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Dynamics of mean-field bosons at positive temperature

Marco Caporaletti, Andreas Deuchert, Benjamin Schlein

TL;DR

The article develops a rigorous mean-field analysis of a trapped Bose gas at positive temperature, focusing on the dynamics after removing the trap. It shows that the leading-order one-particle density matrix of the evolving state remains that of an ideal gas, with the condensate replaced by the Hartree minimizer, while the thermal cloud evolves essentially freely; fluctuations around this leading order are governed by a Hartree–Fock–Bogoliubov framework. The authors introduce a fluctuation dynamics on a doubled Fock space, derive its generator, and prove Gronwall-type bounds that control the growth of excitations and establish convergence of the full many-body dynamics to the effective Hartree/free picture for large $N$ under suitable temperature and interaction assumptions. The results rely on a detailed interplay between Araki–Woods purification, Bogoliubov transformations, and truncated fluctuation dynamics to manage the growth of excitations, yielding precise, albeit double- or triple-exponential in time, error estimates. Collectively, the work provides a rigorous underpinning for the coexistence and separate evolution of a macroscopic condensate and a dilite thermal cloud in a positive-temperature mean-field Bose gas, with implications for understanding real-time BEC dynamics in finite-temperature settings.

Abstract

We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in arXiv:2009.00992 that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in $N$, as $N \to \infty$, by the one of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree-Fock-Bogoliubov equations to define a fluctuation dynamics.

Dynamics of mean-field bosons at positive temperature

TL;DR

The article develops a rigorous mean-field analysis of a trapped Bose gas at positive temperature, focusing on the dynamics after removing the trap. It shows that the leading-order one-particle density matrix of the evolving state remains that of an ideal gas, with the condensate replaced by the Hartree minimizer, while the thermal cloud evolves essentially freely; fluctuations around this leading order are governed by a Hartree–Fock–Bogoliubov framework. The authors introduce a fluctuation dynamics on a doubled Fock space, derive its generator, and prove Gronwall-type bounds that control the growth of excitations and establish convergence of the full many-body dynamics to the effective Hartree/free picture for large under suitable temperature and interaction assumptions. The results rely on a detailed interplay between Araki–Woods purification, Bogoliubov transformations, and truncated fluctuation dynamics to manage the growth of excitations, yielding precise, albeit double- or triple-exponential in time, error estimates. Collectively, the work provides a rigorous underpinning for the coexistence and separate evolution of a macroscopic condensate and a dilite thermal cloud in a positive-temperature mean-field Bose gas, with implications for understanding real-time BEC dynamics in finite-temperature settings.

Abstract

We study the time evolution of an initially trapped weakly interacting Bose gas at positive temperature, after the trapping potential has been switched off. It has been recently shown in arXiv:2009.00992 that the one-particle density matrix of Gibbs states of the interacting trapped gas is given, to leading order in , as , by the one of the ideal gas, with the condensate wave function replaced by the minimizer of the Hartree energy functional. We show that this structure is stable with respect to the many-body evolution in the following sense: the dynamics can be approximated in terms of the time-dependent Hartree equation for the condensate wave function and in terms of the free evolution for the thermally excited particles. The main technical novelty of our work is the use of the Hartree-Fock-Bogoliubov equations to define a fluctuation dynamics.
Paper Structure (23 sections, 16 theorems, 298 equations)

This paper contains 23 sections, 16 theorems, 298 equations.

Table of Contents

  1. Introduction and main results
  2. Background and summary
  3. Notation
  4. The model
  5. Equilibrium states of trapped Bose gases
  6. Construction of the initial data
  7. Main result
  8. Proof strategy and organization of the article
  9. Properties of the 1-pdm of the ideal gas and of Hartree minimizers
  10. The Hartree--Fock--Bogoliubov equations
  11. Well-posedness
  12. Approximation of the HFB dynamics
  13. Fluctuation dynamics
  14. Weyl operators and Bogoliubov transformations
  15. Construction of the fluctuation dynamics
  16. ...and 8 more sections

Key Result

Theorem 1.2

We assume that $v \in L^1(\mathbb{R}^3) \cap W^{1,p}(\mathbb{R}^3)$ with $p>3$ satisfies $v(x) = v(-x)$ for a.e. $x \in \mathbb{R}^3$ and that $\hat{v} \in L^1(\mathbb{R}^3)$, where $\hat{v}$ denotes the Fourier transform of v. Moreover, we assume that the pair $(\phi,\gamma)$ satisfies Assumption a where $\Vert \cdot \Vert_{\mathcal{L}^1}$ denotes the trace norm. The function $\phi_t$ is the solu

Theorems & Definitions (36)

  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['prop:gaphi']}
  • Theorem 3.1
  • Proposition 3.2
  • ...and 26 more