Large-scale limits of interacting particle systems

IHÉS, France / October 4-8, 2021

**Nicolas Rougerie** (ENS de Lyon)

*Bose gases at positive temperature and non-linear Gibbs measures.*

I shall discuss a certain mean-field limit for the positive temperature
equilibria (Gibbs states) of the interacting Bose gas. This serves as a
toy model for the understanding of the phase transition towards a
Bose-Einstein condensate when the temperature is lowered. The limit
model we rigorously derive is a classical field theory, based on a
certain non-linear Gibbs measure. The latter is a natural invariant of
the non-linear Schrödinger equation approximating the dynamics of the
Bose gas. It is also the basic ingredient of the Euclidean approach to
constructive quantum field theory, as well as the large-time asymptote
for the stochastic non-linear heat equation. A difficulty is that this
non-linear Gibbs measure lives on low regularity distributional spaces,
so that the non-linearity has to be understood in a renormalized sense.
I shall put emphasis on the control of the renormalization procedure at
the level of the quantum many-body model.

Based on joint work with Mathieu Lewin and Phan Thành Nam