Generalised BBGKY hierarchy for near-integrable dynamics

Abstract

We study quantum and classical many-body Hamiltonian systems that combine integrable contact interactions with generic long-range two-body potentials. We show that the dynamics of local observables 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. Starting from an ansatz for the state at time $t$, which we call the correlated fluid-cell ensemble, we derive this hierarchy and prove that it reproduces exactly the time evolution of one- and multi-point correlation functions in perturbed integrable models, at all times. We 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.