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Documents  Chazottes, Jean-René | enregistrements trouvés : 15

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Research talks;Dynamical Systems and Ordinary Differential Equations;Number Theory

The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

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Outreach;Mathematics Education and Popularization of Mathematics

David Ruelle est professeur honoraire de Physique Théorique à l’Institut des Hautes Études Scientifiques (IHÉS). Il a reçu la médaille Matteucci en 2004, en 2006 le Prix Henri-Poincaré et en 2014 la Médaille Max-Planck pour l'ensemble de ses travaux.

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Research talks;Mathematical Physics;Probability and Statistics

In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the understanding of the structure of the stationary non equilibrium states. I will discuss these problems in specific examples and from two different perspectives. The stochastic microscopic and combinatorial point of view and the macroscopic variational approach where the microscopic details of the models are encoded just by the transport coefficients. In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the ...

82C05 ; 82C22 ; 60F10

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Research schools

$Let (X,T)$ be a dynamical system preserving a probability measure $\mu $. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), and $\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2$. In the best situation, the bound is $\exp(-C u^2/\sum_{i=0}^{n-1} \mathrm{Lip}_i(F)^2)$.
After explaining how to get such a bound for independent random variables, I will show how to prove it for a Gibbs measure on a shift of finite type with a Lipschitz potential, and present examples of functions $F$ to which one can apply the inequality. Finally, I will survey some results obtained for nonuniformly hyperbolic systems modeled by Young towers.
$Let (X,T)$ be a dynamical system preserving a probability measure $\mu $. A concentration inequality quantifies how small is the probability for $F(x,Tx,\ldots,T^{n-1}x)$ to deviate from $\int F(x,Tx,\ldots,T^{n-1}x) \mathrm{d}\mu(x)$ by an given amount $u$, where $F:X^n\to\mathbb{R}$ is supposed to be separately Lipschitz. The bound on that probability involves a constant $C$ depending only on the dynamical system (thus independent of $n$), ...

37D20 ; 37D25 ; 37A50

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Research talks;Dynamical Systems and Ordinary Differential Equations

The titles of the of the individual lectures are:
1. Operators dynamics versus base space dynamics
2. Dilations and joinings
3. Compact semigroups and splitting theorems

37A30 ; 47A35 ; 47Nxx ; 47A20 ; 47D03

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Research talks;Dynamical Systems and Ordinary Differential Equations

The titles of the of the individual lectures are:
1. Operators dynamics versus base space dynamics
2. Dilations and joinings
3. Compact semigroups and splitting theorems

37A30 ; 47A35 ; 47Nxx ; 47A20 ; 47D03

... Lire [+]

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Research talks;Dynamical Systems and Ordinary Differential Equations

The titles of the of the individual lectures are:
1. Operators dynamics versus base space dynamics
2. Dilations and joinings
3. Compact semigroups and splitting theorems

37A30 ; 47A35 ; 47Nxx ; 47A20 ; 47D03

... Lire [+]

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Research talks;Dynamical Systems and Ordinary Differential Equations;Number Theory

The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Dynamical Systems and Ordinary Differential Equations;Number Theory

The classic mean ergodic theorem has been extended in numerous ways: multiple averages, polynomial iterates, weighted averages, along with combinations of these extensions. I will give an overview of these advances and the different techniques that have been used, focusing on convergence results and what can be said about the limits.

37A05 ; 37A25 ; 37A15

... Lire [+]

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

Research talks;Mathematical Physics;Probability and Statistics

The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

35Q82 ; 60K35 ; 82C22 ; 82C24

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Research talks;Mathematical Physics;Probability and Statistics

The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

35Q82 ; 60K35 ; 82C22 ; 82C24

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Research talks;Mathematical Physics;Probability and Statistics

The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

35Q82 ; 60K35 ; 82C22 ; 82C24

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Research talks;Mathematical Physics;Probability and Statistics

In the last lecture I will apply the macroscopic fluctuation theory to solve specific problems. I will show that several features and behaviors of non equilibrium systems can be deduced within the theory. In particular I will discuss the following issues: the presence of long range correlations in stationary non equilibrium states; the explicit computation of the large deviations rate functional for a few one dimensional stationary non equilibrium states; the existence of dynamical phase transitions in terms of the current flowing across the system, the existence of Lagrangian phase transitions. In the last lecture I will apply the macroscopic fluctuation theory to solve specific problems. I will show that several features and behaviors of non equilibrium systems can be deduced within the theory. In particular I will discuss the following issues: the presence of long range correlations in stationary non equilibrium states; the explicit computation of the large deviations rate functional for a few one dimensional stationary non ...

60F10 ; 82C05 ; 82C22

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Research talks;Mathematical Physics;Probability and Statistics

In this second lecture I will discuss the basic ideas of the macroscopic fluctuation theory as an effective theory in non equilibrium statistical mechanics. All the theory develops starting from a principal formula that describes the distribution at large deviations scale of the joint fluctuations of the density and the current for a diffusive system. The validity of such a formula can be proved for diffusive stochastic lattice gases. I will discuss an infinite dimensional Hamilton-Jacobi equation for the quasi-potential of stationary non equilibrium states, fluctuation-dissipation relationships, the underlying Hamiltonian structure, a relation with work and Clausius inequality, a large deviations functional for the current flowing through a system. In this second lecture I will discuss the basic ideas of the macroscopic fluctuation theory as an effective theory in non equilibrium statistical mechanics. All the theory develops starting from a principal formula that describes the distribution at large deviations scale of the joint fluctuations of the density and the current for a diffusive system. The validity of such a formula can be proved for diffusive stochastic lattice gases. I will ...

60F10 ; 82C05 ; 82C22

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- 361 p.
ISBN 978-3-540-24289-5

Lecture notes in physics , 0671

Localisation : Ouvrage RdC (Dyna)

dynamique réticulaire # mécanique non linéaire # théorie des treillis

70-06 ; 00B25

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