Generalised BBGKY hierarchy for near-integrable dynamics

This paper introduces a generalized BBGKY hierarchy based on quasiparticle densities to exactly describe the non-thermal dynamics of many-body systems combining integrable contact interactions with long-range potentials, successfully explaining experimental observations in dipolar quantum gases and extending the framework to a wide class of strongly interacting systems.

The Gist

The Big Picture: A Crowd That Won't Forget Its Past

Imagine a massive crowd of people moving through a city. In a normal city (a "non-integrable" system), if you bump into someone, you might get pushed, change direction, and eventually, the whole crowd forgets where it started and settles into a random, chaotic shuffle. This is called thermalization—the state where everything is mixed up and calm.

However, some special crowds are "integrable." Imagine a crowd of people who are all on perfectly smooth, frictionless ice skates. If two people bump, they don't just bounce randomly; they swap speeds in a very predictable, mathematical way. Because of this, the crowd never truly "forgets" its initial state. It keeps moving in organized waves forever, never settling down.

The Problem:
Real life isn't perfect ice. Sometimes, people in the crowd also have long-range interactions (like shouting across the street or magnetic attraction) that mess up their perfect ice-skating rules. Scientists wanted to know: How does this crowd eventually settle down when you add these extra, messy interactions?

Previous theories could only explain what happens after a very long time, or they only worked for crowds that were already totally chaotic to begin with. They couldn't explain the messy "middle" part where the crowd is trying to settle but is still holding onto its old habits.

The Solution: The "Generalized BBGKY" (gBBGKY)

The authors of this paper created a new set of rules, which they call the generalized BBGKY hierarchy. Think of this as a new, super-advanced traffic camera system that doesn't just count how many cars are on the road (the average), but also tracks how cars influence each other in groups of two, three, or more.

Here is how they did it, using a creative analogy:

1. The "Correlated Fluid-Cell" Ensemble

Imagine the city is divided into small neighborhoods (fluid cells).

• Old Theory: Assumed each neighborhood was independent. If you knew the average mood of Neighborhood A, you knew everything about it.
• New Theory (gBBGKY): Realizes that Neighborhood A is deeply connected to Neighborhood B and C. Even if they are far apart, a shout in A might echo in B. The authors created a mathematical "ensemble" (a collection of possibilities) that accounts for these long-distance friendships and arguments between the neighborhoods.

2. The Two-Step Dance of Relaxation

The paper discovered that when you break the perfect rules of the crowd, it doesn't relax in one smooth step. It happens in two distinct phases, like a dance:

• Phase 1: The "Kinetic Blocking" (The Stuck Phase)
  In a one-dimensional line (like a single file of people), if two people bump, they just swap places. They can't pass each other. The paper shows that in a perfect line, the crowd gets "stuck" in a pre-thermal state. It looks like it's settling, but it's actually just shuffling in place. This is called kinetic blocking. The crowd is trying to thermalize, but the rules of the line prevent it.

• Phase 2: The "Generalized Thermalization" (The Slow Melt)
  The authors found that the crowd does eventually settle, but only because of a clever trick involving three-way interactions.

  • Imagine Person A and Person B are far apart. They are influenced by a long-range shout (the long-range potential).
  • But to actually change their speed and settle down, they need a third person, Person C, to act as a bridge.
  • Person A bumps into C (a local contact), and C bumps into B. This "relay race" allows the crowd to finally break its perfect rules and start mixing.

  The Surprise: The paper found that this mixing happens much faster in crowds with strong local contact rules (like hard spheres) than in crowds without them. The local "bumping" actually helps the long-range "shouting" do its job.

3. The "Incomplete" Party

Here is the most fascinating part. The paper proves that even when the crowd looks like it has settled down (the average speed and the distance between neighbors look normal), the crowd is not fully thermalized.

• One and Two-Point Functions: These are like the average speed of the crowd and the average distance between neighbors. These settle down quickly.
• Three-Point Functions: This is the relationship between three people at once. The paper shows that these complex, three-way relationships never fully settle down on the same time scale. They retain a "memory" of the initial state.

The Metaphor: Imagine a party where everyone stops dancing and stands still (thermalized). But, if you look closely, you see that groups of three friends are still whispering secrets to each other in a specific pattern that only they understand. The party looks calm from afar, but the deep connections remain "frozen" in a special, non-random state. The authors call this generalized thermalization.

Real-World Validation

The authors didn't just do math; they tested their theory against reality:

1. Computer Simulations: They simulated a gas of hard spheres (like billiard balls) with long-range forces. Their new equations predicted the behavior of these balls perfectly, matching the computer simulation down to the last detail.
2. Cold Atom Experiments: They applied their theory to a real experiment with dipolar quantum gases (atoms with magnetic moments) conducted by other scientists (Tang et al.).
   • The experimenters saw the atoms relax in a specific way.
   • The authors' new equations predicted the exact rate at which this happened.
   • They also showed that their math matched the standard "Fermi's Golden Rule" (a common physics tool) but provided a much deeper explanation of why it worked and what was happening in the short-term "pre-thermal" phase that the old tools missed.

Summary of the Discovery

• The Problem: Old theories couldn't explain how systems with strong local interactions (like hard atoms) relax when disturbed by long-range forces.
• The Fix: A new mathematical framework (gBBGKY) that tracks how groups of particles influence each other over time and distance.
• The Result:
  1. Systems with local contact interactions relax faster than those without them.
  2. The relaxation is incomplete: The crowd settles down on the surface, but deep, complex correlations (three-way relationships) remain frozen in a non-random state.
  3. This explains recent experiments with cold atoms and provides a universal tool to understand how order turns into chaos in complex systems.

In short, the paper gives us a new lens to see how the universe "forgets" its past, revealing that sometimes, even when things look calm, the deep connections between particles are still holding on tight.

Technical Summary

Technical Summary: Generalised BBGKY Hierarchy for Near-Integrable Dynamics

Problem Statement
The study of non-equilibrium many-body physics often encounters systems that are close to, but not exactly, integrable. Integrable models, characterized by stable quasiparticles and extensive conserved charges, govern transport and coherence in low-dimensional systems (e.g., Lieb-Liniger gases, Fermi-Hubbard chains). However, real-world systems often include perturbations, such as long-range interactions (e.g., dipolar forces), which break integrability and drive thermalization.

Existing theoretical frameworks face significant limitations in this regime:

1. Fermi's Golden Rule (FGR) approaches: These rely on explicit knowledge of matrix elements (form factors) between eigenstates of the unperturbed integrable model. While accurate for late-time behavior in free or weakly interacting limits, they struggle with genuinely interacting integrable models where matrix elements depend nontrivially on the state. Furthermore, FGR often fails to capture short-time dynamics and the buildup of correlations, which are crucial for prethermalization.
2. Standard BBGKY Hierarchy: The classical Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) hierarchy is a powerful tool for describing thermalization and dissipation in systems of free particles with weak interactions. However, it has not been successfully extended to unperturbed Hamiltonians that are interacting integrable models. In such systems, the stationary states (Generalized Gibbs Ensembles, GGE) already possess non-trivial local correlations, rendering the standard assumption of uncorrelated fluid cells invalid.

The central challenge is to develop a unified framework capable of describing the time evolution of local observables and multi-point correlations in systems combining strong integrable contact interactions with generic long-range perturbations, covering both short-time prethermalization and late-time thermalization.

Methodology
The authors propose a Generalized BBGKY (gBBGKY) hierarchy formulated in terms of the quasiparticle densities and their correlations of the underlying integrable model. The methodology proceeds through the following steps:

1. Correlated Fluid-Cell Ensemble Ansatz: The authors introduce an ansatz for the state at time $t$, termed the "correlated fluid-cell ensemble." Unlike the standard GGE, which assumes independent local cells, this ensemble includes higher-rank Lagrange parameters ($\beta^{(n)}$) that encode long-range correlations between fluid cells. The density matrix is constructed to maximize entropy subject to constraints on local conserved charges and their connected correlations up to arbitrary order.
2. Derivation of the Hierarchy: Starting from the Heisenberg equation of motion for conserved charge densities, the authors derive an infinite set of coupled equations for correlation functions. By utilizing the properties of the correlated fluid-cell ensemble and performing a cumulant expansion, they express current operators in terms of charge density correlations. This allows the hierarchy to be closed at a finite order (specifically, truncating at three-point functions) while retaining the effects of strong local interactions.
3. Quasiparticle Representation: The hierarchy is recast in terms of quasiparticle rapidities ($\theta$) and their distributions ($\rho_\theta$). This involves introducing dressing operators and effective velocities characteristic of integrable hydrodynamics.
4. Kinetic Limit and Generalized Landau Equation: To extract late-time thermalization dynamics, the authors perform a macroscopic rescaling and a separation of time scales. They derive a "Generalized Landau Equation," a kinetic equation describing the evolution of the one-particle distribution. This derivation accounts for the interplay between ballistic transport, local contact interactions, and long-range scattering.
5. Validation: The theory is validated against:
   • Classical Molecular Dynamics: Simulations of hard spheres (Tonks gas) with long-range interactions.
   • Quantum Experiments: Comparison with experimental data on dipolar quantum gases (specifically $^{162}$Dy atoms in quasi-1D tubes) reported by Tang et al.
   • Fermi's Golden Rule: An independent derivation of the collision integral using FGR for the specific case of perturbed Lieb-Liniger gases, showing exact agreement with the gBBGKY results in the appropriate limit.

Key Contributions and Results

• Generalized BBGKY Hierarchy: The paper successfully extends the BBGKY program to interacting integrable systems. It demonstrates that the hierarchy can be formulated using quasiparticle densities and their correlations, capturing the dynamics of both one-point and multi-point functions at all times.
• Mechanism of Thermalization (Kinetic Blocking vs. Generalized Thermalization):
  • Free Limit ($a=0$): In the absence of contact interactions, thermalization is driven solely by long-range scattering. Due to "kinetic blocking" in 1D (where two-body scattering cannot redistribute momentum effectively), thermalization requires three-body processes, leading to a rate scaling as $V_0^4/\xi^4$.
  • Interacting Limit ($a \neq 0$): The presence of integrable contact interactions ($a$) fundamentally alters the dynamics. Local interactions allow quasiparticles to scatter via the long-range potential and then interact again locally with ballistically propagating particles. This generates three-point functions with specific "delta-function jumps" (discontinuities in spatial derivatives). These jumps induce a diffusive contribution to the two-point function dynamics, leading to a much faster thermalization rate scaling as $(a V_0)^2 / \xi^3$.
• Incomplete (Generalized) Thermalization: The authors find that while one- and two-point functions relax to thermal values on the time scale $\tau \sim \xi^3/(V_0 a)^2$, higher-point correlations (e.g., three-point functions) remain strongly non-thermal. These higher-order correlations preserve long-range structures characteristic of the integrable limit, indicating a form of "generalized thermalization" rather than complete thermalization.
• Quantitative Agreement:
  • For classical hard spheres, the gBBGKY predictions for the evolution of kurtosis and two-point correlations show "perfect agreement" with molecular dynamics simulations across a wide range of interaction strengths.
  • For dipolar quantum gases, the derived Generalized Landau equation reproduces the experimental thermalization rates observed by Tang et al. (Phys. Rev. X 8, 021030 (2018)) with high precision.
• Equivalence to FGR: For the specific case of the Lieb-Liniger model perturbed by long-range potentials, the authors explicitly derive the collision integral using Fermi's Golden Rule (calculating form factors for the density operator). They demonstrate that this result reduces exactly to the collision integral obtained from the gBBGKY hierarchy, providing a rigorous, non-trivial verification of their approach.

Significance and Claims
The paper claims to provide a fully predictive theory for generic near-integrable dynamics in both classical and quantum regimes. Its significance lies in:

1. Unifying Prethermal and Thermal Dynamics: The framework captures the short-time transient phase (prethermalization), where correlations build up rapidly, which is often missed by late-time kinetic theories like FGR.
2. Resolving Kinetic Blocking: It explains how strong local interactions can bypass the kinetic blocking typical of 1D free systems, enabling faster thermalization through a mechanism involving convective diffusion and local scattering.
3. Experimental Relevance: The theory offers a direct theoretical account for recent experiments on dipolar cold-atom gases, where the precise form of interparticle potentials can be uncertain, and provides a tool to interpret thermalization rates in terms of microscopic parameters.
4. Broad Applicability: The framework is applicable to a wide class of systems, including one-dimensional dipolar cold-atom gases, Lennard-Jones molecular fluids, and potentially long-range spin chains and fermionic models.

The authors emphasize that their approach does not require explicit knowledge of all transition amplitudes (unlike FGR) and extends the BBGKY program to regimes with strong local interactions, filling a gap in the theoretical understanding of non-equilibrium many-body physics.