Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Quantum Many-Body States in Dimension d ≤ 3

Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. In Fröhlich et al. (Commun Math Phys 356(3), 883–980, 2017), we prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. We consider bounded defocusing interaction potentials in dimensions d = 1, 2, 3, and we work either on 𝕋 d $$\mathbb {T}^d$$ or on ℝ d $$\mathbb {R}^d$$ with a confining potential. The analogous problem for d = 1 and in higher dimensions with smooth nontranslation-invariant interactions was recently studied by Lewin, Nam, and Rougerie (J Éc Polytech Math 2:65–115, 2015) by means of entropy methods. In our work, we apply a perturbative expansion of the interaction, motivated by ideas from field theory. The terms of the expansion are analysed using a diagrammatic representation and their sum is controlled using Borel resummation techniques. When d = 2,3, we apply a Wick ordering renormalisation procedure. The latter allows us to deal with the rapid growth of the number of particles. Moreover, in the one-dimensional setting our methods allow us to obtain a microscopic derivation of time-dependent correlation functions for the cubic nonlinear Schrödinger equation (Fröhlich et al., Adv Math 353:67–115, 2019). All results presented in this chapter are based on joint work with Jürg Fröhlich (ETH Zürich), Antti Knowles (University of Geneva), and Benjamin Schlein (University of Zürich).