This paper explains how a probability measure used to describe a one‑dimensional nonlinear wave equation can be obtained from a quantum many‑particle system. The measure, called a Gibbs measure, weights field configurations by their energy. In the focusing case the energy is not positive, so the measure only exists after cutting off large total mass. The authors show that the same Gibbs measure appears in a natural limit of quantum thermal states even when the cut‑off is the roughest possible: an indicator function that simply forbids mass above a fixed level.

Concretely, the authors work on the circle and study the focusing quintic nonlinear Schrödinger equation (the local quintic NLS is recovered by taking a delta interaction). They start from a bosonic quantum system on Fock space with a three‑body interaction and a scaled particle number parameter ε. They form the quantum Gibbs state with a truncation that depends only on particle number and then take the semiclassical/mean‑field limit ε → 0. Using a perturbative expansion introduced by Fröhlich, Knowles, Schlein, and Sohinger (2017), together with a Wigner measure method and an inductive argument, they control the explicit terms in the expansion despite the lack of smoothness of the indicator cut‑off.

The main rigorous statement in the excerpt is that, for bounded interaction potentials and for each fixed particle number p, the p‑particle correlation functions of the quantum Gibbs state converge in trace norm to the classical p‑particle correlation functions of the Gibbs measure as ε → 0. The authors also prove the convergence of the quantum relative partition function to the classical partition function. They treat both the nonlocal Hartree version of the model and, in particular, the local quintic case by taking the interaction to be a delta and choosing the cut‑off at the known optimal level from earlier work.