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arXiv:2608.26477

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

Phan Thành Nam, Rongchan Zhu, Xiangchan Zhu

Published Aug 27, 2026• 0 citations
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Abstract

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, 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.

DOI
10.48550/arxiv.2608.26477

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The derivation of the Hartree equation from many-body systems of Bosons in the mean field limit has been very intensively studied in the last couple of years. However, very few results exist showing convergence of the kth marginal of the N-body density matrix to the projection to the k-fold tensor product of the solution of the Hartree equation in stronger trace norms like the energy trace norm, see the work of Michelangeli and Schlein [Commun. Math. Phys. 311(3), 645–687 (2012)] and Lührmann [J. Math. Phys. 53(2), 022105 (2012)]. This issue is from a physical view point very important. The reason is that one can then approximate expectation values of certain observables of the N-body system by means of the Hartree equation, with relaxation of the very restrictive assumption that the observables are bounded operators. Here we consider the non-relativistic case. We prove, assuming only H1-regularity of the initial data, convergence in the energy trace norm without rates, and convergence in any other weaker Sobolev trace norm with rates. Our proof is simple and uses the functional aN introduced by Pickl [Lett. Math. Phys. 97(2), 151–164 (2011)].