Abstracts

Ana Bela Cruzeiro - An entropic minimisation problem for incompressible viscid fluids

In order to study incompressible inviscid fluids, Brenier introduced in the late 80's a relaxation of a geodesic problem addressed by Arnold in 1966. We consider a natural analogue of Brenier's problem, for incompressible viscid fluids, where generalised flows are no more supported by absolutely continuous paths, but by Brownian paths. This new variational problem is an entrop minimisation one.

This is joint work with M. Arnaudon, C. Léonard and J.C. Zambrini.

Johel Beltran – Nucleation phase of condensing zero range process.

In this talk we consider a zero range process where each particle jumps from site x to y at rate r(x,y)g(n) and g(n)=1+b/n. Such g causes that particles prefer to scape quickly from sites with less particles. We fix the set of sites {1,2,...,k} and study a scaling limit for the fraction of particles at each site as the total number of particles N diverges. For b>1, we prove that at time scale N^2, the limiting process is a Feller continuous Markov process with continuous paths on the (k-1)-dimensional simplex. We give in the talk more details on the behavior of this limiting process. This is a joint work with Milton Jara and Claudio Landim.

Leonardo de Carlo - Introduction to the macroscopic fluctuation theory

When for an interacting particles system an hydrodynamic limit is proved, a natural question is that of large deviations from it. For the understanding of nonequilibrium stationary states of thermodynamics, it is interesting to formulate a theory of the stationary large deviations in particles systems. While for the dynamical large deviations the theory of Markov chain gives us the tools to prove the large deviations, in the stationary case there is not a "standard" method. The macroscopic fluctuation theory is a qualitative theory that has been proved to be effecttive to derive the large deviation rate functionals in diffusive boundary driven particles systems.

Fabio Chalub - The Kimura Equation

The Kimura Equation was introduced in the 60's by the Japanese geneticist Motoo Kimura and is considered one of the most important models in population genetics. It is a degenerated partial differential equation of drift diffusion type modeling the evolution of the probability distribution among different genotypes in a population.

In this talk we will derive this equation from basic stochastic models, showing not only that it approximates in all time scales important models as the Moran and the Wright-Fisher models but that it also encloses the well know replicator equation (a first order ordinary differential equation used extensively in evolutionary game theory). We will also show that the correct formulation of the Kimura equation includes two linearly independent conservation laws to be satisfied at all times.

In the final part, we will discuss generalizations and new formulations of the same problem.

Joint work with Max Souza (Brazil), Leonard Monsangeon and Ana Riberio (Portugal).

Chiara Franceschini – The partial simple exclusion process and its bosonic analogue in boundary driven environment.

In this talk I will introduce two interacting particle systems with boundary reservoirs. The first model behaves under the exclusion rule, or, in more physical terms, it has a fermionic nature. In the search of conservative particle systems then it is natural to look for a model where particles are less spread out then the corresponding random walker, so one can think of a bosonic counterpart of the exclusion, where particles have a preference in occupying the same site. Stationary measures are discussed and a self-duality function is constructed for both cases with the aid of Lie algebras. Most of the talk is covered here [1].

[1] Giardinà, C., Redig, F., Vafayi, K. (2010). Correlation inequalities for interacting particle systems with duality. Journal of Statistical Physics, 141(2), 242-263.

Rodrigo Marinho – Hydrodynamic limits for the exclusion process with a slow site.

We consider the simple exclusion process on the one-dimensional torus of length $n$ where particles jump to each of their nearest neighbors at rate one, except if they are at the slow site, where the jump rates now are $\alpha/n^{\beta}$ for fixed values $\alpha \in (0,\infty)$ and $\alpha \in [0,\infty)$. We show that, for any $\alpha>0$: if $\beta \in [0,1)$, then the hydrodynamic behavior of this process is driven by the heat equation; if $\beta=1$, then this phenomenon is driven by the heat equation with Robin boundary conditions. Since the results for $\beta>1$ are known, where the heat equation with Neumann boundary conditions is the hydrodynamic equation, our result proves that there exists a dynamical phase transition at $\beta=1$. We also show an alternative proof for the case $\beta>1$.

Otávio Menezes – Non-equilibrium and stationary fluctuations for the SSEP with slow boundary

We derive the non-equilibrium fluctuations of one-dimensional symmetric simple exclusion processes in contact with slowed stochastic reservoirs. Depending on the strength of the reservoirs, we obtain processes with various boundary conditions. The main ingredient to prove these results is the derivation of precise bounds on the two-point space-time correlation function, which are a consequence of precise bounds on the transition probability of some underlying random walks.

Joint work with Patrícia Gonçalves, Milton Jara and Adriana Neumann

Gabriel Nahum – On the Symmetric Simple Exclusion Process with Extended Current Reservoirs

On this talk we consider the extended version of the SSEP with the so called "Current Reservoirs", first proposed by De Masi et al. in 2011. The model is described as follows: the usual SSEP acts in the box \{1,...,N-1\} - a particle jumps to its neighbor site iff such site is free - coupled with a reservoir of particles at each endpoint, acting on a window of size K. Particles are injected (resp. removed) to the first free site (resp. from the first occupied site) at rates dependent of the position and size of the system. From a technical point of view, the main interest of this models lies in the presence of correlations, and the techniques needed to show the Hydrodynamic/Hydrostatic Limit. We present an overview of the current results for this model: Propagation of Chaos, Hydrodynamic Limit and Matrix Product ansatz, and the main technical difficulties in their proof; and the current development of techniques to overcome these problems - with a special emphasis on the Matrix Product Ansatz.

Renato de Paula – Porous medium model in contact with slow reservoirs.

The porous medium model is an interacting particle system which belongs to the class of Kinetically constrained lattice gases (KCLG). In this talk, we are interested in studying the hydrodynamic limit of this model in contact with slow reservoirs, which guarantees that the evolution of the density of particles of this model is described by the weak solution of the corresponding hydrodynamic equation, namely, the porous medium equation with Dirichlet, Neumann and Robin boundary conditions, depending on the parameter that rules the slowness of the reservoirs.

Stefano Scotta - Hydrodynamics for an exclusion process with long jumps and infinitely extended reservoirs

In this talk, I will present a model of symmetric exclusion process with long jumps in contact with infinitely extended reservoirs. I will show how the hydrodynamics equations associated with this process are influenced by the parameters which define the model. In particular, it is interesting to observe the transition between PDEs characterized by a diffusion operator (the standard Laplacian) and PDEs in which is present the regional fractional Laplacian, when the transition probability has respectively finite and infinite variance. Moreover, we will see how the "strength" of the reservoirs influences not only the PDEs operator but also the boundary conditions. The hydrodynamic limits for this model were proved completely in the case in which the transition has finite variance in [1] and partially in the case in which the variance is infinite in [2]. In a joint work with Cedric Bernardin and Patricia Gonçalves, we are proving the hydrodynamic limit for the remaining regimes.

[1] C. Bernardin, P. Gonçalves, and B. Jimenez-Oviedo. Slow to fast infinitely extended reservoirs for the symmetric exclusion process with long jumps. To appear in MPRF, available online at ArXiv, 2017.

[2] C. Bernardin, P. Gonçalves, and B. Jimenez-Oviedo. A microscopic model for a one parameter class of fractional laplacians with dirichlet boundary conditions. Available onnline at ArXiv, 2018.