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

I thank both referees for their useful comments on the paper. I have adjusted the abstract and introduction a little bit, and added ref 73. I have also modified the text (and added one more ref) to answer all of the referee’s comments as follows — all modifications are in blue in the text:

Referee 1

“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.”

Thank you for this question. Linearly degenerate systems are defined by a condition on the Euler current, and, as has been shown in previous works, imply that no shocks are produced. Here, I consider systems where no shocks are produced, as this seems to be the more important notion; these, then, include linearly degenerate systems, but there might be more. To answer the question, I do not know if the absence of shock production imply absence of quartic nonlinearity. Yes, such singularities can lead to logarithmic corrections to diffusion, but at least at for the $1/\ell$ expansion and for a finite number of modes (or an infinity of modes with well-controlled velocities and filling), the absence of cubic nonlinearity seems to be sufficient to make everything finite — for finite noise (bare) diffusion, one gets finite macroscopic diffusion. I added a comment on this question in the conclusion.

“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.”

Thank you. I had seen conditions of T-symmetry, but I had not seen PT-symmetry being used, however this may be my lack of knowledge. But I am certainly ready to accept that it is discussed in textbooks, as it is a natural notion, so I will cite one. I suppose current ref [49] is what the referee meant by Chaikin & Lubensky.

“"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?”

I mean here connected correlation functions - so the system may be in an ordered phase (where connected correlation functions may still vanish exponentially, and do away from critical points), I have clarified this.

Referee 2

“The notations can be difficult to follow. For instance, understanding all the equalities in an equation like (53) currently requires to go look for the different definitions that are scattered through the article.”

Yes that’s true - however I did try to be consistent and to keep everything as precise as possible, which is not so easy given the various notions here (the microscopic observables, their representation in the effective theory of fluctuating conserved densities, their averages in maximal entropy states, their averages in the long-wavelenth non-equilibrium state, their averages in the effective theory of fluctuating conserved densities). But I agree that Eq 53 is particularly cryptic. In fact, it is perhaps not a bad place there to recall the notation somewhat, and point to the previous equations. I have repeated there the meaning of the symbols, for clarity.

“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?”

Yes indeed this is crucial. I have added “It is crucial here that the volume be taken to infinity, as otherwise the equivalence does not hold, a point that will become important below.” on page 7. In fact, these corrections are important at the diffusive scale. This was not made clear enough in the paper. I have now added a few sentences on page 10, explaining that the finite-size effects on the microcanonical averages are what are responsible for the ``regularisation” of the nonlinear Euler current in the fluctuating theory, and are essential in order to render the theory well-defined (at least in no-shock systems).

“2) On page 7, around Eq. (10): why is the fact that the map β––→⟨q–⟩β is invertible related to the positivity covariance matrix?”

This is because then the Jacobian of this map is non-singular by eq 9; I have added a sentence just after eq 10.

“3) In Eq. (14), the author could clarify the meaning of O(ℓ−∞)”

It means that the corrections decay faster than any power of $\ell$ - I have adde an explanation just after eq 14.

“1) Include a summary of the notations and definitions used in the article (for the observables, averagings, ...). That would greatly help the reader.”

This is a great idea. I’ve added a un-numbered section before the introduction with the important notations, and references to equations in the paper.

“ 2) typos: - p. 10, after (27) "noise, They" → "noise. They" - p. 12, "conserve density" → "conserved density" - p. 12, "Physcailly" → "Physically" - p. 13, "not bee too large" → "not be too large" - p. 25, "intregable" → "integrable" - p. 26, "scatteing" → "scattering" - p. 29, "ligthness" → "lightness" - p. 33, "disccused" $\to" "discussed" - p. 33, "via because of” ” Thank you, corrected.