Lisbon WADE — Webinar in Analysis and Differential Equations  

21/01/2021, 14:00 — 15:00 — Online
Charles Collot, Cergy Paris Université

On the stability of equilibria for infinitely many particles

We study the evolution of a system of particles. Instead of the usual Hartree equation for density matrices, we consider the following equivalent model, proposed by de Suzzoni, of a Hartree type equation but for a random field:$$iX_t=-\Delta X +(w*\mathbb E(|X|^2))X.$$Above, $X:[0,T]\times \mathbb R^d\times \Omega$ is a time-dependent random field, $w$ a pair interaction potential, $*$ the convolution product and $\mathbb E$ the expectation. This equation admits equilibria which are random Gaussian fields whose laws are invariant by time and space translations. They are hence not localised and represent an infinite number of particles. We give a stability result under certain hypotheses, by showing that small perturbations scatter as $t\rightarrow \pm \infty$ to linear waves. This is joint work with de Suzzoni.