This book chapter is published open access.
We consider the linear Schrödinger equation describing quantum (bosonic) particles at equilibrium and study its behavior as tends to infinity. We place the system in the mean-field regime, in which the particles are very tightly packed but interact weakly. In this limit we prove that they become essentially independent and identically distributed according to a nonlinear partial differential equation. Our main tool is the quantum de Finetti theorem, an abstract result about how independence can arise due to symmetry in such systems.
By considerably increasing the randomness in the system, we can also obtain nonlinear Gibbs measures. Those are probability measures over an infinite-dimensional space, which play a major role in different areas of mathematics. The two- and three-dimensional cases are particularly challenging due to the necessity of using a renormalization procedure to cancel infinities.