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Derivation of renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann equations in an interacting Bose gas

Thomas Chen, Michael Hott

TL;DR

The paper addresses deriving renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann dynamics for an interacting Bose gas near Bose-Einstein condensation. It develops a renormalization framework employing Weyl and Bogoliubov transformations to separate fast HFB fluctuations from slow Boltzmann-type kinetics, yielding sharp error bounds and extending validity to times up to $t\sim(\log N)^2$. The main results establish renormalized HFB equations for the condensate and fluctuations, plus explicit cubic and quartic Boltzmann collision kernels that drive the long-time dynamics, with rigorous control of remainder terms. This provides a rigorous, unconditional derivation of coupled HFB-QBE dynamics in a mesoscopic regime and suggests extensibility to smaller $1/N$ orders, with potential impact on the theoretical understanding of non-equilibrium Bose gases.

Abstract

Our previous work [37] presented a rigorous derivation of quantum Boltzmann equations near a Bose-Einstein condensate (BEC). Here, we extend it with a complete characterization of the leading order fluctuation dynamics. For this purpose, we correct the latter via an appropriate Bogoliubov rotation, in partial analogy to the approach by Grillakis-Machedon et al. [59], in addition to the Weyl transformation applied in [37]. Based on the analysis of the third order expansion of the BEC wave function, and the second order expansions of the pair-correlations, we show that through a renormalization strategy, various contributions to the effective Hamiltonian can be iteratively eliminated by an appropriate choice of the Weyl and Bogoliubov transformations. This leads to a separation of renormalized Hartree-Fock-Bogoliubov (HFB) equations and quantum Boltzmann equations. A multitude of terms that were included in the error term in [37] are now identified as contributions to the HFB renormalization terms. Thereby, the error bound in the work at hand is improved significantly. To the given order, it is now sharp, and matches the order or magnitude expected from scaling considerations. Consequently, we extend the time of validity to $t\sim (\log N)^2$ compared to $t\sim (\log N/\log \log N)^2$ before. We expect our approach to be extensible to smaller orders in $\frac1N$.

Derivation of renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann equations in an interacting Bose gas

TL;DR

The paper addresses deriving renormalized Hartree-Fock-Bogoliubov and quantum Boltzmann dynamics for an interacting Bose gas near Bose-Einstein condensation. It develops a renormalization framework employing Weyl and Bogoliubov transformations to separate fast HFB fluctuations from slow Boltzmann-type kinetics, yielding sharp error bounds and extending validity to times up to . The main results establish renormalized HFB equations for the condensate and fluctuations, plus explicit cubic and quartic Boltzmann collision kernels that drive the long-time dynamics, with rigorous control of remainder terms. This provides a rigorous, unconditional derivation of coupled HFB-QBE dynamics in a mesoscopic regime and suggests extensibility to smaller orders, with potential impact on the theoretical understanding of non-equilibrium Bose gases.

Abstract

Our previous work [37] presented a rigorous derivation of quantum Boltzmann equations near a Bose-Einstein condensate (BEC). Here, we extend it with a complete characterization of the leading order fluctuation dynamics. For this purpose, we correct the latter via an appropriate Bogoliubov rotation, in partial analogy to the approach by Grillakis-Machedon et al. [59], in addition to the Weyl transformation applied in [37]. Based on the analysis of the third order expansion of the BEC wave function, and the second order expansions of the pair-correlations, we show that through a renormalization strategy, various contributions to the effective Hamiltonian can be iteratively eliminated by an appropriate choice of the Weyl and Bogoliubov transformations. This leads to a separation of renormalized Hartree-Fock-Bogoliubov (HFB) equations and quantum Boltzmann equations. A multitude of terms that were included in the error term in [37] are now identified as contributions to the HFB renormalization terms. Thereby, the error bound in the work at hand is improved significantly. To the given order, it is now sharp, and matches the order or magnitude expected from scaling considerations. Consequently, we extend the time of validity to compared to before. We expect our approach to be extensible to smaller orders in .
Paper Structure (32 sections, 28 theorems, 354 equations)

This paper contains 32 sections, 28 theorems, 354 equations.

Table of Contents

  1. Statements and Declarations
  2. Acknowledgements
  3. Introduction
  4. Summary of previous results
  5. New contributions
  6. Statement of results
  7. Initial state
  8. Fluctuation dynamics
  9. Main result
  10. Renormalized HFB equations
  11. Boltzmann collision kernels
  12. Main theorem
  13. Sketch of the proof
  14. Derivation of leading order terms
  15. Density expansion
  16. ...and 17 more sections

Key Result

Proposition 2.7

Assume that $\hat{v}\in L^1_{\sqrt{1+E}}\cap L^\infty({\Lambda^*})$, and that $v\geq0$. Let $(\phi_0,\Gamma_0,\Sigma_0)\in{\mathcal{X}}^1$, such that $\Gamma^{T}_0\geq0$ and $|\Sigma^{T}_0|^2\leq (\Gamma^{T}_0+1)\Gamma^{T}_0$. Let $(\phi,\Gamma,\Sigma)\in C^0_t([0,T_0),{\mathcal{X}}^1)\cap C^1_t([0,

Theorems & Definitions (59)

  • Definition 2.1: Quasifree state
  • Remark 2.2
  • Definition 2.3: Number conserving state
  • Definition 2.4: Translation invariance
  • Remark 2.5
  • Definition 2.6: Mild solution
  • Proposition 2.7: Global well-posedness
  • Theorem 2.8
  • Remark 2.9
  • Corollary 2.10
  • ...and 49 more