Hydrodynamics without Averaging -- a Hard Rods Study

This paper demonstrates that generalized hydrodynamics can accurately describe single coarse-grained samples of the integrable hard rods model without relying on local equilibrium averaging, thereby revealing the absence of intrinsic diffusion and clarifying that hydrodynamic behavior in this system does not require local thermalization.

The Gist

Imagine you are watching a massive, chaotic dance floor filled with thousands of people (particles) bumping into each other. You want to predict where the crowd will be in an hour.

In physics, there are two main ways to do this:

1. The Microscopic View: You track every single person's exact position and every time they bump into someone. This is incredibly detailed but impossible to calculate for a huge crowd.
2. The Hydrodynamic View: You ignore individuals and look at the "flow" of the crowd. You ask, "Is the crowd denser here? Are they moving faster there?" This is like looking at a river: you don't track every water molecule; you just look at the current.

For a long time, physicists believed that to get the Hydrodynamic View right, you had to start by averaging over many different possible starting crowds. They assumed that if you took a "typical" random crowd, the math would work out.

This paper says: "Wait a minute. Let's try something different."

The author, Friedrich Hübner, proposes a radical new idea called "Hydrodynamics without Averaging." Instead of imagining a random crowd, let's pick one single, specific, deterministic crowd and watch it evolve. We don't average out the randomness; we embrace the specific details of that one group.

Here is the breakdown of the paper's journey, using simple analogies:

1. The Problem: The "Pixelation" Issue

To turn a crowd of individuals into a smooth flow (hydrodynamics), you have to blur the picture. Imagine taking a high-resolution photo of the dance floor and turning it into a low-resolution grid (like a Minecraft world).

• The Old Way: You assume the grid cells are filled with an "average" person.
• The New Way: You look at the actual people in each grid cell and count them.

The author asks: If we do this with just one specific crowd, does the smooth flow math still work? And how much error does the "blurring" (coarse-graining) introduce?

2. The Test Case: The "Hard Rods"

To test this, the author uses a model called Hard Rods. Imagine a 1D line of rigid sticks (rods) sliding around. When they hit each other, they bounce off instantly.

• Why this model? It's a "perfect" system. We know exactly how every single stick moves (microscopic), and we also know the exact formula for how the crowd flows (hydrodynamic). It's the perfect laboratory to test the theory.

3. The Big Discovery: No "Intrinsic" Diffusion

In normal fluids (like water), if you have a smooth flow, tiny random jitters eventually cause the flow to spread out or "diffuse" (like a drop of ink spreading in water). This is called intrinsic diffusion.

The author found something surprising with the Hard Rods:

• On a single, specific crowd: There is NO intrinsic diffusion. The flow stays perfectly sharp. If you start with a smooth wave, it stays a smooth wave forever. The "blurring" we see in real life isn't because the particles are naturally messy; it's because of how we look at them.
• The "Diffusion from Convection" Illusion: Previously, physicists thought they saw diffusion in these systems. The author explains that this was an illusion caused by averaging. When you average over many different starting crowds, the differences between them get transported by the flow, looking like diffusion. But for any single crowd, that diffusion doesn't exist.

Analogy: Imagine a line of runners.

• Intrinsic Diffusion: The runners are naturally jittery and start spreading out on their own.
• Diffusion from Convection: The runners are perfectly steady. But if you take a photo of 100 different races where the runners started at slightly different spots, and you average the photos, the result looks like a blurry, spreading cloud. The blurriness isn't in the runners; it's in your averaging method.

4. The "Observer" Effect

The paper highlights that hydrodynamics depends on how you look at the system (the coarse-graining).

• If you look at the crowd through a "fluid cell" (a grid box), the error in your prediction depends on the size of the box.
• If you look through a "smooth blur" (like a soft-focus lens), the error is different.
• Crucial Point: The author proves that for any single deterministic crowd, the standard hydrodynamic equations (Euler equations) are actually more accurate than the equations that try to add "diffusion" corrections. The "diffusion" term is only needed if you are averaging over many different starting points.

5. Entropy and "Thermalization"

In physics, "entropy" is a measure of disorder. Usually, we think systems naturally become more disordered over time (thermalization).

• The author argues that for these specific systems, the entropy doesn't actually increase for a single deterministic crowd. The particles just keep moving in their perfect, predictable patterns.
• The reason it looks like they are becoming disordered (thermalizing) is that our "grid" or "lens" (coarse-graining) eventually becomes too blurry to see the individual patterns. We lose the information, so it seems like the system has become random. It's not that the system changed; it's that our view of it got fuzzy.

Summary: Why This Matters

This paper is a wake-up call for physicists. It suggests that:

1. Randomness isn't always real: Sometimes, the "noise" and "diffusion" we see in equations are just artifacts of how we average data, not properties of the particles themselves.
2. Determinism is powerful: You can understand the flow of a complex system by watching just one specific, non-random instance of it, provided you understand the limits of your "lens" (coarse-graining).
3. The "Diffusion" Correction: The famous "diffusive correction" to hydrodynamic equations (which usually adds a term to make the math work) is actually just a mathematical trick to account for the fact that we are averaging over different starting conditions. If you look at a single reality, you don't need that correction.

In a nutshell: The author took a magnifying glass to the math of fluid flow, removed the "statistical averaging" fog, and found that the underlying reality is much sharper, more deterministic, and less "messy" than we thought. The messiness was in the averaging, not the physics.

Technical Summary

1. Problem Statement

The paper addresses fundamental questions regarding the validity and limitations of hydrodynamic approximations in many-body systems, specifically within the context of Generalized Hydrodynamics (GHD) for integrable models (using the 1D hard rods model as a testbed).

• The Traditional Approach: Standard hydrodynamics relies on averaging over an ensemble of initial states (typically Local Equilibrium or Local GGE states). This assumes that local observables relax to a Generalized Gibbs Ensemble (GGE) and that the system is locally in equilibrium.
• The Limitations:
  • Entropy Paradox: True thermalization (increasing entropy) contradicts the unitary/deterministic evolution of isolated quantum or classical systems, which preserve information.
  • Diffusive Corrections: It is unclear whether the diffusive corrections ($1/\ell$) observed in hydrodynamic equations arise from intrinsic diffusion (physical noise/entropy production) or "diffusion from convection" (the transport of initial state fluctuations).
  • Ambiguity: The traditional approach obscures the distinction between intrinsic physical effects and artifacts introduced by ensemble averaging.

Core Question: Can hydrodynamics emerge from a single, deterministic configuration without ensemble averaging? If so, what are the limits of its accuracy, and is there intrinsic diffusion?

2. Methodology: "Hydrodynamics without Averaging"

The author proposes a new paradigm where the initial state is a single, fixed, deterministic configuration of particles. The methodology involves three main steps:

1. Coarse-Graining: Since the microscopic density is a sum of delta functions (spiky), it must be coarse-grained over a mesoscopic scale ($\Delta x, \Delta p$) to obtain a smooth initial density $\rho(x,p)$ for the hydrodynamic equations. The paper investigates two schemes:
   • Fluid Cell Averaging: Dividing phase space into boxes and assigning a constant density based on particle count.
   • Smoothing: Convoluting the microscopic density with a smooth kernel (e.g., Gaussian).
2. Hydrodynamic Evolution: The coarse-grained density is evolved using the Euler-scale GHD equations (which are exact for hard rods in the $\ell \to \infty$ limit).
3. Comparison: The hydrodynamic prediction for a smooth observable $\Upsilon$ is compared against the exact microscopic evolution of the same single configuration. The error is quantified as $\xi = \Upsilon_{\text{hydro}} - \Upsilon_{\text{micro}}$.

The study analyzes the scaling of this error as the system size $\ell \to \infty$ and the coarse-graining scale $\Delta x \sim \ell^{\mu-1}$ varies.

3. Key Contributions and Results

A. Error Scaling and "Phase Transition"

The paper derives and numerically verifies the scaling of the hydrodynamic error $\xi \sim \ell^{-\nu}$. A critical finding is a "phase transition" in the error scaling depending on the coarse-graining exponent $\mu$:

• Small Cells ($\mu < 1/2$): The error is dominated by statistical fluctuations (randomness of particle positions within cells).
  • Scaling: $\nu = \frac{3}{2} - \mu$.
• Large Cells ($\mu > 1/2$): The error is dominated by systematic error (Taylor expansion truncation within the cell).
  • Scaling: $\nu = 2 - 2\mu$.
• Implication: The minimum error achievable by any hydrodynamic theory based on these coarse-graining schemes is bounded by $O(\Delta x^2)$. Since $\Delta x \gg 1/\ell$, the theory cannot capture corrections smaller than $O(1/\ell^2)$. This suggests that hydrodynamic theories cannot naturally capture dispersive corrections ($O(1/\ell^2)$) without additional degrees of freedom.

B. Absence of Intrinsic Diffusion

By analyzing a single deterministic sample, the author proves that intrinsic diffusion is absent in the hard rods model.

• The Euler-scale GHD equations are accurate even at the diffusive scale ($1/\ell$) for a single trajectory.
• There is no entropy production or "Navier-Stokes" like diffusion term arising from the microscopic dynamics itself.
• Significance: This clarifies that the "diffusive" terms found in previous literature are not intrinsic to the hard rods dynamics but are artifacts of the averaging procedure.

C. Origin of "Diffusive" Corrections: Diffusion from Convection

The paper reconciles its findings with recent literature (e.g., [18, 22]) that did find diffusive corrections.

• Mechanism: When one does average over an ensemble of initial states, the initial microscopic fluctuations (randomness) are transported non-linearly by the hydrodynamic flow.
• Result: This transport of initial randomness ("diffusion from convection") generates an effective $1/\ell$ correction to the averaged density.
• Conclusion: The diffusive GHD equation derived in previous works is actually an equation for the averaged density, driven entirely by the transport of initial correlations, not by intrinsic dissipation. The correction is time-reversible and depends on two-point correlation functions.

D. Entropy and Thermalization

The paper re-evaluates entropy increase:

• Deterministic View: For a single trajectory, entropy is conserved (Euler GHD is time-reversible).
• Coarse-Grained View: Entropy increase (thermalization to a GGE) occurs only because the coarse-graining procedure loses information about the specific microstate.
• Time Scale: Thermalization happens on a time scale $T \sim 1/\Delta p$ (dictated by the coarse-graining resolution), which is much shorter than the diffusive time scale $T \sim \ell$.
• Implication: Thermalization in integrable models is an artifact of the observer's limited resolution (coarse-graining), not a fundamental dynamical process.

E. Robustness to Initial States

Numerical simulations were performed using three distinct ensembles:

1. Local Equilibrium States (LES): Physical thermal states.
2. Poisson Point Process: Uncorrelated states.
3. Ginibre Ensemble: Highly unphysical states with strong eigenvalue repulsion (non-GGE).

• Finding: The hydrodynamic approximation works equally well for all three, including the unphysical Ginibre states. This demonstrates that local equilibrium is not a prerequisite for hydrodynamics to emerge; only "local genericity" (sufficient randomness within fluid cells) is required.

4. Significance and Impact

1. Clarification of Diffusion: The paper resolves the confusion regarding diffusive corrections in integrable systems. It establishes that "diffusion" in hard rods is purely a kinematic effect of transporting initial fluctuations, not a dissipative mechanism.
2. Validation of BMFT: The results provide a rigorous foundation for Ballistic Macroscopic Fluctuation Theory (BMFT), showing that it correctly describes the evolution of correlations by treating initial randomness as the source of diffusive-scale effects.
3. Limits of Hydrodynamics: It highlights a fundamental limitation: standard hydrodynamic theories based on density coarse-graining have a hard error floor ($O(\Delta x^2)$). Capturing higher-order corrections (like dispersion) requires more than just density equations; it requires tracking the internal structure of the coarse-grained cells.
4. New Paradigm: "Hydrodynamics without averaging" offers a cleaner, more fundamental way to study hydrodynamic emergence, separating intrinsic dynamics from statistical artifacts. It suggests that for integrable models, the Euler equations are exact for individual trajectories, and "diffusion" is an emergent property of the observer's averaging.

5. Conclusion

The paper concludes that for the hard rods model, Euler hydrodynamics is valid at the diffusive scale for single deterministic trajectories. The apparent diffusive corrections found in averaged theories are entirely due to the transport of initial state randomness ("diffusion from convection"). Furthermore, entropy increase and thermalization are shown to be consequences of the coarse-graining procedure (measurement imprecision) rather than intrinsic dynamical relaxation. This work calls for a re-evaluation of how hydrodynamic corrections are derived and suggests that future theories must account for the specific coarse-graining scheme used.