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A representation formula for large deviations rate functionals of invariant measures on the one dimensional torus

Alessandra Faggionato, Davide Gabrielli (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider a generic diffusion on the 1D torus and give a simple representation formula for the large deviation rate functional of its invariant probability measure, in the limit of vanishing noise. Previously, this rate functional had been characterized by M. I. Freidlin and A. D. Wentzell as solution of a rather complex optimization problem. We discuss this last problem in full generality and show that it leads to our formula. We express the rate functional by means of a geometric transformation...

Derivation of Langevin dynamics in a nonzero background flow field

Matthew Dobson, Frédéric Legoll, Tony Lelièvre, Gabriel Stoltz (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We propose a derivation of a nonequilibrium Langevin dynamics for a large particle immersed in a background flow field. A single large particle is placed in an ideal gas heat bath composed of point particles that are distributed consistently with the background flow field and that interact with the large particle through elastic collisions. In the limit of small bath atom mass, the large particle dynamics converges in law to a stochastic dynamics. This derivation follows the ideas of [P. Calderoni,...

Macroscopic models of collective motion and self-organization

Pierre Degond, Amic Frouvelle, Jian-Guo Liu, Sebastien Motsch, Laurent Navoret (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss it in view...

Markovian perturbation, response and fluctuation dissipation theorem

Amir Dembo, Jean-Dominique Deuschel (2010)

Annales de l'I.H.P. Probabilités et statistiques

We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is,...

The mean-field limit for the dynamics of large particle systems

François Golse (2003)

Journées équations aux dérivées partielles

This short course explains how the usual mean-field evolution PDEs in Statistical Physics - such as the Vlasov-Poisson, Schrödinger-Poisson or time-dependent Hartree-Fock equations - are rigorously derived from first principles, i.e. from the fundamental microscopic models that govern the evolution of large, interacting particle systems.

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