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Effective Dynamics for Weakly Interacting Bosons in an Iterated High-Density Thermodynamic Limit

Daniele Ferretti, Kalle Koskinen

TL;DR

This work analyzes the time evolution of a weakly interacting Bose gas on a 3D torus in a high-density thermodynamic limit with coupling scaled as $1/\rho$. Adopting a grand canonical Fock-space framework and Weyl coherence, it derives a fluctuation dynamics around a time-dependent Hartree condensate and proves that, for each fixed time in a finite interval, the one-particle density matrix converges to the Hartree condensate projector in the iterated limit $L\to\infty$ followed by $\rho\to\infty$. The convergence is quantified in trace (and Hilbert-Schmidt) norm, with a rate governed by the initial quasi-vacuum’s tail and energy quasi-self-consistency, under quasi-complete Bose-Einstein condensation. The results establish a rigorous link between microscopic many-body dynamics and nonlinear Hartree evolution in a thermodynamic torus setting, including controlled fluctuations and energy consistency, and highlight the importance of limit ordering in deriving effective condensate dynamics. Overall, the paper advances mathematical understanding of Bose gases in the high-density regime by providing a robust, quantitative derivation of Hartree dynamics as the emergent macroscopic description.

Abstract

We study the time evolution of weakly interacting Bose gases on a three-dimensional torus of arbitrary volume. The coupling constant is supposed to be inversely proportional to the density, which is considered to be large and independent of the particle number. We take into account a class of initial states exhibiting quasi-complete Bose-Einstein condensation. For each fixed time in a finite interval, we prove the convergence of the one-particle reduced density matrix towards the projection onto the normalised order parameter describing the condensate - evolving according to the Hartree equation - in the iterated limit where the volume (and therefore the particle number), and subsequently the density go to infinity. The rate of convergence depends only on the density and on the decay of both the expected number of particles and the energy of the initial quasi-vacuum state.

Effective Dynamics for Weakly Interacting Bosons in an Iterated High-Density Thermodynamic Limit

TL;DR

This work analyzes the time evolution of a weakly interacting Bose gas on a 3D torus in a high-density thermodynamic limit with coupling scaled as . Adopting a grand canonical Fock-space framework and Weyl coherence, it derives a fluctuation dynamics around a time-dependent Hartree condensate and proves that, for each fixed time in a finite interval, the one-particle density matrix converges to the Hartree condensate projector in the iterated limit followed by . The convergence is quantified in trace (and Hilbert-Schmidt) norm, with a rate governed by the initial quasi-vacuum’s tail and energy quasi-self-consistency, under quasi-complete Bose-Einstein condensation. The results establish a rigorous link between microscopic many-body dynamics and nonlinear Hartree evolution in a thermodynamic torus setting, including controlled fluctuations and energy consistency, and highlight the importance of limit ordering in deriving effective condensate dynamics. Overall, the paper advances mathematical understanding of Bose gases in the high-density regime by providing a robust, quantitative derivation of Hartree dynamics as the emergent macroscopic description.

Abstract

We study the time evolution of weakly interacting Bose gases on a three-dimensional torus of arbitrary volume. The coupling constant is supposed to be inversely proportional to the density, which is considered to be large and independent of the particle number. We take into account a class of initial states exhibiting quasi-complete Bose-Einstein condensation. For each fixed time in a finite interval, we prove the convergence of the one-particle reduced density matrix towards the projection onto the normalised order parameter describing the condensate - evolving according to the Hartree equation - in the iterated limit where the volume (and therefore the particle number), and subsequently the density go to infinity. The rate of convergence depends only on the density and on the decay of both the expected number of particles and the energy of the initial quasi-vacuum state.

Paper Structure

This paper contains 20 sections, 24 theorems, 321 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. The Model
  4. Our main result, Theorem \ref{['th:gamma1Convergence']}
  5. Background and Related Works
  6. Outline of the Strategy
  7. Setting and Statement of Results
  8. Second Quantisation
  9. Quasi-Complete Bose-Einstein Condensates
  10. Main Results
  11. Generator of the Fluctuation Dynamics
  12. Properties of Quasi-Complete Bose-Einstein Condensates
  13. The Hartree Equation on the Torus
  14. Global Well-Posedness
  15. The Hartree Equation in Momentum Space
  16. ...and 5 more sections

Key Result

Lemma 2.1

Assume the potential satisfies the decay condition eq:potentialDecay with $\hat{V} _\infty\!\geq\mspace{-1.5mu} 0$ andCalling for such a decay for the Fourier transform of $V_\infty$ implies $V_\infty$ is at least $C^{\mspace{0.75mu}4}(\mathbb{R}^3)$.$\delta_2\!>\mspace{-1.5mu} 4$. Then, given the w admits a unique solution $t\longmapsto\Psi^{\mspace{1.5mu} t}_{\!\varrho,\mspace{1.5mu} L}\!\in C^{

Theorems & Definitions (76)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1: Quasi-Vacuum States
  • Remark 2.3
  • Definition 2.2: Quasi-Coherent States
  • Remark 2.4
  • Definition 2.3: Quasi-Complete BEC
  • Remark 2.5
  • Definition 2.4: Energy Quasi-Self-Consistency
  • Remark 2.6
  • ...and 66 more