  We prove that nonlinear Gibbs measures can be obtained from the corresponding
many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where
the temperature T diverges and the interaction behaves as 1/T. We proceed by
characterizing the interacting Gibbs state as minimizing a functional counting
the free-energy relatively to the non-interacting case. We then perform an
infinite-dimensional analogue of phase-space semiclassical analysis, using fine
properties of the quantum relative entropy, the link between quantum de Finetti
measures and upper/lower symbols in a coherent state basis, as well as
Berezin-Lieb type inequalities. Our results cover the measure built on the
defocusing nonlinear Schr{\"o}dinger functional on a finite interval, as well
as smoother interactions in dimensions d\textgreater{}1.
