  In a recent paper, in collaboration with Mathieu Lewin and Phan Th{\`a}nh
Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like
functionals could be derived from many-body quantum mechanics, in a mean-field
limit. This text summarizes these findings. It focuses on the simplest, but
most physically relevant, case we could treat so far, namely that of the
defocusing cubic NLS functional on a 1D interval. The measure obtained in the
limit, which (almost) lives over H^{1/2} , has been previously shown to be
invariant under the NLS flow by Bourgain.
