Gibbs measures of nonlinear Schr\"odinger equations as limits of many-body quantum states in dimensions $d \leq 3$

TL;DR

This paper demonstrates that Gibbs measures for nonlinear Schrödinger equations can be derived as high-temperature limits of many-body quantum states in dimensions 1 to 3, involving renormalization techniques and perturbative expansions.

Contribution

It establishes a rigorous connection between many-body quantum thermal states and nonlinear Schrödinger Gibbs measures in low dimensions, including renormalization procedures.

Findings

Gibbs measures are obtained as limits of quantum states at high temperature.

Renormalization of chemical potential is necessary in dimensions 2 and 3.

The proof employs diagrammatic perturbation theory and Borel resummation.

Abstract

We prove that Gibbs measures of nonlinear Schr\"odinger equations arise as high-temperature limits of thermal states in many-body quantum mechanics. Our results hold for defocusing interactions in dimensions $d =1,2,3$. The many-body quantum thermal states that we consider are the grand canonical ensemble for $d = 1$ and an appropriate modification of the grand canonical ensemble for $d =2,3$. In dimensions $d =2,3$, the Gibbs measures are supported on singular distributions, and a renormalization of the chemical potential is necessary. On the many-body quantum side, the need for renormalization is manifested by a rapid growth of the number of particles. We relate the original many-body quantum problem to a renormalized version obtained by solving a counterterm problem. Our proof is based on ideas from field theory, using a perturbative expansion in the interaction, organized by using a diagrammatic representation, and on Borel resummation of the resulting series.