Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d\le 3$

Abstract

We prove that Gibbs measures of nonlinear Schrödinger 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.

References in corpus (9)

• Random data Cauchy theory for supercritical wave equations I: Local theory
• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction
• Random data Cauchy theory for supercritical wave equations II : A global existence result
• Quantum many-body fluctuations around nonlinear Schrödinger dynamics
• On the rate of convergence for the mean field approximation of many-body quantum dynamics
• Bogoliubov correction to the mean-field dynamics of interacting bosons
• Beyond mean field: On the role of pair excitations in the evolution of condensates
• The Gross-Pitaevskii hierarchy on general rectangular tori

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• Classical field theory limit of many-body quantum Gibbs states in 2D and 3D
• Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three
• A path-integral analysis of interacting Bose gases and loop gases
• Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases
• Differential Equations of Quantum Mechanics