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

This episode covers a paper by Saikat Santra, Maciej Lebek, and Milosz Panfil on the crossover from generalized hydrodynamics to conventional hydrodynamics in nearly integrable quantum systems, using a relaxation time approximation. The central puzzle is how quantum systems with infinitely many conservation laws—special rules that keep quantities unchanged over time—eventually behave like everyday fluids described by universal equations. In plain terms, these systems start with particles zipping along without slowing down much, but weak interactions make them spread out diffusively at long times.

So this paper is basically asking why quantum setups that seem special, with all those extra protections, still end up following the same fluid rules we see in water or air?

Yes, exactly. Integrable systems have infinite conserved charges—like an endless list of invariants that let particles travel ballistically, without friction. But in nearly integrable ones, a small perturbation adds collisions, modeled here by the relaxation time approximation. Think of it like a gentle nudge: over a single timescale τ, distributions shift toward a local thermal state, preserving only particle number, momentum, and energy. This leads to a clear crossover: short times follow generalized hydrodynamics, long times shift to Navier-Stokes equations.

Right, and the practical side—like in cold atom experiments with Lieb-Liniger gases—is that initial ballistic spread turns diffusive?

That's the challenge. In those experiments, weak interactions barely disturb the integrable core at first, so transport looks ballistic. But eventually, the collisions dominate, forcing the system into standard viscous flow. This work analytically pins down the transport coefficients—like viscosity—from the integrable Drude weights.

Okay, so the key is spotting those space-time scales where the infinite conservations fade, leaving just the three basics.

Precisely. They diagonalize the collision operator using the GHD charge basis—zero eigenvalues for the first three conserved modes, uniform gap for the rest. It's like a highway with infinite frictionless lanes narrowing to three with uniform drag. This lets them solve for Navier-Stokes analytically and identify the crossover. It sets up predictions testable in cold atoms.

So those orthonormal charges make the collision operator diagonal—zeros for the first three, gap for the rest. How does that lead straight to formulas for the fluid properties like viscosity?

The first three charges track particle number, momentum, and energy—quantities the collisions can't change. All higher ones get damped equally by the 1/τ gap, so the system projects onto just those three at large scales, matching the standard fluid equations. Researchers call these equations 'Navier-Stokes,' which describe everyday flows like air or water with two key numbers: viscosity, the fluid's internal drag, and thermal conductivity, how heat diffuses through it.

Okay, so viscosity and conductivity split into parts from the integrable side and the breaking side. Does the diagonal setup give exact expressions for the breaking parts?

Yes. The integrable parts come from known diffusion in the charge basis, while the breaking parts solve an equation where the collision operator acts on special functions, orthogonal to the conserved charges. Because it's diagonal, the solution sums over higher modes simply: each contributes proportionally to τ. This yields viscosity from the irreducible momentum Drude weight and conductivity from energy.

So the infinite charges collapse the math into something solvable, pulling transport numbers directly from the ballistic Drude weights. But how do they spot the actual switch in space and time scales?

They linearize the dynamics around a uniform thermal state—small ripples in charges—and Fourier transform to get a matrix blending collisions, advection, and diffusion. Its spectrum shows three gapless modes at small k, like sound and heat waves in fluids, matching Navier-Stokes dispersion. At large k, modes fill the GHD picture. The crossover k_c is where fluid waves hit the 1/τ gap, marking when infinite conservations yield to the three basics.

Huh. So in experiments, a local pulse of particles spreads ballistically short-term, diffusively long-term, with this k_c predicting the handoff.

Exactly. Figure two plots the real spectrum: gapless trio below k_c, gapped rest above, confirming the NS sector emerges naturally. This pins testable scales for cold atom gases, where weak interactions drive the shift without full chaos.

That spectrum in figure two sounds clear—gapless modes below k_c behaving like fluid waves, gapped above. But how does that play out in the actual time evolution of a perturbation, say a local bump in density?

They solve the linearized equations exactly in Fourier space, getting charge ripples as sums over eigenvalues and a rotation matrix times initial state. For initial bumps with small scales—momentum k_ini much less than k_c—the three conserved charges follow Navier-Stokes waves right away, while higher charges just fade exponentially at rate 1/τ. The matrix R decouples them cleanly, like separate tracks. Figure three shows: solid lines from full equations match circle points from Navier-Stokes perfectly.

Okay, so big-picture initial states—ones with higher momentum components—don't start in the fluid regime?

Correct. High-k parts decay first on GHD timescales, faster than τ. They homogenize the state by damping ripples above k_c. What's left is small-k, where thermalization kicks in—non-conserved charges decay, leaving conserved ones to evolve as Navier-Stokes. Figure three's bottom row confirms: for k_ini above k_c, early times match pure GHD—no collisions yet—but by t around 10τ, it converges to Navier-Stokes fields like density and energy.

So two steps: first, chop off high-momentum tails; second, shed extra charges. And that late-time match is solid across setups?

Yes—the paper numerically evolves both full GHD-Boltzmann and Navier-Stokes, using transport coefficients from earlier formulas. Agreement is excellent for t much larger than τ, validating the coefficients like viscosity from Drude weights. For generic starts, Navier-Stokes is the universal late fate: everything smooths to uniform. This pins testable predictions, like how long until diffusive spread dominates in cold atoms.

Makes sense why experiments see ballistic then diffusive—k_c sets the handoff scale from infinite charges to three-mode fluids.

So at those late times, well beyond τ, what do the actual measurements look like—say, correlations between densities at different spots?

They look at two-point functions, which track how a wiggle in density at one spot connects to another spot over time and distance—like ripples spreading from a pebble drop. At early times, these follow the infinite-charge picture; later, for the three conserved quantities, they match a version with built-in randomness. Researchers describe this as fluctuating hydrodynamics: fluids aren't perfectly smooth but have tiny thermal jitters, like molecules bumping around, following bell-shaped patterns whose spread depends on viscosity and conductivity. Figure four confirms it—the density-density link decays for non-conserved charges on timescale τ, but conserved ones settle into these diffusive and sound-wave humps.

Right, so the randomness strength ties back to those transport numbers from the Drude weights. But does this hold only for linear cases, or do bigger effects kick in later?

The linear version works well here because interactions are weak, keeping things simple over times much longer than τ but shorter than some microscopic scale. Stronger nonlinear terms in the flow—curving paths for bigger waves—matter only at enormous scales, far beyond typical experiments.

Huh, so the uniform damping doesn't just smooth charges—it sets up these noisy, realistic fluid signals from the integrable starting point.

Exactly. By diagonalizing collisions in the charge basis, they derive transport from Drude weights alone, no full chaos needed, and map the full path from ballistic quasiparticles to everyday fluids.

So overall, these quantum systems with their endless protections don't stay special forever—they hand off to the same fluid rules that describe air or water, and this work maps exactly when and how. But are there spots where this simplification might not hold up?

Yes, a key limitation is the relaxation time approximation itself—it assumes all non-conserved modes relax over one uniform timescale τ, while real collisions have varied rates depending on particle energies. Also, the linear regime around uniform states misses nonlinear effects in the hydrodynamic equations, which become relevant only at ultra-large scales, far beyond typical lab times. The paper notes this keeps the approach valid for weak perturbations in near-integrable setups.

Okay, so single τ smooths things for analysis, but reality has more variety—and big waves could curve the flow differently later on. Still, for labs tuning those weak breaks in cold atoms, this gives concrete targets—like when to expect the diffusive shift.

Exactly. It predicts transport in tunable quantum simulators, letting researchers engineer custom fluid behaviors by dialing integrability-breaking strength. The upshot is clearer tests against experiments like Lieb-Liniger gases, advancing control over quantum many-body flows.

Huh. So this isn't just theory—it's a practical map from special quantum rules to everyday fluids, with scales you can measure in the lab. A meaningful step for those systems. Thanks, Sam—clear as always. And thanks for listening to ResearchPod.