Vedran Sohinger

Gibbs measures of 1D quintic nonlinear Schrödinger equations as limits of many-body quantum Gibbs states

Gibbs measures of nonlinear Schrödinger equations (NLS) 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. We study the problem of the derivation of Gibbs measures as mean-field limits of Gibbs states in many-body quantum mechanics.

In earlier joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein, we studied this problem for variants of the cubic NLS with defocusing (positive) interactions. The latter models physically correspond to pair interactions of bosons. In these works, the problem was studied in dimensions d=1,2,3.

In this talk, I will explain how one can obtain an analogous result for the 1D quintic NLS, which corresponds to three-body interactions of bosons. In this setting, we consider focusing interactions, due to which we need to add a truncation in the mass and rescaled particle number. Our methods allow us to obtain a microscopic derivation of the time-dependent correlation functions for the 1D quintic NLS. This is joint work with Andrew Rout.