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Hydrodynamic noise in one dimension: projected Kubo formula and how it vanishes in integrable models
by Benjamin Doyon
Submission summary
| Authors (as registered SciPost users): | Benjamin Doyon |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2506.05279v2 (pdf) |
| Date submitted: | Aug. 27, 2025, 12:17 p.m. |
| Submitted by: | Benjamin Doyon |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
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.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
1) Determines the exact corrections to the deterministic hydrodynamic (large scale) theory for a large class of 1D interacting system.
2) Physically discusses the origin of noise and characterises its covariance in terms of a new "projected Kubo formula".
3) Identifies the diffusive corrections and derives an extended Einstein relation that links the noise covariance to the bare diffusion coefficients.
4) Proves a recent conjecture that noise vanishes for integrable systems.
Weaknesses
Report
This is a very interesting work, that brings new mathematical and physical understanding to the emergence of hydrodynamic theory (including its corrections) at large scales. It contains a lot a results and thus takes a lot of time to read, but it is definitely worth reading. The article contains very interesting physical discussions, in particular in Section 3, which greatly help understanding the physics behind the equations. It definitely meets the criteria for publication in SciPost Physics. Nevertheless, I think the readability can be increased easily (in particular in regards of the numerous notations), see below.
Minor comments/suggestions:
1) The equivalence between microcanonical and canonical ensembles is used several times. But for finite systems of length L (size of the fluid cell), one expects corrections. Are these corrections subleading and thus irrelevant? Could the author add a comment on this point?
2) On page 7, around Eq. (10): why is the fact that the map $\underline{\beta} \to \langle \underline{q} \rangle_{\beta}$ is invertible related to the positivity covariance matrix?
3) In Eq. (14), the author could clarify the meaning of $\mathcal{O}(\ell^{-\infty})$ which is used here for the first time and can be a bit surprising, in particular since there is also a $\mathcal{O}(\ell^{-2})$ in the same equation.
Requested changes
1) Include a summary of the notations and definitions used in the article (for the observables, averagings, ...). That would greatly help the reader.
2) typos: - p. 10, after (27) "noise, They" $\to$ "noise. They" - p. 12, "conserve density" $\to$ "conserved density" - p. 12, "Physcailly" $\to$ "Physically" - p. 13, "not bee too large" $\to$ "not be too large" - p. 25, "intregable" $\to$ "integrable" - p. 26, "scatteing" $\to$ "scattering" - p. 29, "ligthness" $\to$ "lightness" - p. 33, "disccused" $\to" "discussed" - p. 33, "via because of"
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Report
This paper is concerned with the noise in the fluctuating hydrodynamics of integrable systems. It provides several proofs that the bare noise is absent in these systems (although there are indirect mechanisms, uncovered in previous work, that effectively produce noise through nonlinear coupling to other modes). This confirms previous conjectures made by the author and others.
The paper is well written, with an emphasis on precision at the cost of readability. I only have a few clarifying questions:
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The concept of "linearly degenerate systems" plays an important role throughout. They seem to be defined as systems where shocks are not produced at the Euler scale. This implies no cubic nonlinearity and hence no emergence of KPZ. Does this also imply no quartic nonlinearity? These can arise in systems with particle-hole symmetry (see, e.g., [36]), which are still marginal terms in the fluctuating hydrodynamics, and thus lead to logarithmic corrections to diffusion.
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Above Eq. (6): I suspect the consequence of PT symmetry on diffusive fluctuating hydrodynamics long predates [41] - for example in textbooks by Forster, or Chaikin & Lubensky, or others. It is also pervasive in part of the recent litterature on fluctuating hydrodynamics.
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"We assume that β i ’s are away from phase transitions, so that correlations are short-range in these states" - is it then also assumed that we are not in an ordered phase?
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)