Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1605.07095 · Communications in Mathematical Physics

Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Many-Body Quantum States in Dimensions $${d \leqslant 3}$$ d ⩽ 3

Jürg Martin Fröhlich, Antti Knowles, Benjamin Schlein, Vedran Sohinger

Published Oct 5, 2017• 27 citations
arXivHTMLPDF
DOI
10.1007/s00220-017-2994-7

Related papers

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

Jürg Martin Fröhlich, Antti Knowles, Benjamin Schlein +1 more

•Dec 31, 2017•17 citations

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

Vedran Sohinger

Springer eBooks•Jan 1, 2020•5 citations

The nonlinear Schrödinger equation in one dimension as a solvable model displaying certain basic features of the many-body problem in the thermodynamical limit

Francesco Calogero, A. Degasperis

Il Nuovo cimento della Società italiana di fisica. B/Il Nuovo cimento B•Jun 1, 1976•1 citation

Expansion of the Many-body quantum Gibbs state of the Bose–Hubbard model on a finite graph

Zied Ammari, Shahnaz Farhat, Sören Petrat

We consider the many-body quantum Gibbs state for the Bose–Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.

arXiv:2405.04055•Documenta Mathematica•Mar 20, 2025•0 citations

Limit Gibbs state for some classes of one-dimensional systems of quantum statistical mechanics

Yuri Suhov

Communications in Mathematical Physics•Jun 1, 1978•13 citations

A microscopic derivation of Gibbs measures for the 1D focusing cubic nonlinear Schrödinger equation

Andrew Rout, Vedran Sohinger

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states.Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting.Our methods are based on a perturbative expansion of the interaction, similarly as in [1].Due to the presence of the truncation, the obtained series have infinite radius of convergence.We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions.Within this framework, we also study time-dependent correlation functions.This is the first such known result in the focusing regime.

arXiv:2206.03392•Communications in Partial Differential Equations•Aug 3, 2023•7 citations
Gibbs Measures of Nonlinear Schrödinger Equations as Limits of Many-Body Quantum States in Dimensions d⩽3{d \leqslant 3}d⩽3 d ⩽ 3
Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions d⩽3d \leqslant 3d⩽3
We consider the many-body quantum Gibbs state for the Bose–Hubbard model on a finite graph at positive temperature. We scale the interaction with the inverse temperature, corresponding to a mean-field limit where the temperature is of the order of the average particle number. For this model it is known that the many-body Gibbs state converges, as temperature goes to infinity, to the Gibbs measure of a discrete nonlinear Schrödinger equation, i.e., a Gibbs measure defined in terms of a one-body theory. In this article we extend these results by proving an expansion to any order of the many-body Gibbs state with inverse temperature as a small parameter. The coefficients in the expansion can be calculated as vacuum expectation values using a recursive formula, and we compute the first two coefficients explicitly.
In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states.Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting.Our methods are based on a perturbative expansion of the interaction, similarly as in [1].Due to the presence of the truncation, the obtained series have infinite radius of convergence.We treat the case of bounded, L 1 and delta function interaction potentials, without any sign assumptions.Within this framework, we also study time-dependent correlation functions.This is the first such known result in the focusing regime.