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arxiv: 2506.05279 · v3 · submitted 2025-06-05 · ❄️ cond-mat.stat-mech · math-ph· math.MP

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

Benjamin Doyon This is my paper

Pith reviewed 2026-05-19 10:37 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech math-phmath.MP
keywords hydrodynamic noiseintegrable systemsprojected Kubo formulaballistic hydrodynamicsone-dimensional fluctuationsbare diffusionmacroscopic fluctuation theory
0
0 comments X p. Extension

The pith

In integrable one-dimensional systems, hydrodynamic noise and bare diffusion vanish at all orders under a suitable current gauge, so the Ballistic Macroscopic Fluctuation Theory supplies the complete hydrodynamic description.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a projected Kubo formula for the covariance of hydrodynamic noise that removes contributions from ballistic long-range correlations. Nonlinearities are handled by point-splitting regularization, producing a well-defined stochastic PDE in the ballistic scaling limit. In integrable models, this projected noise is shown to vanish identically, eliminating bare diffusion terms. The result generalizes recent hydrodynamic equations to include both long-range correlations and diffusion while preserving ordinary diffusion for two-point functions. Consequently the Ballistic Macroscopic Fluctuation Theory becomes the all-order theory for such systems.

Core claim

Hydrodynamic noise is given by a projected Kubo formula in which ballistic long-range correlation effects have been subtracted; nonlinearities are tamed by point-splitting. In integrable systems this noise vanishes at all orders once an appropriate gauge for the currents is chosen. The Ballistic Macroscopic Fluctuation Theory therefore furnishes the complete hydrodynamic theory, including the asymptotic expansion of connected correlation functions organized by cumulants.

What carries the argument

The projected Kubo formula, which extracts the noise covariance by removing ballistic long-range correlations from the standard expression while preserving the Einstein relation to bare diffusion.

If this is right

  • Fluctuating hydrodynamics is expressed as a stochastic PDE whose solutions organize the asymptotic expansion of connected correlation functions via cumulants.
  • The equation for average densities incorporates both long-range correlations and bare diffusion, generalizing earlier ballistic results.
  • Two-point functions obey a standard diffusion equation whose matrix is fixed by the projected Kubo formula.
  • In integrable models the theory reduces to the Ballistic Macroscopic Fluctuation Theory without diffusive corrections at every order.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The vanishing of noise suggests that leading fluctuations in integrable systems remain purely ballistic, with diffusive spreading appearing only at higher orders or through boundary effects.
  • The same projection technique might be used to regularize fluctuating hydrodynamics in other systems that possess long-range correlations but are not fully integrable.
  • Direct comparison of measured current fluctuations against the projected Kubo expression in finite integrable chains would provide a quantitative test of the all-order claim.

Load-bearing premise

The systems admit no shocks, as occurs in linearly degenerate and integrable cases, so that diffusive scaling of corrections to ballistic motion survives the presence of nonlinearities.

What would settle it

A numerical measurement of current-current correlations in a clean integrable chain that yields a nonzero bare diffusion coefficient at asymptotically large scales would contradict the vanishing result.

Figures

Figures reproduced from arXiv: 2506.05279 by Benjamin Doyon.

Figure 1
Figure 1. Figure 1: A fluid cell of length L over time T. There are any more interactions within the cell than there are boundary effects, leading to a separation of scales between fluctuations of arbitrary local observables and fluctuations of conserved densities. gives us a framework from which we will evaluate à la Kubo, in the next section, all constitutive elements in terms of equilibrium correlation functions, in linear… view at source ↗
Figure 2
Figure 2. Figure 2: A fluid cell with the typical particle trajectories within it over a time [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
read the original abstract

Hydrodynamic noise is the Gaussian process that emerges at larges scales of space and time in many-body systems. It is justified by the central limit theorem, and represents degrees of freedom forgotten when projecting coarse-grained observables onto conserved quantities. It is the basis for fluctuating hydrodynamics, where it appears along with bare diffusion terms related to the noise covariance by the Einstein relation. In one spatial dimension, nonlinearities are relevant and may modify the corrections to ballistic behaviours by superdiffusive effects. But in systems where no shocks appear, such as linearly degenerate and integrable systems, the diffusive scaling of these corrections stays intact. Nevertheless, anomalies remain. We show that in such systems, the noise covariance is given by a modification of the Kubo formula, where effects of ballistic long-range correlations have been projected out, and that nonlinearities are tamed by a point-splitting regularisation. With these ingredients, we obtain a well-defined hydrodynamic fluctuation theory in the ballistic scaling of space-time, as a stochastic PDE. It describes the asymptotic expansion in the inverse variation scale of connected correlation functions, self-consistently organised via a cumulant expansion. The resulting anomalous hydrodynamic equation for average densities takes into account both long-range correlations and bare diffusion, generalising recent results. Despite these anomalies, two-point functions satisfy an ordinary diffusion equation, with diffusion matrix determined by the Kubo formula. In integrable systems, we show that hydrodynamic noise, hence bare diffusion, must vanish, as was conjectured recently, and argue that under an appropriate gauge of the currents, this is true at all orders. Thus the Ballistic Macroscopic Fluctuation Theory give the all-order hydrodynamic theory for integrable models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript derives a projected Kubo formula for the covariance of hydrodynamic noise in one-dimensional systems by removing ballistic long-range correlations, regularizes nonlinearities via point-splitting, and obtains a stochastic PDE description of fluctuating hydrodynamics in the ballistic scaling. It shows that the noise (and thus bare diffusion) vanishes in integrable models, and argues that this vanishing extends to all orders in the cumulant expansion under an appropriate gauge choice for the currents, implying that Ballistic Macroscopic Fluctuation Theory supplies the complete all-order hydrodynamic theory for such systems.

Significance. If the central claims hold, the work supplies a parameter-free derivation of hydrodynamic fluctuations grounded in the central limit theorem and Einstein relation, resolves anomalies in 1D fluctuating hydrodynamics for linearly degenerate and integrable systems, and strengthens the status of BMFT as the exact hydrodynamic description for integrable models by addressing a recent conjecture on noise vanishing.

major comments (1)
  1. [abstract and integrable systems discussion] The extension of the vanishing of hydrodynamic noise to all orders in the cumulant expansion (abstract and the section discussing integrable systems) is argued to follow from the existence of an 'appropriate gauge' of the currents. However, no explicit construction of this gauge is provided, nor is there a demonstration that such a gauge can be defined order-by-order while preserving both the form of the stochastic PDE and the vanishing of the projected noise covariance. This leaves the all-order claim unsupported by the leading-order calculation.
minor comments (2)
  1. [derivation of stochastic PDE] Clarify the precise definition of the point-splitting regularization in the stochastic PDE and how it ensures well-definedness at the level of the cumulant expansion.
  2. [two-point functions] The statement that two-point functions satisfy an ordinary diffusion equation despite anomalies in the average densities would benefit from an explicit cross-reference to the relevant equation or subsection.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, their positive assessment of its significance, and for identifying a point where the all-order claim requires further clarification. We address the major comment below and will revise the manuscript to strengthen the presentation of the argument.

read point-by-point responses

  1. Referee: [abstract and integrable systems discussion] The extension of the vanishing of hydrodynamic noise to all orders in the cumulant expansion (abstract and the section discussing integrable systems) is argued to follow from the existence of an 'appropriate gauge' of the currents. However, no explicit construction of this gauge is provided, nor is there a demonstration that such a gauge can be defined order-by-order while preserving both the form of the stochastic PDE and the vanishing of the projected noise covariance. This leaves the all-order claim unsupported by the leading-order calculation.

    Authors: We thank the referee for this precise observation. The manuscript provides a complete leading-order derivation showing that the projected Kubo formula for the noise covariance vanishes identically in integrable models, owing to the infinite tower of conserved charges that permit a gauge choice in which the relevant current correlators are zero. The all-order statement is presented as an argument that the same gauge freedom—corresponding to the addition of terms proportional to total derivatives or other conserved quantities—can be exercised order by order in the cumulant expansion without changing the structure of the stochastic PDE or the definition of the projection. While this reasoning follows from the structure of the theory, we acknowledge that an explicit recursive construction of the gauge at each order is not supplied. In the revised version we will expand the integrable-systems section with a more detailed sketch of the recursive gauge choice, showing that the projection operator remains well-defined and that the noise covariance continues to vanish at every order while preserving the form of the fluctuating hydrodynamic equations. revision: yes

Circularity Check

1 steps flagged

Projected Kubo formula and noise vanishing derived independently from CLT and Einstein relation; minor self-citation on prior conjecture

specific steps
  1. self citation load bearing [Abstract]
    "In integrable systems, we show that hydrodynamic noise, hence bare diffusion, must vanish, as was conjectured recently, and argue that under an appropriate gauge of the currents, this is true at all orders."

    The vanishing result is presented as shown in the current work, yet the phrasing invokes a recent conjecture (likely prior work by the same author or close collaborators) before supplying the projected-Kubo argument; while not load-bearing for the leading-order claim, the self-reference on the conjecture contributes the minor circularity score.

full rationale

The derivation begins from the central limit theorem to justify the Gaussian hydrodynamic noise process and applies a projection to remove ballistic long-range correlations, yielding a modified Kubo formula for the noise covariance. This is tied to the Einstein relation for bare diffusion without reduction to fitted inputs. In integrable models the vanishing of noise follows from the structure of the projected formula. The all-order extension is argued via an appropriate gauge choice on currents rather than shown by explicit construction at each cumulant order, but this does not reduce the leading result to a tautology. The reference to a recent conjecture is a minor self-citation that is not load-bearing because the present work supplies the explicit projected-Kubo derivation and the gauge argument. No self-definitional steps, no fitted parameters renamed as predictions, and the overall chain remains self-contained against external benchmarks such as the central limit theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard assumptions in fluctuating hydrodynamics and integrability, without introducing new free parameters or entities; the projection is a modification of existing Kubo formula.

axioms (2)
  • domain assumption Hydrodynamic noise emerges from the central limit theorem at large scales
    Stated as justification for the Gaussian process.
  • domain assumption In systems with no shocks such as linearly degenerate and integrable systems, the diffusive scaling remains intact
    Key for the anomalies to be handled without superdiffusion.

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discussion (0)

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