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# Documents  82C31 | enregistrements trouvés : 15

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## Large-time behavior in (hypo)coercive ODE-systems and kinetic models Arnold, Anton | CIRM H

Post-edited

Research talks

In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic transport equations with a relaxation operator that drives the phase space distribution towards the spatially local equilibrium, a Gaussian with the same macroscopic parameters. Due to the absence of dissipation w.r.t. the spatial direction, convergence to the global equilibrium is only possible thanks to the transport term that mixes various positions. Hence, such models are hypocoercive.
We shall prove exponential convergence towards the equilibrium with explicit rates for several linear, space periodic BGK-models in dimension 1 and 2. Their BGK-operators differ by the number of conserved macroscopic quantities (like mass, momentum, energy), and hence their hypocoercivity index. Our discussion includes also discrete velocity models, and the local exponential stability of a nonlinear BGK-model.
The third part of the talk is concerned with the entropy method for (non)symmetric Fokker-Planck equations, which is a powerful tool to analyze the rate of convergence to the equilibrium (in relative entropy and hence in L1). The essence of the method is to first derive a differential inequality between the first and second time derivative of the relative entropy, and then between the entropy dissipation and the entropy. For hypocoercive Fokker-Planck equations, i.e. degenerate parabolic equations (with drift terms that are linear in the spatial variable) we modify the classical entropy method by introducing an auxiliary functional (of entropy dissipation type) to prove exponential decay of the solution towards the steady state in relative entropy. The obtained rate is indeed sharp (both for the logarithmic and quadratic entropy). Finally, we extend the method to the kinetic Fokker-Planck equation (with nonquadratic potential).
In this talk we discuss the convergence to equilibrium in conservative-dissipative ODE-systems, kinetic relaxation models (of BGK-type), and Fokker-Planck equation. This will include symmetric, non-symmetric and hypocoercive evolution equations. A main focus will be on deriving sharp decay rates.
We shall start with hypocoercivity in ODE systems, with the ”hypocoercivity index” characterizing its structural complexity.
BGK equations are kinetic ...

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## An introduction to molecular dynamics Stoltz, Gabriel | CIRM H

Post-edited

Research talks;Partial Differential Equations;Probability and Statistics

The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs measure using appropriate stochastic dynamics of Langevin type. Rigorous ergodicity results as well as elements on the estimation of numerical errors are provided. The last part is devoted to the computation of transport coefficients such as the mobility or autodiffusion in fluids, relying either on integrated equilibrium correlations à la Green-Kubo, or on the linear response of nonequilibrium dynamics in their steady-states. The aim of this two-hour lecture is to present the mathematical underpinnings of some common numerical approaches to compute average properties as predicted by statistical physics. The first part provides an overview of the most important concepts of statistical physics (in particular thermodynamic ensembles). The aim of the second part is to provide an introduction to the practical computation of averages with respect to the Boltzmann-Gibbs ...

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## Mean-field analysis of an excitatory neuronal network: application to systemic risk modeling? Delarue, François | CIRM

Post-edited

Research talks;Partial Differential Equations;Mathematics in Science and Technology

Inspired by modeling in neurosciences, we here discuss the well-posedness of a networked integrate-and-fire model describing an infinite population of companies which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the debt of a company increases when some of the others default: precisely, the loss it receives is proportional to the instantaneous proportion of companies that default at the same time. From a mathematical point of view, the coefficient of proportionality, denoted by a, is of great importance as the resulting system is known to blow-up when a takes large values, a blow-up meaning that a macroscopic proportion of companies may default at the same time. In the current talk, we focus on the complementary regime and prove that existence and uniqueness hold in arbitrary time without any blow-up when the excitatory parameter is small enough. Inspired by modeling in neurosciences, we here discuss the well-posedness of a networked integrate-and-fire model describing an infinite population of companies which interact with one another through their common statistical distribution. The interaction is of the self-excitatory type as, at any time, the debt of a company increases when some of the others default: precisely, the loss it receives is proportional to the instantaneous proportion ...

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## Semiclassical methods for Langevin dynamics at low temperature Laurent, Michel | CIRM H

Multi angle

Research talks

Semiclassical methods have shown to be very efficient to get quantitative description of metastability of Langevin dynamics. In this talk we try to explain the main ideas of this approach in both reversible and non-reversible cases.

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## Space-time covariance of KPZ growth models Ferrari, Patrik | CIRM H

Multi angle

Research talks;Mathematical Physics;Probability & Statistics

For some growth models in the Kardar-Parisi-Zhang universality class, the large time limit process of the interface profile is well established. Correlations in space-time are much less understood. Along special space-time lines, called characteristics, there is a sort of ageing. We study the covariance of the interface process along characteristic lines for generic initial conditions. Joint work with A. Occelli (arXiv:1807.02982).

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## The KPZ fixed point - Lecture 2 Remenik, Daniel | CIRM H

Multi angle

Research School;Mathematical Physics;Probability and Statistics

In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion process (TASEP). Then I will present a Fredholm determinant formula for its distribution (at a fixed time) and show how all the main properties of the fixed point (including the Markov property, space and time regularity, symmetries and scaling invariance, and variational formulas) can be derived from the formula and the construction, and also how the formula reproduces known self-similar solutions such as the $Airy_1andAiry_2$ processes.
The second part of the course will be devoted to explaining how the KPZ fixed point can be computed starting from TASEP. The method is based on solving, for any initial condition, the biorthogonal ensemble representation for TASEP found by Sasamoto '05 and Borodin-Ferrari-Prähofer-Sasamoto '07. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.
Based on joint work with K. Matetski and J. Quastel.
In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion ...

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## The KPZ fixed point - Lecture 1 Remenik, Daniel | CIRM H

Multi angle

Research School;Mathematical Physics;Probability and Statistics

In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion process (TASEP). Then I will present a Fredholm determinant formula for its distribution (at a fixed time) and show how all the main properties of the fixed point (including the Markov property, space and time regularity, symmetries and scaling invariance, and variational formulas) can be derived from the formula and the construction, and also how the formula reproduces known self-similar solutions such as the $Airy_1andAiry_2$ processes.
The second part of the course will be devoted to explaining how the KPZ fixed point can be computed starting from TASEP. The method is based on solving, for any initial condition, the biorthogonal ensemble representation for TASEP found by Sasamoto '05 and Borodin-Ferrari-Prähofer-Sasamoto '07. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.
Based on joint work with K. Matetski and J. Quastel.
In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion ...

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## Computing qualitatively correct approximations of balance laws:exponential-fit, well-balanced and asymptotic-preserving Gosse, Laurent | Springer 2013

Ouvrage

- xix; 340 p.
ISBN 978-88-470-2891-3

SIMAI Springer series , 0002

Localisation : Ouvrage RdC (GOSS)

équations aux dérivées partielles # mathématiques de l'ingénieur # asymptotic -preserving # équation de balance # schéma de Godunov # convergence faible # solutions visqueuses # modèles cinétiques # transfert radiatif # chimiotaxie # équations de semi-conducteurs # lissage exponentiel # équations cinétiques non linéaires # évaluations de l'erreur # modèle de Boltzmann-Poisson # équation de Klein-Kramers # modèle de Burgers/Fokker-Planck # stabilité équations aux dérivées partielles # mathématiques de l'ingénieur # asymptotic -preserving # équation de balance # schéma de Godunov # convergence faible # solutions visqueuses # modèles cinétiques # transfert radiatif # chimiotaxie # équations de semi-conducteurs # lissage exponentiel # équations cinétiques non linéaires # évaluations de l'erreur # modèle de Boltzmann-Poisson # équation de Klein-Kramers # modèle de Burgers/Fokker-Planck # ...

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## Levy processes, integral equations, statistical physics: connections and interactions Sakhnovich, Lev A. | Birkhäuser 2012

Ouvrage

- ix; 245 p.
ISBN 978-3-0348-0355-7

Operator theory: advances and applications , 0225

Localisation : Collection 1er étage

processus de Levy # équation intégrale # analyse mathématique # probabilités # analyse numérique

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## Free energy computations:a mathematical perspective Lelièvre, Tony ; Rousset, Mathias ; Stoltz, Gabriel | Imperial College Press 2010

Ouvrage

- xii; 458 p.
ISBN 978-1-84816-247-1

Localisation : Ouvrage RdC (LELI)

physique # analyse numérique # energie libre # thermodynamique statistique # méthode adaptative

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## Stochastic models, information theory, and Lie groups. Volume 2: analytic methods and modern applications Chirikjian, Gregory S. | Birkhäuser 2012

Ouvrage

- xxvii; 433 p.
ISBN 978-0-8176-4943-2

Applied and numerical harmonic analysis

Localisation : Ouvrage RdC (CHIR)

modèles stochastiques # processus stochastiques # équation différentielle ordinaire stochastique #
groupe de Lie # calcul de variations # théorie des codes # équation de Euler-Poincaré # théorie de l'information

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## Stochastic Models, Information Theory, and Lie Groups, Volume 1 : Classical Results and Geometric Methods Chirikjian, Gregory S. | Birkhäuser 2009

Ouvrage

- xxi, 380 p.
ISBN 978-0-8176-4802-2

Applied and Numerical Harmonic Analysis

Localisation : Ouvrages RdC (CHIR)

probabilités # processus stochastique # processus de diffusion # équation différentielle stochastique # modèle stochastique # géométrie différentielle # théorie de l'information # groupe de Lie

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## The user's approach to topological methods in 3D dynamical systems Natiello, Mario A. ; Solari, Hernan G. | World Scientific 2007

Ouvrage

- 127 p.
ISBN 978-981-270-380-4

Localisation : Ouvrage RdC (NATI)

dynamique chaotique # fonctions de Lyapunov #
stabilité # système physique dynamique # analyses des séries temporelles # méthode stochastique # mécanique statistique # solutions récurrantes d'une EDO # analyse de données expérimentales

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## Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten laplacians Helffer, Bernard ; Nier, Francis | Springer 2005

Ouvrage

- 209 p.
ISBN 978-3-540-24200-0

Lecture notes in mathematics , 1862

Localisation : Collection 1er étage

opérateur hypoelliptique # opérateur de Fokker-Planck # laplacien de Witten

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## Mathematical theory of nonequilibrium steady states :on the frontier of probability and dynamical systems Jiang, D. Q. ; Qian, Min ; Qian, M. P. | Springer 2004

Ouvrage

- 280 p.
ISBN 978-3-540-20611-8

Lecture notes in mathematics , 1833

Localisation : Collection 1er étage

physique statistique # production d'entropie # irréversibilité # équilibre # chaîne de Markov # processus de diffusion # système dynamique hyperbolique # fluctuation de Gallavotti-Cohen

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