Simulating generalised fluids via interacting wave packets evolution

This paper introduces an efficient simulation framework that models Generalized Hydrodynamics as a gas of interacting semiclassical wave packets, enabling fast large-scale studies of quasi-integrable systems with integrability-breaking perturbations while revealing that long-range correlations can persist indefinitely even when local observables appear thermalized.

The Gist

Imagine a crowded dance floor where everyone is moving to a very specific, complex rhythm. In the world of physics, this is like a one-dimensional system of particles (like atoms in a thin tube) that are "integrable." This means they follow strict, predictable rules where they bounce off each other without ever truly losing their individual energy or getting "messy."

For a long time, scientists had a great way to describe the average movement of this crowd, called Generalized Hydrodynamics (GHD). Think of GHD as a weather forecast for the dance floor: it tells you where the crowd is dense and where it's sparse, and how the "wind" of movement flows.

The Problem:
Real life isn't perfect. Sometimes, the dance floor isn't perfectly flat (external traps), or the dancers bump into things they shouldn't (integrability-breaking perturbations). When these small imperfections happen, the old "weather forecast" (GHD) breaks down. It becomes incredibly hard to calculate, and it fails to predict the tiny, chaotic fluctuations that happen when the system tries to settle down (thermalize). It's like trying to predict a storm using a map that ignores the wind gusts.

The New Solution: The "Ghost" Dancers
The authors of this paper propose a clever new way to simulate these systems. Instead of trying to solve complex math equations for the whole crowd, they imagine the system as a gas of semi-classical wave packets.

Here is the creative analogy:
Imagine the real, interacting dancers are hard to track because they constantly push and pull each other. The authors suggest we pretend these dancers are actually "ghost" dancers (called "bare particles") who walk in straight lines, never touching each other.

However, there's a magic trick:

1. We track these ghost dancers moving in straight lines.
2. We then apply a mathematical "lens" or mapping to translate their straight-line positions into the actual, wiggly positions of the real dancers.
3. This mapping accounts for the fact that when real dancers get close, they effectively "shift" each other's positions (like hard rods bouncing off one another).

Why is this cool?

• It's Fast: Tracking straight lines is easy for a computer. The complex "bouncing" is handled by the math lens at the end, not by simulating every collision in real-time.
• It Handles Chaos: If you add a bump in the dance floor (an external potential) or change the rules slightly, you just change how the ghost dancers move. The math lens automatically adjusts to show you how the real crowd reacts.
• It Captures the "Fluff": Old methods ignored the tiny, random jitters (fluctuations). This new method naturally includes them, just like how a real crowd has people shuffling their feet, not just marching in lockstep.

The Big Surprise: The "Long-Range Hangover"
The researchers used this new tool to study what happens when the dance floor is curved (like a bowl or a trap). They expected the crowd to eventually calm down and look like a random, thermal mess (equilibrium).

They found something surprising:

• The "Face" Looks Calm: If you look at the crowd from far away (checking just the average speed or density), it looks like it has settled down and reached a peaceful, thermal state.
• The "Memory" Remains: However, if you look closely at how different parts of the crowd are connected (correlations), they are still linked over very long distances. It's as if the crowd remembers a specific dance move they did a long time ago, even though they look relaxed.

The Conclusion:
The paper shows that even when a system looks like it has "thermalized" (reached a steady, random state), it might actually be stuck in a long-lived, far-from-equilibrium state because of these hidden, long-range connections. The "ghost dancer" simulation proves that true relaxation takes much longer than previously thought, especially in confined spaces.

In short: They built a faster, smarter way to simulate crowded quantum systems by tracking "ghosts" instead of "real" particles, and discovered that these systems hold onto their memories much longer than we thought.

Technical Summary

Technical Summary: Simulating Generalised Fluids via Interacting Wave Packets Evolution

Problem Statement
Generalized Hydrodynamics (GHD) provides a universal framework for describing transport in one-dimensional integrable and quasi-integrable systems. While standard Euler-scale GHD accurately captures the evolution of average densities and currents, it fails to describe fluctuations and becomes numerically intractable when integrability-breaking perturbations (such as external trapping potentials or many-body interactions) are introduced. Existing extensions, such as diffusive GHD involving Navier-Stokes corrections, rely on the assumption that long-range correlations between fluid cells are negligible. However, recent theoretical work indicates that intermediate-time ballistic transport generates persistent long-range correlations that invalidate the local equilibrium assumption underlying standard diffusive hydrodynamics. Solving the full hierarchy of equations required to capture these correlations (involving two-point and higher-order functions) is computationally prohibitive, particularly for quasi-integrable systems.

Methodology: The Wave Packet Gas (WPG)
The authors propose an alternative, particle-based representation of GHD known as the Wave Packet Gas (WPG). This approach models the generalized fluid as a gas of semiclassical, interacting wave packets (solitonic-like particles) whose trajectories are mapped onto those of non-interacting "bare" particles.

1. Bare-Interacting Mapping: The core of the method is a transformation relating interacting coordinates $x_i$ to bare coordinates $X_i$. For generic integrable models, this is defined by the equation:
   $$X_i = x_i + \frac{1}{2} \sum_{k} a_{ik} \text{sign}(x_i - x_k)$$
   where $a_{ik}$ is the model-dependent scattering shift (e.g., constant for hard rods, momentum-dependent for Lieb-Liniger). The bare particles evolve ballistically with velocity $v^{\text{bare}}(\theta_i)$, while the interacting particles experience effective forces derived from the mapping.
2. Handling Integrability Breaking: When integrability-breaking terms (e.g., an external potential $V(x)$) are present, the mapping transforms the interacting Hamiltonian into a system of bare particles subject to a state-dependent, non-local effective potential. The equations of motion for the bare particles become:
   $$\dot{X}_i = v^{\text{bare}}(\theta_i), \quad \dot{\theta}_i = -\left. \partial_x V(x) \right|_{x=x_i}$$
   This requires transforming between bare and interacting coordinates at each time step to evaluate the force, effectively encoding the many-body complexity into the single-particle dynamics.
3. Statistical Sampling: To recover hydrodynamic behavior, the method samples an ensemble of initial states. For classical systems, bare positions are initialized via a Poisson point process, and rapidities are sampled from the target distribution. For quantum systems, Fermi-Dirac or Bose-Einstein statistics are incorporated via phase-space binning and occupation probabilities. The interacting coordinates are then obtained by inverting the mapping (often via gradient descent minimization of a convex surface).
4. Fluctuations and Correlations: Unlike previous "flea gas" algorithms that failed to capture correct diffusive terms, the WPG naturally embeds the correct initial fluctuations. By averaging over the ensemble, the method reproduces not only the Euler-scale evolution but also higher-order hydrodynamic corrections, including diffusive spreading and two-point correlation functions, without explicitly solving coupled integro-differential equations.

Key Contributions and Results

• Efficient Simulation of Quasi-Integrable Systems: The WPG algorithm successfully simulates systems with integrability-breaking perturbations (external potentials and two-body interactions) where solving partial differential equations for diffusive GHD becomes numerically unstable due to the formation of strong gradients and "turbulent" phases.
• Reproduction of Known Limits: The method quantitatively reproduces standard GHD results in integrable limits and matches diffusive Navier-Stokes predictions at short times ($t \ll \ell$), serving as a non-trivial benchmark.
• Persistence of Long-Range Correlations: The authors demonstrate that in systems confined by external potentials (harmonic, quartic, or cosine traps), local one-point observables (such as rapidity distributions) may appear thermalized (converging to a Gaussian) on diffusive time scales ($t \sim \ell^2$). However, two-point correlation functions (density-density and energy-energy) remain finite and exhibit long-range behavior (correlation lengths $\sim O(10\ell)$) even at these times.
• Thermalization Timescales: The results indicate that true thermalization, defined by the decay of all correlations, does not occur at diffusive time scales. Instead, the relaxation of two-point functions requires longer time scales ($t \sim \ell^3$) involving three-point correlations, which are not captured by standard diffusive hydrodynamics.
• Robustness to Initial Conditions: The study shows that the generation and persistence of long-range correlations are sensitive to initial states. Protocols such as "chopping" a thermal cloud (creating a Newton's cradle setup) amplify these non-thermal correlations, making them detectable in state-of-the-art experimental platforms.

Significance and Claims
The paper claims that the WPG framework provides a transparent and efficient route to simulate generalized fluids, offering a natural generalization of hard-rod particles that automatically embeds the exact fluctuating-hydrodynamics extension of GHD.

The primary significance lies in the demonstration that true thermalization is not reached at diffusive time scales in the presence of weak integrability breaking. The authors argue that while local observables may suggest a thermal state, the system sustains far-from-equilibrium long-range correlations for arbitrarily long times. This challenges previous assumptions regarding the thermalization of trapped integrable systems and suggests that the hierarchy of correlations (involving three-point and higher functions) plays a pivotal role in the relaxation process.

The method is presented as a versatile tool for exploring phenomena in 1D fluids, interpolating between strictly integrable models and regimes dominated by realistic perturbations, including complex two-body integrability-breaking potentials relevant to cold-atom experiments (e.g., dipolar interactions). The authors conclude that their findings necessitate a reassessment of thermalization in quasi-integrable systems and highlight the potential for using tailored initial conditions to probe these long-range correlation dynamics experimentally.