influence of an external field. We present the results for the hydrodynamic limit and the equilibrium fluctuations in the weakly asymmetric regime. The goal is to derive coupled stochastic Burgers' equations as scaling limits of the fluctuation density fields. The stochastic Burgers' equation is a gradient version of the KPZ equation.

Renato de Paula - Phase transition of a porous medium equation with Robin boundary conditions - slides

The aim of this talk is to show, using interacting particle systems tools, that the weak solution of a porous medium equation (PME) with Robin boundary conditions $\rho^{\kappa}$ converges to a weak solution of a PME with Neumann boundary conditions when $\kappa \to 0$, and to a weak solution of a PME with Dirichlet boundary conditions when $\kappa \to \infty$.