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PreprintarXiv licencearXiv

Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D

  • Phan ThĂ nh Nam
  • Rongchan Zhu
  • Xiangchan Zhu
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Abstract

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus 𝕋d, d=2,3, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

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Cite

BibTeXRIS
BibTeX
@misc{nam2026dynamical,
  title = {{Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D}},
  author = {Phan ThĂ nh Nam and Rongchan Zhu and Xiangchan Zhu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.26477},
  eprint = {2608.26477},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.26477},
}
RIS
TY  - UNPB
TI  - Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D
AU  - Phan ThĂ nh Nam
AU  - Rongchan Zhu
AU  - Xiangchan Zhu
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.26477
UR  - https://arxiv.org/abs/2608.26477
AB  - In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus 𝕋^d, d=2,3, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.
ER  -