In this paper, we introduce a novel method for deriving higher order corrections to the mean-field description of the dynamics of interacting bosons. More precisely, we consider thedynamics ofNd-dimensional bosons for largeN. The bosons initially form a Bose–Einsteincondensate and interact with each other via a pair potential of the form(N−1)−1Ndβv(Nβ·)forβ∈[0,14d).WederiveasequenceofN-body functions which approximate the true many-body dynamics inL2(RdN)-norm to arbitrary precision in powers ofN−1. The approximatingfunctions are constructed as Duhamel expansions of finite order in terms of the first quantisedanalogue of a Bogoliubov time evolution.

The derivation of mean-field limits for quantum systems at zero temperature has attracted many researchers in the last decades. Recent developments are the consideration of pair correlations in the effective description, which lead to a much more precise description of both spectral properties and the dynamics of the Bose gas in the weak coupling limit. While mean-field results typically lead to convergence for the reduced density matrix only, one obtains norm convergence when considering the pair correlations proposed by Bogoliubov in his seminal 1947 paper. In this article, we consider an interacting Bose gas in the case where both the volume and the density of the gas tend to infinity simultaneously. We assume that the coupling constant is such that the self-interaction of the fluctuations is of leading order, which leads to a finite (nonzero) speed of sound in the gas. In our first main result, we show that the difference between the N-body and the Bogoliubov description is small in L2 as the density of the gas tends to infinity and the volume does not grow too fast. This describes the dynamics of delocalized excitations of the order of the volume. In our second main result, we consider an interacting Bose gas near the ground state with a macroscopic localized excitation of order of the density. We prove that the microscopic dynamics of the excitation coming from the N-body Schrödinger equation converges to an effective dynamics which is free evolution with the Bogoliubov dispersion relation. The main technical novelty are estimates for all moments of the number of particles outside the condensate for large volume, and in particular control of the tails of their distribution.

The derivation of effective equations for interacting many body systems has seen a lot of progress in the recent years. While dealing with classical systems, singular potentials are quite challenging (Hauray and Jabin in Annales scientifiques de l’École Normale Supérieure, 2013, Lazarovici and Pickl in Arch Ration Mech Anal 225(3):1201–1231, 2017) comparably strong results are known to hold for quantum systems (Knowles and Pickl in Comm Math Phys 298:101–139, 2010). In this paper, we wish to show how techniques developed for the derivation of effective descriptions of quantum systems can be used for classical ones. While our future goal is to use these ideas to treat singularities in the interaction, the focus here is to present how quantum mechanical techniques can be used for a classical system and we restrict ourselves to regular two-body interaction potentials. In particular we compute a mean field limit for the Hamilton Vlasov system in the sense of (Fröhlich et al. in Comm Math Phys 288:1023–1058, 2009; Neiss in Arch Ration Mech Anal. https://doi.org/10.1007/s00205-018-1275-8) that arises from classical dynamics. The structure reveals strong analogy to the Bosonic quantum mechanical ensemble of the many-particle Schrödinger equation and the Hartree equation as its mean field limit (Pickl in arXiv:0808.1178v1, 2008).