Generalised BBGKY hierarchy for near-integrable dynamics

TL;DR

The paper develops a generalised BBGKY (gBBGKY) framework to describe near-integrable dynamics, where an interacting integrable base is perturbed by a weak long-range potential. By introducing the correlated fluid cells ensemble, it derives a hierarchy for charge densities and their correlations, and reformulates it in terms of quasiparticle occupations with dressing and effective velocities. A generalized Landau equation is extracted to capture late-time dynamics, revealing kinetic blocking in 1D and a generalized thermalization rate that scales as (aV_0)^2/\xi^3, while higher-point correlations remain non-thermal on those time scales. The framework is validated via classical hard-sphere simulations and applied to dipolar 1D quantum gases, where it reproduces experimental observations and aligns with Fermi’s Golden Rule calculations in appropriate limits. Overall, gBBGKY provides a predictive, broadly applicable tool to understand prethermalization, generalized thermalization, and ballistic transport in near-integrable many-body systems.

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.

Figures (8)

• Figure 1: Schematic evolution of (a) BBGKY and (b) gBBGKY for the two-point functions $g^{(2)}$, the two insets show, respectively, (c) the long-range potential for free particles and (d) the combination of long-range and short-range interactions in the interacting integrable case. In (a), two particles drawn from the momentum distribution $\rho(\theta)$ scatter via $V(r)$ and become correlated on the time scale $\xi^2/V_0^2$, this is the prethermalisation stage where correlations build up but no extensive mixing occurs, they then scatter again and generate three-point correlations $g^{(3)}$, which control the thermalisation stage on time scales $(\xi/V_0)^4$. In (b), with integrable local interactions in homogeneous systems, the two-point correlation is generated by the large-scale interaction $V(r)$, while the three-point function is produced by a different and faster mechanism. Particles propagate and scatter through local interactions, shown as red circles, providing an effective diffusive bath for the two-point functions, and leading to the thermalisation of one and two-point functions, namely generalised thermalisation, on time scales $\xi^3/(V_0 a)^2$.
• Figure 2: Schematic representation of the correlated fluid cells ensemble of Eq. \ref{['eq: longrange_gge_state']}. Fluid cells of size $\xi$ can be correlated over long distances by the external potential $V$, or by the interplay of contact interactions and ballistic dynamics in the system.
• Figure 3: Diagrams representing the right-hand sides of the hierarchy equations \ref{['eq: evo_eq_implicit_deltas_1']}, \ref{['eq: evo_eq_implicit_deltas_2']} and \ref{['eq: evo_eq_implicit_deltas_3']}. Dashed lines represent the current operator $j_{i_1,0}(x_1)$, while solid lines represent charge operators $q_{i_k}(x_k)$, with $k$ indicated by the number above the line. Vertical lines represent connected correlations between operators. Each diagram contains a single vertex associated with the potential $V(x-x')$, which couples a current to an additional operator $q_0$. To translate a diagram into an expression, one takes products of connected correlation functions according to the vertical lines, multiplies by $V(x_1-x_{m+1})$, where $m+1$ labels the extra charge operator, and integrates over $x_{m+1}$. To draw diagrams at the $m$-th level, one considers one dashed line from which an extra solid line emanates, together with $m-1$ additional solid lines, and then enumerates all possible connections, or partitions, between the $m+1$ lines. The case with no connections is excluded, as it corresponds to the disconnected part of the correlator.
• Figure 4: Schematic representation of the two thermalisation mechanisms leading to the two terms in eq. \ref{['eq: Landau Equation']}. On the left is represented the dynamics giving the self + cross term: two correlated particles at a distance $\xi$ experience fluctuations due to convective waves carried by the other quasiparticles, giving cross diffusion, and they also scatter among each other, giving a self-diffusion to the two-point functions. On the right, the dynamics giving the cross term: two correlated particles never interact with contact interactions; therefore, they only experience the convective density waves through the system, giving standard diffusion to the two-point functions.
• Figure 5: Examples of microscopic trajectories of hard spheres with long-range interaction given by the potential \ref{['eq:potentialHR']}, both with $a=0$ (free particles) and finite $a=1$. In the latter case we can notice in the trajectories with finite $V_0$ the interplay of local scatterings and long-range interaction.