## Séminaire Orléans

**Analysis of the relaxation time of a large bistable particle system at low temperature**

Giacomo Di Gesu (CERMICS)

Thursday 08 October 2015 14:00 - Orléans - Salle de Séminaire

**Résumé :**

A large system of strongly coupled diffusions on unbounded state space moving in a double-well potential is considered. This system can be seen as a spatially discrete approximation of the stochastically perturbed Allen-Cahn equation on the one-dimensional torus, which is a basic and widely studied stochastic partial differential equation. In the small temperature regime the typical picture of a so-called metastable dynamics emerges: the system quickly reaches a local equilibrium in one of the two wells, depending on its initial condition; this state endures for a very long time, until a sufficiently large stochastic fluctuation enables the system to overcome the energetic barrier separating the two wells and thus to distribute according to the global equilibrium. I will discuss some basic features of the model and then present some analytical results, obtained in collaboration with Dorian Le Peutrec, which quantify the mentioned slowdown in the relaxation to equilibrium. More specifically, these results concern the asymptotic behaviour of the spectral gap and log-Sobolev constant in regimes of low temperature and large number of particles.

**Liens :**