Generalised BBGKY hierarchy for near-integrable dynamics

We study quantum and classical many-body Hamiltonian systems that combine integrable contact interactions with generic long-range two-body potentials. Starting from an ansatz for the state at time $t$, which we call the correlated fluid-cell ensemble, we show that the dynamics of local observables at macroscopic times and length scales can be cast into a generalized Bogoliubov-Born-Green-Kirkwood-Yvon (gBBGKY) hierarchy formulated in terms of the quasiparticle densities of the underlying integrable model and their correlations. We derive this hierarchy and validate these predictions against microscopic molecular-dynamics simulations, finding perfect agreement. At late times, the one-particle distribution relaxes via a Boltzmann-type scattering integral encoding the interplay between integrable contact processes and long-range collisions, whereas higher-point correlations remain strongly non-thermal on thermalization time scales, indicative of a form of incomplete or generalised thermalisation. Focusing on long-range dipolar quantum gases, where the relevant matrix elements can be obtained explicitly, we show that our collision integral reduces exactly to the Fermi golden rule result and provide a complete theoretical account of the experimental observations of Tang et al. (Phys.Rev.X 8, 021030 (2018)). More broadly, our framework extends the BBGKY program to regimes with strong local interactions, and applies to a wide class of experimentally relevant systems, from one-dimensional dipolar cold-atom gases to Lennard--Jones molecular fluids.