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Large deviation principles for the Gross Pitaevskii Gibbs measure at low temperature

Liam Packer, Kihoon Seong, Philippe Sosoe

TL;DR

This work proves a large deviation principle for the focusing Gross–Pitaevskii Gibbs measure in the low-temperature limit under a mixed canonical in energy and microcanonical in mass conditioning. The analysis proceeds via the grand canonical ensemble, employing the Laplace principle and Gamma-convergence to obtain a rate function $J^D(\phi)=H(\phi)-\inf_{M(\phi)=D}H(\phi)$, while showing that diverging Wick renormalization constants cancel in the rate function. Consequently, the conditional Gibbs measure concentrates on the soliton manifold $\mathcal{M}^D$, i.e., the minimizers of $H$ under $M(\phi)=D$, and the long-time GP dynamics are captured by fluctuations around these ground states. The results extend prior mixed-ensemble LDPs to unbounded quartic interactions and singular conditioning, leveraging confinement by the harmonic potential to obtain a nontrivial infinite-volume measure and making the macroscopic behavior robust to renormalization effects.

Abstract

We prove the large deviation principle for the conditional Gibbs measure associated with the focusing Gross Pitaevskii equation in the low temperature regime. This conditional measure is of mixed type, being canonical in energy and microcanonical in particle number. In particular, our result extends the large deviation principle for the mixed ensemble studied by Ellis, Jordan, Otto, and Turkington to a more singular setting, where the interaction potential is unbounded and the conditional event involves diverging renormalization constants. As a consequence of the large deviation principle, the Gibbs measure concentrates along the soliton manifold in the low temperature limit.

Large deviation principles for the Gross Pitaevskii Gibbs measure at low temperature

TL;DR

This work proves a large deviation principle for the focusing Gross–Pitaevskii Gibbs measure in the low-temperature limit under a mixed canonical in energy and microcanonical in mass conditioning. The analysis proceeds via the grand canonical ensemble, employing the Laplace principle and Gamma-convergence to obtain a rate function , while showing that diverging Wick renormalization constants cancel in the rate function. Consequently, the conditional Gibbs measure concentrates on the soliton manifold , i.e., the minimizers of under , and the long-time GP dynamics are captured by fluctuations around these ground states. The results extend prior mixed-ensemble LDPs to unbounded quartic interactions and singular conditioning, leveraging confinement by the harmonic potential to obtain a nontrivial infinite-volume measure and making the macroscopic behavior robust to renormalization effects.

Abstract

We prove the large deviation principle for the conditional Gibbs measure associated with the focusing Gross Pitaevskii equation in the low temperature regime. This conditional measure is of mixed type, being canonical in energy and microcanonical in particle number. In particular, our result extends the large deviation principle for the mixed ensemble studied by Ellis, Jordan, Otto, and Turkington to a more singular setting, where the interaction potential is unbounded and the conditional event involves diverging renormalization constants. As a consequence of the large deviation principle, the Gibbs measure concentrates along the soliton manifold in the low temperature limit.
Paper Structure (20 sections, 17 theorems, 218 equations)

This paper contains 20 sections, 17 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Large deviation principle (LDP) for the mixed ensemble
  3. Main result
  4. Motivation and comments on the literature
  5. Large deviation principle for the mixed ensemble
  6. Gross-Pitaevskii Gibbs measure and dynamical problem
  7. Translation invariant Gibbs measures
  8. Structure of the proof
  9. Notations and preliminary results
  10. Notations
  11. Harmonic oscillator operator
  12. Gaussian measure associated with the harmonic oscillator
  13. Variational characterization of the minimizers
  14. Tools from stochastic analysis
  15. Large deviation principles for the grand canonical ensemble
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $\mathcal{S}=H^{-\eta}(\mathbb{R})$ for any $\eta>0$, or $L^p(\mathbb{R})$ for any finite $p>2$. Then, there exists $D^*>0$ such that for every $D \ge D^*$, the mixed ensemble $\rho^D_{\varepsilon,r}$ in Gibbs4 satisfies a large deviation principle on $\mathcal{S}$ with rate function $J^D$ and s

Theorems & Definitions (39)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5
  • Remark 2.6
  • ...and 29 more