Hydrodynamic noise in one dimension: projected Kubo formula and how it vanishes in integrable models

This paper establishes a well-defined hydrodynamic fluctuation theory for one-dimensional systems by deriving a projected Kubo formula for noise covariance and demonstrating that in integrable models, hydrodynamic noise and bare diffusion vanish to all orders, thereby confirming that Ballistic Macroscopic Fluctuation Theory fully describes their hydrodynamics.

The Gist

The Big Picture: Predicting the Weather of Atoms

Imagine you are trying to predict the weather. You don't track every single air molecule; that's impossible. Instead, you look at "fluid cells"—chunks of the atmosphere—and measure the average temperature, pressure, and wind speed in those chunks. This is Hydrodynamics.

In most systems (like a gas in a box), if you zoom out far enough, the chaotic motion of individual particles averages out into smooth, predictable waves. This is the "Euler scale." But if you look closer, or if the system is very long and thin (one-dimensional), things get messy. The smooth waves start to wobble, and "noise" (random jitters) appears.

This paper is about understanding that noise and how it behaves in a very special type of system: Integrable Systems.

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1. The Problem: The "Forgotten" Particles

When we simplify a complex system into smooth waves, we throw away a lot of information. We forget the tiny, fast movements of individual particles.

• The Analogy: Imagine a crowded dance floor. If you look from a balcony, you see a smooth flow of people moving left and right. But if you zoom in, you see people bumping into each other, stepping on toes, and dancing randomly.
• The Noise: In physics, this "random dancing" that we forgot about doesn't just disappear. It comes back as Hydrodynamic Noise. It's like a static hiss on a radio. Usually, this noise causes diffusion (smearing out), like a drop of ink spreading in water.

2. The Special Case: The "Perfectly Organized" Dance Floor

The paper focuses on Integrable Systems. These are rare, mathematical "perfect" systems (like certain quantum spin chains or hard rods) where particles interact in a way that they never truly get chaotic. They pass through each other like ghosts, or bounce off in a perfectly predictable way.

• The Conjecture: Scientists recently guessed that in these "perfect" systems, the noise should vanish. If the dance floor is perfectly organized, maybe there is no random jittering at all?
• The Mystery: But wait, we know these systems still have "diffusion" (smearing) in their two-point correlations (how two points talk to each other). So, where is the noise coming from?

3. The Solution: A New Way to Measure (The Projected Kubo Formula)

The author, Benjamin Doyon, develops a new mathematical tool to measure this noise. He calls it the "Projected Kubo Formula."

Here is the analogy:

• The Old Way (Standard Kubo): Imagine trying to measure the noise in a room by listening to everything. You hear the music, the talking, the traffic outside, and the wind. It's a mess.
• The New Way (Projected): Doyon says, "Let's filter out the music." In physics terms, he filters out the long-range correlations caused by the "waves" of the system. These waves are so strong they look like noise, but they aren't random; they are deterministic.
• The Result: Once you subtract these "wave noises," you are left with the true random noise.

4. The Big Discovery: Silence in Integrable Systems

When Doyon applies his new filter to Integrable Systems, he finds something amazing:

• The Noise is Zero.
• The Analogy: It's like tuning a radio to a station that is so perfectly clear that the static hiss completely disappears.
• Why? Because in these special systems, the "randomness" is actually just the particles following a very strict, infinite set of rules (commuting flows). If you choose your "gauge" (your way of measuring the current) correctly, you can see that the particles are never truly "jittering" randomly. They are just following a complex, but deterministic, choreography.

What does this mean?
It means that for these special systems, the "bare diffusion" (the raw smearing caused by noise) is zero. The diffusion we see in experiments isn't caused by random noise; it's caused by the long-range "echoes" of the initial state bouncing around.

5. The "Point-Splitting" Trick

To make the math work, the author uses a trick called Point-Splitting Regularization.

• The Analogy: Imagine trying to measure the temperature of a single point in a fluid. If you measure it exactly at the point, the math blows up because the fluid is too dense there.
• The Fix: Instead of measuring at one point, you measure at two points that are infinitesimally close to each other (like $x$ and $x + \epsilon$). You average them.
• Why it matters: This prevents the math from breaking down. It allows the author to write down a clean, well-defined equation (a Stochastic PDE) that describes how the system evolves, even when there are "shocks" or sudden changes.

6. The Takeaway: Why Should We Care?

1. For Integrable Systems: We now have a complete theory. We know that noise vanishes, and the "diffusion" we see is actually a different kind of effect (long-range correlations). This confirms recent guesses and gives us a "perfect" description of how these quantum systems behave at large scales.
2. For General Systems: The paper provides a new, robust framework for understanding how noise emerges in any 1D system. It separates the "real" random noise from the "fake" noise caused by wave interactions.
3. The "No-Shock" Rule: The theory works best for systems that don't form "shocks" (sudden, violent breaks in the flow, like a sonic boom). Integrable systems are "shock-free," which is why the math is so clean.

Summary in One Sentence

Benjamin Doyon has figured out that in perfectly ordered, one-dimensional quantum systems, the "static noise" we expect to see actually disappears, and he has built a new mathematical filter to prove that what looks like random diffusion is actually just the system's waves echoing in a very specific, predictable way.

Technical Summary

1. Problem Statement

The paper addresses the theoretical description of hydrodynamic noise and bare diffusion in one-dimensional many-body systems at large scales (hydrodynamic limit). While standard fluctuating hydrodynamics posits that coarse-grained observables evolve according to stochastic partial differential equations (SPDEs) driven by Gaussian white noise, the precise microscopic origin and magnitude of this noise in systems with ballistic transport remain subtle.

Specifically, the paper tackles:

• The nature of noise in "no-shock" systems: In systems where hydrodynamic modes do not self-interact to form shocks (e.g., linearly degenerate and integrable systems), superdiffusive effects (like those in the KPZ universality class) are absent. However, it was unclear whether diffusive corrections and noise persist.
• The validity of the "no-noise" conjecture: Recent conjectures suggested that in integrable systems, hydrodynamic noise and bare diffusion vanish entirely, implying that initial fluctuations do not affect coarse-grained currents beyond the Euler scale.
• The definition of constitutive relations: How to rigorously define the noise covariance and diffusion coefficients in terms of microscopic correlation functions (Kubo formulas) when long-range correlations and non-linear fluxes are present.

2. Methodology

The author develops a rigorous framework based on the Ballistic Macroscopic Fluctuation Theory (BMFT) extended to the diffusive scale ($1/\ell$ corrections, where $\ell$ is the macroscopic scale parameter).

• Nonlinear Fluctuating Boltzmann-Gibbs Principle: The core methodology is a generalized version of the Boltzmann-Gibbs principle. It expresses any coarse-grained local observable $o(x,t)$ as a sum of three components:
  1. Microcanonical part: A function of fluctuating conserved densities, $o_{reg}(q)$.
  2. Diffusive part: A term proportional to spatial gradients, $-\frac{1}{2\ell} \hat{D} \partial_x q$.
  3. Noise part: A stochastic term, $\frac{1}{\ell} \xi_o$.
• Point-Splitting Regularization: To handle singularities arising from equal-point correlations in the microcanonical average, the author introduces a point-splitting regularization. This involves averaging observables over slightly displaced spatial coordinates, effectively removing short-range singularities.
• Projection onto Quadratic Charges: The derivation utilizes the Hilbert space structure of the Green-Kubo formula. The author identifies a "scattering subspace" ($H_{scat}$) spanned by quadratic charges (products of normal modes). The noise covariance is derived by projecting out this subspace from the standard Kubo formula.
• Integrable System Analysis: For integrable models, the author employs two distinct arguments:
  1. Algebraic: Using the known exact Onsager matrix for integrable systems and showing it equals the projected Kubo formula, implying the remainder (noise) is zero.
  2. First-Principles/Heuristic: Utilizing the infinite family of commuting time flows (higher Hamiltonians) in integrable systems. By averaging currents over these flows, the variance of the noise is shown to vanish as the number of flows increases.

3. Key Contributions and Results

A. The Projected Kubo Formula

The paper derives an exact expression for the noise covariance $\hat{L}_{o_1, o_2}$ in no-shock systems (specifically linearly degenerate systems). It is given by a projected Kubo formula:
$$\hat{L}_{o_1, o_2} = L_{o_1, o_2} - \frac{1}{2} \sum_{I, I': v_I \neq v_{I'}} \frac{\langle o_1^-, Q_I, Q_{I'} \rangle_c \langle o_2^-, Q_I, Q_{I'} \rangle_c}{|v_I - v_{I'}|}$$
where:

• $L_{o_1, o_2}$ is the standard Green-Kubo integral.
• The subtracted term represents the contribution of quadratic charges (scattering subspace).
• This subtraction accounts for long-range correlations generated by the non-linear transport of initial fluctuations (Euler scale effects).
• Significance: This clarifies that the "bare" noise is what remains after subtracting the deterministic, long-range correlations induced by the Euler dynamics.

B. Point-Splitting Regularization

The author proves that the microcanonical average of currents must be regularized via point-splitting to be well-defined in the hydrodynamic expansion.
$$[q_i(x) q_j(x)]_{reg} = \lim_{\epsilon \to 0} \frac{1}{2} [q_i(x+\epsilon)q_j(x-\epsilon) + q_i(x-\epsilon)q_j(x+\epsilon)]$$
This regularization is crucial for making the stochastic PDE well-defined despite delta-correlated initial conditions and noise.

C. Vanishing of Noise in Integrable Systems

The paper provides a rigorous proof that hydrodynamic noise vanishes in integrable systems at the diffusive scale ($O(\ell^{-1})$).

• Result: $\hat{L}_{j_i, j_k} = 0$ for currents in integrable models.
• Implication: Consequently, the bare diffusion matrix $\hat{D}$ also vanishes.
• Gauge Freedom: The author argues that by choosing an appropriate "gauge" (averaging currents over higher commuting flows), the noise vanishes at all orders in the hydrodynamic expansion ($O(\ell^{-n})$). This confirms that for integrable models, the Ballistic Macroscopic Fluctuation Theory (BMFT) is the complete hydrodynamic theory without needing stochastic terms.

D. Extended Einstein Relation

The paper establishes an extended Einstein relation connecting the bare diffusion matrix $\hat{D}$ to the projected noise covariance $\hat{L}$:
$$\hat{L}_{o, j_i} = \sum_k \hat{D}^k_o C_{ki}$$
This relation holds for arbitrary observables, not just currents, and is derived from the self-consistency of the fluctuating hydrodynamic equations.

E. Normal Diffusion for Two-Point Functions

Despite the vanishing of bare noise in integrable systems, the paper shows that two-point correlation functions still satisfy a standard diffusion equation with a normal diffusion constant.

• The diffusion matrix $D$ is related to the full (un-projected) Onsager matrix via the Einstein relation.
• This occurs because the "bare" diffusion and the contribution from long-range correlations (via the regularized microcanonical part) combine perfectly to yield the full Onsager matrix.
• This resolves a paradox: while the noise vanishes, the diffusive scaling of correlations remains, driven by the interplay of initial state fluctuations and non-linear transport.

4. Significance and Impact

1. Resolution of the Integrable Noise Conjecture: The paper definitively proves the conjecture that integrable systems possess no hydrodynamic noise at the diffusive scale. This implies that the dynamics of coarse-grained currents in integrable models are deterministic at the leading and first sub-leading orders, governed solely by the Euler equations and long-range correlations.
2. Unification of Hydrodynamic Theories: It unifies the concepts of Ballistic Macroscopic Fluctuation Theory (BMFT) and standard Fluctuating Hydrodynamics. It shows that BMFT is the leading order, and the "noise" is a specific correction that can be projected out in integrable systems.
3. Microscopic Origin of Noise: The work provides a clear microscopic mechanism for hydrodynamic noise: it arises from the central limit theorem applied to "forgotten" degrees of freedom, but its magnitude is reduced by the projection of long-range correlations (quadratic charges).
4. Generalization to Non-Integrable Systems: For generic linearly degenerate (non-integrable) systems, the noise does not vanish. The derived projected Kubo formula provides the correct constitutive relation for these systems, distinguishing them from integrable ones.
5. Methodological Rigor: The introduction of point-splitting regularization and the rigorous derivation of the projected Kubo formula provide a robust mathematical toolkit for analyzing higher-order corrections in hydrodynamic expansions, potentially applicable to higher dimensions and other scaling regimes (e.g., KPZ).

In summary, Doyon's work establishes a complete, self-consistent fluctuating hydrodynamic theory for one-dimensional no-shock systems, proving that integrability imposes a unique constraint where hydrodynamic noise vanishes, while simultaneously explaining how normal diffusion emerges in two-point functions through the cancellation of bare and correlation-induced terms.