The modeling of cold atoms systems has known an increasing interest in the theoretical physics community, after the first experimental realizations of Bose Einstein condensates, some twenty years ago. In the recent years, mean field models taking account of thermal fluctuations, aiming to describe condensates close to critical condensation temperature have been introduced, as e.g. the so called “Stochastic Projected Gross Pitaevskii equation”.

We will describe some mathematical results about the global existence of solutions and long time dynamics of a non-truncated version of the model — which is a complex Ginzburg Landau equation with a confining harmonic potential and additive space-time white noise. We take advantage in particular of the fact that the Gibbs measure is invariant for both the reversible and non reversible dynamics. Additionally, we will describe some numerical methods aiming to approximate the equilibrium of the (truncated) equation, or to simulate transitions dynamics between two metastable states composed of different numbers of vortices.

This is a joint work with A. Debussche, R. Fukuizumi and R. Poncet. The work was supported by the JSPS KAKENHI Grant Number 15K04944 and 16KT0127, the ANR project ANR-12 MONU-0007 BECASIM, and the French government “Investissements d'Avenir” program ANR-11-LABX-0020-01.