Ballistic macroscopic fluctuation theory

Dear Editor,

we thank you for sending us correspondence on our manuscript and we thank the Referee for their report. We are grateful that the Referee recognizes the importance of our article and recommends it for publication upon addressing their comments, which we do below. We indicate here the changes we performed in the revised version of the manuscript. The revised parts of the manuscript are highlighted in blue.

We hope that the Referees will be satisfied by our revision and that the manuscript will be found ready for publication in Scipost Physics.

Yours Sincerely,

the Authors.

"The paper develops a formalism for describing macroscopic fluctuations for the hydrodynamics of an arbitrary number of conserved quantities which possess ballistic modes. The main output of the approach is a formal setup for computing the full counting statistics of charges and currents, and dynamical correlation functions, both in the presence of a slowly space-dependent initial Gibbs ensemble. These quantities are then computed explicitly in the case of integrable systems.

Determining the general structures governing the emergent dynamics of many-body systems, classical and quantum, is presently a very active area of research. The paper is quite well-written and certainly contains results that can lead to further research on the topic. I therefore recommend the paper for publication, after the following remarks/questions have been addressed:"

We thank the Referee for the supportive report and the careful reading of our manuscript. In the following, we provide a point-by-point reply to all the questions contained in their reports.

1) This formalism strongly resembles the MSR approach to fluctuating hydrodynamics, which also uses an action formulation. What are the differences between this and the MSR approach, and what are the advantages of the present framework?

Reply: The BMFT and the Martin-Siggia-Rose (MSR) formalism are, indeed, similar, though there are also fundamental differences between the two approaches. The common idea between the MSR and the BMFT lies in the fact that in both cases one writes down path integral formulae in terms of fields associated to the slow system variables. The fundamental difference between the BMFT and the MSR refers, however, to the scale the two theories refer to. The BMFT is a macroscopic theory as it is based on the macroscopic large-scale deterministic Euler equation. The MSR is a mesoscopic theory, where the fast degrees of freedom are kept into account in a coarse-grained way through the noise term. The MSR is therefore based on the mesoscopic stochastic Langevin equation. The advantage of the BMFT in comparison with the MSR is that the former allows for an exact evaluation of path integrals via saddle point, while in the latter path integrals have to be evaluated perturbatively. We further briefly discuss here the similarities and the differences between the two approaches.

In the case of the BMFT the slow variables, the emergent degrees of freedom, are the conserved densities. Fluctuations of the conserved densities stem from the initial condition, while their evolution is deterministic according to the Euler equation. In the MSR, one has, instead, a coarse-grained field $\psi(x,t)$, which obeys the stochastic Langevin equation. Fluctuations of the field $\psi(x,t)$ in this case stem from the noise in the Langevin equation (in addition to the fluctuations possibly coming from the initial condition). In both the cases, the fluctuating fields obey the associated equations of motions (Euler equation in BMFT and Langevin equation in MSR), which results in a delta function constraint over the equation of motion into the path integral. This constraint is in both cases handled resorting to the Laplace representation of the delta function. This leads to the introduction of an auxiliary field $\widetilde\psi(x,t)$, which in the MSR formalism is usually named ``response field''. In the BMFT the equivalent of the response field is the field $H(x,t)$ introduced in Eq.~(47). The difference between the MSR functional and the BMFT one is that the former is quadratic in the response field $\widetilde{\psi}$, while the latter is linear in $H(x,t)$. This difference comes from the noise contained into the Langevin equation, whose integration generates the quadratic term in $\widetilde{\psi}$.

Fundamentally, there is no macroscopic-scale parameter in the MSR functional since it is formulated for the mesoscopic Langevin equation. The MSR action is usually taken as a starting point for approximation schemes such as perturbation theory and renormalization group schemes. The BMFT action that we propose, on the contrary, is a macroscopic fluctuation theory and it is therefore entirely controlled by the scale parameter $\ell$, i.e., the macroscopic variation length of the initial state. The BMFT therefore goes one step further than MSR by realizing that the path integrals are in fact dominated by the saddle point of the action when taking the limit $\ell\to\infty$.

The BMFT path integrals for current fluctuations and correlation functions are consequently dominated by the saddle point and no perturbative approximation is needed to evaluate them. This is clearly an advantage with respect to the MSR, as it reduces calculations to a variational problem, which amounts to solving a coupled differential equation. Thus in principle the path integrals can be evaluated exactly (in a non-perturbative way) provided that the BMFT equations can be solved. In many cases solving the BMFT equations is difficult, but there are some cases where that's possible, including integrable systems, as we demonstrate in the present article.

We have commented about the fundamental differences between the MSR and the BMFT formalism in the manuscript after Eq.~(57). We have therein also emphasized the non-perturbative character of the BMFT results in contrast to the case of the MSR formalism. The added part of the text is highlighted in blue. We have also added in the bibliography of the revised manuscript Refs.~[79-82] regarding the MSR formalism.

2) The presence of non-local correlations in the Euler scaling limit is quite intriguing. Three- and higher-point functions should include the same type of information. Is the advantage of the BMFT in that it allows to infer this scaling somehow more systematically? Also, how are diffusive/noisy corrections expected to alter the result?

Reply: The scaling of the two and higher point functions is entirely set by the Euler scaling since the long-range correlations discussed in the manuscript are a universal hydrodynamic effect. We indeed expect that higher point functions also incorporate the effect of long-range correlations. In particular, we expect equal-time $n$-point connected correlation functions to display long-range correlations of strength $\ell^{1-n}$ over macroscopic times $t\sim \ell$ and macroscopic distances of extent $x\sim \ell$. This scaling is set by the Euler-scaling definition in Eq.~(21). The advantages of the BMFT is that it allows us to exactly infer this scaling of the correlations, as shown explicitly by Eq.~(50). The BMFT furthermore allows to compute these $n$-point correlation functions exactly, and in particular within non-stationary states presenting fluid motion, by solving the associated BMFT equations (Eqs.~(101) of the manuscript).

The diffusive corrections should be evaluated from the multiscale fluctuating hydrodynamic theory, which we propose in Sec.~3.3 of the manuscript (see Eq.~(62) in particular). In particular, one should look at fluctuations around the saddle-point solution, which gives the Euler-scale $\ell \to \infty$ result. From Eq.~(63), we see that diffusive corrections are suppressed with a factor $\ell^{-1}$ with respect to the ballistic component of the current. We therefore expect that this amounts to sub-leading corrections decaying as $\ell^{-\alpha}$, with $\alpha>1$, to the ballistic two-point long-range correlations, which decay instead as $\ell^{-1}$ as explained in the manuscript. Clearly the establishment of the multiscale fluctuating hydrodynamic theory of Sec.~3.3 still requires a systematic study. Similarly, pinpointing the ensuing diffusive corrections to correlations and fluctuations is a very interesting topic which needs to be worked out in detail and that we leave to future studies.

3) The authors mention that non-local correlations require the presence of multiple conserved quantities. I'm confused by this statement: one can have a ballistic hydrodynamic mode with only a single conserved charge (at the cost of breaking time reversal), wouldn't such systems also possess non-local correlations in the same scaling limit?

Reply: The necessary requirement of multiple conservation laws for the presence of long-range correlations is intuitively clear. Conservation laws are, indeed, in one-to-one correspondence with hydrodynamic normal modes. With only one normal mode, and therefore only one conserved quantity, all the hydrodynamic modes travel at the same velocity in the initial state. As a result, they don't separate macroscopically on macroscopic time (under ballistic scalint $x\sim t$). Long-range correlations therefore cannot build up over time when only one normal mode is present. In the manuscript, we have explicitly shown this in Eqs.~(112)-(113) by starting from the evolution equation for the two-point, equal time, correlation function obtained from hydrodynamic projections.

In the Appendix E, we have also briefly discussed the totally asymmetric exclusion process (TASEP). This model is an example of a system that has only one conservation law that exhibits ballistic transport. We have therein argued (see Eq.~(199)) that for systems with a one-component (only one conserved charge) Euler hydrodynamic long-range correlations cannot develop in time.

It is therefore essential to have more than one conservation law on top of an inhomogeneous initial condition and interactions for the long-range correlations we discover in this paper to be present. The three conditions we have stated for long-range spatial correlations on ballistic times, which were not known before, are in fact one important result of the formalism that we have developed.

4) Finally, there is a minor typo on p. 29 above eq. (84): "T(o1)T(o1)" $->$ "T(o1)T(o2)".

Reply: We thank the Referee for pointing out this typo in the manuscript. We have corrected the typo in the revised version of the manuscript.

1. We added after Eq.~(57) a discussion of the similarities and the differences between the BMFT and the Martin, Siggia, Rose formalism. We have therein emphasized the advantage of the BMFT compared to the Martin, Siggia, Rose, as it allows to evaulate path integrals exactly by saddle point. In the Martin, Siggia, Rose, path integrals can, instead, only be evaluated approximately by perturbation theory. We have also added Refs.~[79-82] in the bibliography concerning the Martin, Siggia, Rose formalism.

2. We fixed the typo above Eq.~(84).