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From bosonic canonical ensembles to non-linear Gibbs measures

van Duong Dinh, Nicolas Rougerie

Abstract

We study the mean-field limit of the 1D bosonic canonical ensemble in a superharmonic trap. This is the regime with temperature proportional to particle number, both diverging to infinity, and correspondingly scaled interactions. We prove that the limit model is a classical field theory based on a non-linear Schr{ö}dinger-Gibbs measure conditioned on the L2 mass, thereby obtaining a canonical analogue of previous results for the grand-canonical ensemble. We take advantage of this set-up with fixed mass to include focusing/attractive interactions/non-linearities in our study.

From bosonic canonical ensembles to non-linear Gibbs measures

Abstract

We study the mean-field limit of the 1D bosonic canonical ensemble in a superharmonic trap. This is the regime with temperature proportional to particle number, both diverging to infinity, and correspondingly scaled interactions. We prove that the limit model is a classical field theory based on a non-linear Schr{ö}dinger-Gibbs measure conditioned on the L2 mass, thereby obtaining a canonical analogue of previous results for the grand-canonical ensemble. We take advantage of this set-up with fixed mass to include focusing/attractive interactions/non-linearities in our study.

Paper Structure

This paper contains 21 sections, 22 theorems, 544 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Quantum and classical models
  4. Semi-classical limit.
  5. Proof strategy
  6. Possible extensions
  7. The Gibbs measure conditioned on mass
  8. More on definitions
  9. Dependence on the mass
  10. From the free canonical state to the fixed mass Gaussian measure
  11. Classical limit of the relaxed free canonical Gibbs state
  12. Convergence of the relaxed Gibbs state to the canonical Gibbs state
  13. Free energy upper bound
  14. Trial state and reduction to a finite dimensional estimate
  15. Semiclassics for the finite dimensional canonical ensemble.
  16. ...and 6 more sections

Key Result

Theorem 2.5

Let the interacting canonical Gibbs state $\Gamma_{mT,T,g}^c$ be defined as in eq:C ens with the particle number set as $N = mT, m >0$. Let the corresponding reduced density matrices be as in eq:def red mat. We fix $m>0,g \geq 0$ and let $T\to \infty$. Then, for any $k\geq 1$ strongly in the trace-class $\mathfrak S ^1 (\mathfrak h^k)$. In particular, the interacting Gibbs measure with fixed mass

Figures (1)

  • Figure 1: Derivation of the fixed mass Gaussian measure

Theorems & Definitions (44)

  • Definition 2.2: The reference Gaussian measure
  • Definition 2.3: Fixed mass Gaussian measure
  • Definition 2.4: The non-linear Gibbs measure
  • Theorem 2.5: Mean-field limit of the bosonic canonical ensemble
  • Proposition 3.1: Density matrices of the conditioned Gaussian measure
  • Proposition 3.2: Gibbs measure conditioned on mass
  • Lemma 3.3: Density functions of the $L^2$ norm
  • proof
  • proof : Proof of Proposition \ref{['theo-fix-mass-meas']}
  • proof : Proof of Proposition \ref{['pro:fixed mass meas']}
  • ...and 34 more