Crossover from generalized to conventional hydrodynamics in nearly integrable systems under relaxation time approximation

This paper investigates the transition from generalized to conventional hydrodynamics in nearly integrable systems by employing a relaxation time approximation for the collision term, explicitly deriving Navier-Stokes transport coefficients and characterizing the crossover scales through the dynamics of charge densities and correlation functions.

The Gist

Imagine a crowded dance floor where everyone is following a strict, perfect set of rules. In this perfect world, if you push a dancer, they glide across the floor forever without bumping into anyone or losing energy. They never stop, they never slow down, and they never forget their original path. In physics, we call this an Integrable System. It's a mathematical fantasy where particles are like ghosts that pass through each other perfectly.

However, the real world isn't perfect. There are bumps, friction, and people bumping into each other. In physics, we call this breaking integrability. When you add these "imperfections" (like extra interactions between particles), the perfect gliding stops. Eventually, the dancers start bumping, slowing down, and the whole crowd settles into a chaotic, warm, thermal state.

This paper is about the transition (the "crossover") between that perfect, ghost-like dancing and the messy, realistic dancing we see in the real world.

Here is the breakdown of their discovery using simple analogies:

1. The Two Rules of the Dance Floor

The authors study a system that is almost perfect but has a tiny bit of chaos added to it. They look at two different ways to describe the dance:

• Generalized Hydrodynamics (GHD): This is the "Perfect World" rulebook. It assumes the dancers are ghosts. It tracks infinitely many rules (conservation laws). If you know the position of every dancer, you can predict exactly where they will be forever. It's incredibly detailed but ignores the fact that people eventually get tired and stop.
• Navier-Stokes (NS) Hydrodynamics: This is the "Real World" rulebook. It's the standard physics we use for water, air, and traffic. It only cares about three things: Mass (how many people), Momentum (how fast they are moving), and Energy (how hot the room is). Everything else (like individual dance moves) gets forgotten and turns into heat.

2. The "Relaxation Time" Shortcut

Usually, calculating how a system moves from the "Perfect World" to the "Real World" is a nightmare. You have to calculate every single collision between every particle, which is like trying to track every single handshake in a stadium.

The authors used a clever shortcut called the Relaxation Time Approximation (RTA).

• The Analogy: Imagine a room full of people talking. If you shout a secret, it takes a certain amount of time for the secret to spread and for everyone to stop talking about it and return to normal chatter.
• The Shortcut: Instead of tracking every single conversation, the authors just say: "Okay, let's assume it takes exactly $\tau$ (tau) seconds for the system to forget its weird, perfect rules and return to normal."
• They treat this time $\tau$ as a "reset button." Before $\tau$ seconds pass, the system acts like the perfect ghosts (GHD). After $\tau$ seconds, it acts like normal fluid (NS).

3. The Big Discovery: The Crossover

The paper answers the question: When does the system stop acting like ghosts and start acting like a normal fluid?

They found that the answer depends on how far you look and how long you wait.

• The Short-Term/Close-Up View: If you zoom in on a small area and watch for a very short time (less than $\tau$), the system looks like the perfect GHD. The particles are still gliding ballistically. The "collision term" (the friction) hasn't kicked in yet.
• The Long-Term/Far-Out View: If you wait a long time (much longer than $\tau$) or look at a very large scale, the "friction" wins. The system forgets all its infinite rules and settles into the standard Navier-Stokes equations. The infinite conservation laws decay, leaving only Mass, Momentum, and Energy.

The "Crossover Point":
The authors calculated a specific "crossover speed" (or scale). Think of it like a speed limit.

• If the particles are moving slowly (low momentum), they quickly hit the "friction" and behave like normal fluid.
• If they are moving very fast (high momentum), they can zip past the friction for a while, behaving like the perfect ghosts of GHD.
• The paper gives a formula to tell you exactly where this line is drawn.

4. What Happens to the "Extra" Charges?

In the perfect world, there are many "charges" (like specific dance moves) that never disappear.

• Conserved Charges: These are the big three (Mass, Momentum, Energy). They survive the transition. They are the ones that end up in the Navier-Stokes equations.
• Non-Conserved Charges: These are the weird, specific rules of the perfect world. The paper shows that these charges die out exponentially. They decay like a dying echo. Once the time passes $\tau$, these specific details vanish completely, leaving only the standard fluid behavior.

5. Why Does This Matter?

This is crucial for modern physics, especially with cold atomic gases (atoms cooled to near absolute zero).

• Scientists can create these "nearly perfect" systems in the lab.
• They can tweak the interactions to make the system slightly imperfect.
• This paper gives them a map. It tells them: "If you look at your experiment for 1 second, you will see GHD. If you wait 10 seconds, you will see standard fluid dynamics. Here is exactly how the transition happens."

Summary in a Nutshell

Imagine a river.

• GHD is the river flowing perfectly straight, with every drop of water knowing exactly where to go forever.
• NS (Navier-Stokes) is the river after it hits a rocky patch, creating eddies, heat, and turbulence.
• This Paper explains exactly how long it takes for the smooth, perfect river to turn into the messy, turbulent river when you introduce a few rocks. They found that if you wait long enough, the river forgets its perfect path and just flows like water always does. They also figured out the math to predict exactly when that switch happens.

Technical Summary

1. Problem Statement

Integrable quantum systems possess infinitely many conserved quantities, leading to non-thermalizing behavior described by Generalized Hydrodynamics (GHD). However, real-world systems are never perfectly integrable; they are subject to weak integrability-breaking perturbations (e.g., long-range interactions or external potentials).

• The Challenge: When integrability is broken, the system eventually thermalizes, and its long-time, large-scale dynamics should be described by conventional hydrodynamics (specifically the Navier-Stokes equations).
• The Gap: The transition (crossover) from GHD to conventional hydrodynamics is governed by a collision integral in the Boltzmann equation. Calculating this integral exactly is extremely difficult because it typically requires perturbative treatments (Fermi's Golden Rule or gBBGKY hierarchy) involving complex multi-particle scattering processes, particularly in 1D where three-body collisions dominate.
• The Goal: To understand the mechanism and characteristic scales of the crossover from GHD to Navier-Stokes (NS) hydrodynamics in nearly integrable systems without relying on the intractable exact collision integral.

2. Methodology

The authors adopt a Relaxation Time Approximation (RTA) for the collision operator, a standard simplification in kinetic theory.

• Model Framework: They consider a nearly integrable system described by a Hamiltonian $H = H_0 + \lambda V$, where $H_0$ is integrable and $\lambda V$ breaks integrability. The dynamics are governed by a modified GHD equation:
  $$\partial_t \rho_p + \partial_x (v \rho_p) = \frac{1}{2}\partial_x (D \partial_x \rho_p) + I_{RTA}[\rho_p]$$
  where $\rho_p$ is the quasiparticle distribution, $v$ is the effective velocity, $D$ is the diffusion kernel, and $I_{RTA}$ is the collision term.
• Relaxation Time Approximation (RTA): Instead of computing the complex collision integral, they use:
  $$I_{RTA}[\rho_p] = \frac{\rho_p^{th} - \rho_p}{\tau}$$
  Here, $\rho_p^{th}$ is a local thermal (Gibbs) state determined by the conservation of the first three charges (particle number, momentum, energy), and $\tau$ is the characteristic relaxation time.
• Linear Regime Analysis: The study focuses on linear perturbations around a homogeneous thermal background. They employ Fourier-Laplace transforms to convert the partial differential equations into an eigenvalue problem.
• Thermodynamic Bethe Ansatz (TBA): All thermodynamic quantities (Drude weights, susceptibilities, specific heats) are computed using TBA for the underlying integrable model (specifically the Lieb-Liniger model is used as a concrete example).

3. Key Contributions

A. Analytical Derivation of Transport Coefficients

The authors explicitly compute the transport coefficients (viscosity $\zeta$ and thermal conductivity $\kappa$) for the emergent Navier-Stokes equations.

• They decompose the coefficients into two parts:
  1. Diffusive ($D$): Arising from integrable interactions (GHD diffusion).
  2. Collisional ($I$): Arising from the integrability-breaking term (RTA).
• Key Result: They derive closed-form analytical expressions for the collisional contributions ($\zeta_I, \kappa_I$) in terms of the Drude weights and thermodynamic matrices of the integrable model.
  $$\zeta_I = \varrho \tau \omega_1^{coll}, \quad \kappa_I = \varrho c_V \tau \omega_{th}^{coll}$$
  where $\omega^{coll}$ are specific combinations of Drude weights and sound velocities. This allows for precise predictions of NS transport coefficients based solely on the integrable model's data and the phenomenological $\tau$.

B. Characterization of the GHD-NS Crossover

The paper identifies the specific space-time scales governing the transition:

• Crossover Momentum ($k_c$): Defined as the momentum scale where the NS dispersion relations (sound/heat modes) intersect with the gap introduced by the RTA collision term ($\tau^{-1}$).
  $$k_c = \min \left[ (D_{\pm}\tau)^{-1/2}, (D_{th}\tau)^{-1/2} \right]$$
• Regimes:
  • $k \ll k_c$ (Long scales): The system exhibits standard NS hydrodynamics with sound and heat modes.
  • $k \gg k_c$ (Short scales): The system behaves according to GHD (ballistic transport with diffusion).
• Decoupling Mechanism: They demonstrate that for $t \gg \tau$, non-conserved charges (higher than the first three) decay exponentially ($e^{-t/\tau}$), while conserved charges evolve according to NS equations.

C. Dynamics of Correlation Functions

The authors analyze two-point correlation functions to show how the crossover manifests in observables:

• Early Times ($t \ll \tau$): Correlators follow GHD predictions (ballistic propagation of quasiparticles).
• Late Times ($t \gg \tau$):
  • Conserved Charges: Correlators converge to the form predicted by fluctuating hydrodynamics (Gaussian fields with diffusive broadening and sound peaks).
  • Non-conserved Charges: Correlators decay exponentially to zero.

4. Results

• Transport Coefficients Behavior: Using the Lieb-Liniger model, they plotted viscosity and thermal conductivity against interaction strength $c$.
  • Most coefficients ($\zeta_D, \zeta_I, \kappa_D$) show non-monotonic behavior, peaking at intermediate interactions and vanishing in the free limits ($c \to 0$ and $c \to \infty$).
  • Thermal conductivity due to collisions ($\kappa_I$) behaves differently, showing monotonic increase with interaction strength.
• Numerical Verification: Numerical simulations of the linearized GHD-Boltzmann equation confirm that:
  • If the initial perturbation has a momentum scale $k_{ini} < k_c$, the system immediately follows NS dynamics.
  • If $k_{ini} > k_c$, the system initially follows GHD dynamics, but after a time $t \sim \tau$ (or longer depending on $k$), the high-momentum modes decay, and the system relaxes into the NS regime.
• Universal Fate: Regardless of the initial state, the long-time dynamics of conserved quantities are universally described by Navier-Stokes equations with the derived transport coefficients.

5. Significance

• Bridging Theory and Experiment: The work provides a practical framework for analyzing experiments with cold atoms where integrability is weakly broken. It offers a way to predict transport coefficients without solving the full, intractable collision integral.
• Validation of RTA: It demonstrates that the Relaxation Time Approximation, despite its simplicity, correctly captures the essential physics of the crossover from integrable to non-integrable hydrodynamics, including the emergence of standard conservation laws.
• Theoretical Clarity: The paper clarifies the hierarchy of scales in nearly integrable systems, explicitly defining the "hydrodynamic window" where linear fluctuating hydrodynamics is valid before nonlinear effects (which appear at even larger scales) take over.
• Analytical Tractability: By expressing complex transport coefficients in terms of known integrable model data (Drude weights), the authors make the study of non-equilibrium dynamics in nearly integrable systems analytically accessible.

In summary, this paper successfully maps the complex landscape of nearly integrable hydrodynamics, showing how the "memory" of infinite conservation laws fades over time $\tau$ and space scale $1/k_c$, giving way to the universal laws of Navier-Stokes fluid dynamics.