Mini-courses:
Cedric Bernardin (University of Nice)
Macroscopic energy diffusion in chains of coupled oscillators and related models
I will discuss the problem of normal and anomalous diffusion of energy in systems of coupled oscillators perturbed by a stochastic noise conserving energy. This course will be a review of the results obtained during the last ten years for these models.
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Vincent Giovangigli (École Polytechnique, Paris)
Mathematical Modeling of Reactive Flows
Multicomponent reactive flow models can be derived from the kinetic theory of gases. We discuss the mathematical structure of the resulting systems of partial differential equations. The mathematical properties are notably related to that of the underlying kinetic model. We address in particular the evaluation of multicomponent transport coefficients, the structure of thermochemistry, the Cauchy problem, existence of deflagration waves and numerical simulations. We also discuss related models which have a similar hyperbolic-parabolic structure.
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Plenary talks:
Adriana Neumann (UFRGS, Brazil)
Large deviations for the slowed exclusion process
In this work we study the asymptotic behavior of slowed simple exclusion process.
The name "slowed" is due to the presence of a slow bond. While an usual bond has conductance equal to one, the slow bond has a lower conductance. The time-evolution of the spatial density of particles, usually called the empirical measure, converges under diffusive scaling to a weak solution of the heat equation with Robin's boundary conditions.
This is known as hydrodynamic limit and corresponds to a law of large numbers for the empirical measure. Weak solutions of the equation mentioned above are called hydrodynamic profiles. Since the empirical measure is random, there is some deviation of this convergence to the hydrodynamic profile. In other words, the empirical measure can converge to another profile, with small probability. The large deviations principle is the characterization of this probability, that is exponentially small as a function of that profile.
This is a joint work with Tertuliano Franco (UFBA).
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André de Oliveira Gomes (Humboldt Universität Zu Berlin)
First Exit Times for Levy Driven Diffusions in R^d
We overview the state of art of the First Exit Times problem for Alpha-stable Levy driven Diffusions. Then we present the methodology employed to study the first exit time problem for exponentially light tails Levy Diffusions from bounded domains in R^d. Some comments about work in progress in infinite dimensions will be sketched.
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Alessia Nota (La Spienza, Italy)
A diffusion limit for a test particle in a random distribution of scatterers
We consider a point particle moving in a random distribution of obstaclesdescribed by a potential barrier. We show that, in a weak-coupling regime, under a suitable rescaling suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation. (Joint work with G. Basile, M.Pulvirenti) We present also some recent results on Lorentz gas in contact with particle reservoirs. (Work in progress, joint work with G. Basile, F. Pezzotti, M. Pulvirenti)
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Damir Madjarevic (University of Novi Sad, Serbia)
Shock structure and temperature overshoot in macroscopic multi-temperature model of binary mixtures
The present study deals with the shock wave profiles in the macroscopic multi-temperature model of binary gaseous mixtures. For that purpose we have adopted the hyperbolic model developed within the framework of extended thermodynamics. It is assumed that the mass difference between the constituents has the most prominent influence on the shock structure. Systematic analysis of results is carried out with special regard to the shock thickness and temperature overshoot of the heavier constituent using a large set of values for parameters. We found that temperature overshoot varies non-monotonically with mass ratio of the constituents. Also, the dependence of shock thickness parameter on mass ratio, Mach number and equilibrium concentration is analyzed. Shock profiles were compared with the experimental results in the case of helium-argon mixture. The influence of the different types of dissipation on the shock structure is considered by extending the original hyperbolic system with diffusion terms.
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Eric Luçon (Université René Descartes, Paris)
Fluctuation SPDE for interacting diffusions in a quenched random environment
I will talk about a class of models of mean-field interacting diffusions in a random environment. This includes models of synchronization (Kuramoto model) as well as models of interacting neurons (FitzHugh-Nagumo oscillators). The particularity of those models is
that the particles live in a random environment modeling the fact that the behavior of each particle may not be the same from one particle to another. I will discuss the fluctuations in large population of the empirical measure of the particles around its deterministic limit (McKean-Vlasov equation), the main difficulty being that we deal with quenched fluctuations (that is, for a fixed realization of the environment). If time allows, I will consider spatial generalizations of this model, which exhibit anomalous scaling.
This part is joint work with W. Stannat (TU Berlin).
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Fernando Pestana da Costa (CAMGSD & Universidade Aberta, Lisbon)
On a model of cluster annihilation
We consider a model of cluster annihilation, and point out its difference relative to more common models of cluster coagulation and fragmentation, such as the Smoluchowski's equation. We briefly present some recent results on the behaviour of solutions and point to some work still in progress.
This is joint work with J.T. Pinto (IST) and R. Sasportes (UAb).
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Florian Vollering (University of Göttingen)
Symmetric exclusion as a random environment: hydrodynamic limit and large deviations
We consider a slowly moving one-dimensional continuous-time random walk with transition rates which depend on an underlying autonomous simple symmetric exclusion process. This model represents an example of a slowly non-uniform mixing environment. I will present a hydrodynamic limit theorem and large deviations for the environment seen from the position of the random walk. The main difficulties in proving the hydrodynamic limit stem from the very local interaction of the random walk with the environment and not knowing the invariant distribution of the environment seen from the particle. For the large deviation principle we need to first identify the rate function of the random walk and then prove sufficient continuity.
Joint work with Luca Avena, Tertuliano Franco and Milton Jara.
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François Huveeners (Univ. Paris Dauphine)
Lack of energy transfer for chains of oscillators in the weak coupling limit
I'll discuss some mathematical results on energy localization in one-dimensional chains of oscillators. Quenched disorder is known to reduce or suppress transport in some chains, as is the case for the disordered harmonic chain, a system equivalent to Anderson's Hamiltonian. Randomness stemming from the Gibbs state potentially plays a role similar to that of quenched disorder, leading to slow energy dissipation. In this talk, I'll report on recent investigations on that phenomenon.
Joint work with W. De Roeck.
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Maria C. Carvalho (CMAF, University of Lisbon)
Invariant Measures for a Kinetic Equation Modelling Schooling of Fish
We study a Boltzmann type equation for densities on the unit circle arising from a binary interaction model for schooling of fish. The uniform density is always invariant, but in certain parameter ranges, it is not stable. We construct non-uniform invariant densities and study their stability.
Joint work with E. Carlen, P. Degond and B. Wennberg
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Maria Joana Torres (CMAT, University of Minho)
Stability of non-deterministic systems
A space of non-deterministic dynamical systems of Markov type on compact manifolds is considered. This is a natural space for stochastic perturbations of maps. For such systems, both the combinatorial stability, of the periodic attractors, and the spectral stability, of the invariant measures, are characterized.
Some work in progress to prove a version of Smale's Spectral Decomposition Theorem for smooth non-deterministic systems and to analyze their stability is addressed.
Joint work with Pedro Duarte (CMAF, University of Lisbon)
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Maria João Oliveira (CMAF & Universidade Aberta, Lisbon)
Asymptotic Scaling of Self-Repelling Fractional Brownian Motion
Mathematically, self-repulsion can be modelled using the self-intersection local time of Brownian motion (Varadhan, Westwater). Its effect on the scaling property of paths has been extensively studied (and is still not completely understood in three dimensions). Here we extend the model to fractional Brownian motion and propose an extension of the scaling law to general Hurst indices and dimensions.
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Patrícia Gonçalves (PUC - Rio de Janeiro and CMAT - University of Minho)
The symmetric slowed exclusion process
In this talk we will consider the symmetric slowed exclusion, which is an interacting particle system, where particles jump on the one-dimensional lattice at rate 1, except at a bond where the rate is slowed down in order to block the passage of particles across that bond. I will present some results related to the phase transition at the level of hydrodynamics and of the equilibrium fluctuations for the density.
This is a joint work with Tertuliano Franco (Univ. Bahia) and Adriana Neumann (UFRGS).
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Piedade Ramos (CMAT, University of Minho)
Dynamical properties of a cosmological model with diffusion
In cosmology, galaxies can be represented as particles of a fluid. In the work presented here we further assume that these particles undergo velocity diffusion. The model is developed within the Einstein theory of general relativity with a non zero cosmological scalar field, which plays the role of the background medium in which diffusion takes place. We consider in particular spatially homogeneous and isotropic solutions with non negative scalar factor. The equations for the scalar factor, the entropy of the fluid and the cosmological scalar field constitute an Abel ODE system that we were unable to solve analytically. We rewrite this system in terms of new dynamical variables and use the methods of dynamical systems theory to analyze the resulting system of equations. In particular, we study the dynamics of the spatially flat solutions as well as the asymptotic behavior of these spatially flat solutions in the past and in the future.
This is a joint work with Simone Calogero (Chalmers University of Technology, Sweden) and Ana Jacinta Soares (CMAT, University of Minho).
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Renatos Santos (Université Lyon 1)
Random walks in dynamic random environments
We will discuss models of random walks in dynamic random environments given by interacting particle systems. General limit theorems are currently available for such models only under certain mixing conditions on the random environment, all of which require uniform mixing. This leaves out several interesting examples such as zero-range, contact and exclusion processes, for which only partial results are available. We will give an overview of known results in both uniform and non-uniform mixing cases.
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Rui Vilela Mendes (CMAF & IPFN, Lisbon)
Stochastic solutions of nonlinear PDE's and an extension of superprocesses
Stochastic solutions provide new rigorous results for nonlinear PDE's and, through its local non-grid nature, are a natural tool for parallel computation. A review is made of the results so far obtained by this technique to generate solutions of charged kinetic equations. There are two different approaches for the construction of stochastic solutions: MacKean's and superprocesses. When restricted to measures, superprocesses can only be used to generate solutions for a limited class of nonlinear PDE's. A new class of superprocesses, superprocesses on signed measures and on distributions, is proposed to extend the stochastic solution approach to a wider class of PDE's.
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Stefano Olla (CEREMADE, France)
Diffusion and superdiffusion of energy in one dimensional systems of oscillators
We consider a system of harmonic oscillators with stochastic perturbations of the dynamics that conserve energy and momentum. In the one dimensional unpinned case, under proper space-time rescaling, Wigner distribution of energy converges to the solution of a fractional heat equation (with power 3/4 for the laplacian). For pinned systems or in dimension 3 or higher, we prove normal diffusive behaviour. Similar results are also obtained for space-time energy correlations in equilibrium. This is in agreement with previous 'weak noise' limits, passing through a kinetic equation, and conjectured behaviour for beta-FPU chains (quartic symmetric interaction).
Joint works with Tomasz Komorowski and Giada Basile. _________________________________________________________________________________________________
Srboljub Simic (University of Novi Sad, Serbia)
The Structure of Shock Waves in Dissipative Hyperbolic Models
The lecture is devoted to the shock structure problem in hyperbolic systems of balance laws, where the dissipation is taken into account through relaxation. The aim is to show that these models typically arise by extending the set of state variables, and governing equations as well. Furthermore, the mathematical aspects of the shock structure problem will be discussed. The existence of the physically admissible solution will be related to stability properties of equilibrium states and their transcritical bifurcation. The main examples will be the hyperbolic model of isothermal viscoelasticity, 13 moments equations for monatomic gases and the multi-temperature binary mixture of non-reacting ideal gases. The problems which arise in models with mixed type of dissipation, i.e. both relaxation and difusion, will be indicated.
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Valeria Ricci (University of Palermo, Italy)
Modelling of systems with dispersed phase
We shall describe some models of systems containing dispersed phase and their rigorous derivation as the macroscopic limit of suitablemicroscopic models in which the dispersed phase is associated to a particle-like component.