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arXiv:2408.00593 · Physical Review X

Generalized Bogoliubov-Born-Green-Kirkwood-Yvon Hierarchy for Near-Integrable Dynamics

Leonardo Biagetti, Maciej Łebek, Miłosz Panfil, Jacopo De Nardis

Published May 20, 2026• 2 citations
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Abstract

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 (BBGKY) 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 nonthermal on thermalization timescales, indicative of a form of incomplete or generalized thermalization. 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 []. 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.

DOI
10.1103/sx1x-pyl1

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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 (BBGKY) 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 nonthermal on thermalization timescales, indicative of a form of incomplete or generalized thermalization. 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 []. 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.
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility, and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two minimally invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer timescale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimally invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.
The time evolution of -spin reduced density operators is studied for a class of Heisenberg-type quantum spin models with long-range interactions. In the framework of the quantum Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy, we introduce an unconventional representation, different from the usual cluster expansion, which casts the hierarchy into the form of a second-order recursion. This structure suggests a scaling of the expansion coefficients and the corresponding time scales in powers of N 1/2 with the system size N , implying a separation of time scales in the large-system limit. For special parameter values and initial conditions, we can show analytically that closing the BBGKY hierarchy by neglecting -spin correlations never leads to equilibration, but gives rise to quasi-periodic time evolution with at most independent frequencies. Moreover, for the same special parameter values and in the large- N limit, we solve the complete recursion relation (the full BBGKY hierarchy), observing a superexponential decay to equilibrium in rescaled time τ = tN − 1/2 .
We establish a theoretical framework for exploring the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy of equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and symmetry conserving closure approximation is developed on the basis of a recursively formulated cluster expansion for these finite systems. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. We apply this theoretical framework to both a tunneling and an interaction-quench scenario. Due to our efficient formulation of the theory, we can reach truncation orders as large as twelve and thereby systematically study the impact of the truncation order on the results. While the short-time dynamics is found to be excellently described with controllable accuracy, significant deviations occur on a longer time-scale in sufficiently far off-equilibrium situations. Theses deviations are accompanied by exponential-like instabilities leading to unphysical results. The phenomenology of these instabilities is investigated in detail and we show that the minimal-invasive correction algorithm of the equation of motion can indeed stabilize the BBGKY hierarchy truncated at the second order.
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We establish a theoretical framework for the exploration of the quantum dynamics of finite ultracold bosonic ensembles based on the Born-Bogoliubov-Green-Kirkwood-Yvon (BBGKY) hierarchy for equations of motion for few-particle reduced density matrices (RDMs). The theory applies to zero as well as low temperatures and is formulated in a highly efficient way by utilizing dynamically optimized single-particle basis states and representing the RDMs in terms of permanents with respect to those. An energy, RDM compatibility and, if applicable, symmetry conserving closure approximation is developed on the basis of a cluster expansion for these finite systems, using suitably normalized RDMs and symmetrization operators. The underlying cluster expansion is formulated recursively by means of super-operators, which allows for going to large truncation orders. In order to enforce necessary representability conditions, two novel, minimal-invasive and energy-conserving correction algorithms are proposed, involving the dynamical purification of the solution of the truncated BBGKY hierarchy and the correction of the equations of motion themselves, respectively. For gaining conceptual insights, the impact of two-particle correlations on the dynamical quantum depletion is studied analytically. The theoretical framework provided here is extensively applied and evaluated in our immediate follow-up work [1], involving truncation orders as large as twelve.
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