Ballistic macroscopic fluctuation theory via mapping to point particles

1. The authors derive Ballistic Macroscopic Fluctuation Theory for a class of integrable models by means of a mapping to point particles. They re-derive known results for full counting statistics and long-range correlation functions.

This is a rather technical but interesting piece of work and certainly should be published. However, given that the work appears to be largely concerned with the rederivation of known results, it is not immediately obvious that it fulfils the criteria for SciPost Physics.

I have a number of comments and questions I would ask the authors to address before the paper can be recommended for publication in SciPost Physics.

The paper is very technical and I don't think it will be clear to most readers what the advantage of the new approach of deriving BMFT is compared to Refs [24-26]. I think the authors should explain in more detail what can be done in the new approach which was not possible in the one of Refs [24-26]. The relevant SciPost Physics criterion is ``Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work." How does this work satisfy this criterion?

Response: The key feature of our work is not just a rederivation of known BMFT results, but the effective utilisation of the mapping to point particles for generic integrable models in which quasi-particles can be expressed as a function of the phase space density (Eq. 14). This provides a new framework for studying fluctuation in Ballistic systems and moreover it bridges more clearly BMFT with standard formulation of MFT in terms of fluctuating densities. In contrast to Refs. [24-26], our approach allows access to trajectory level, or particle-resolved observables, as we can express the trajectories as functional of the phase space density profile (Eq. 14). In particular, we can in principle compute the multiple quasi particle position statistics or tagged particle position statistics. These observables are not naturally accessible in the eariler BMFT formalism. Furthermore, by mapping an interacting integrable system to a non-interacting system, the structure of the large deviation function is substantially simplified. While we have focused on rederiving the existing results, an intelligent use of this mapping could provide clearer ways to compute large deviation functions explicitly that might be technically difficult in the earlier approach.

We therefore believe that this work satisfies the SciPost Physics criterion of opening a new pathway, as it reformulates the fluctuating theory for ballistic systems that enables particle-resolved observables.

2. What is the physical importance of the normal-mode two-point function determined in Section 4 (given that the normal mode density is very complicated in terms of the microscopic degrees of freedom)? Can it be measured in experiments? If so the authors should explain. What is its theoretical importance? How difficult is it to evaluate in practice, and what can be learned from it?

Response: The correlations between the normal modes can be used to compute correlations between any generic densities of charges (e.g. density-density, current-current, etc). These are experimentally relevant quantities; they can be measured in most of cold-atom settings (bosonic/fermionic/spins) and in classical systems as well, obviously (wave experiments in optical fibers). The non-trivial point of BMFT is actually to show that due to ballistic propagations, there are long-range correlations developing.

From a theoretical perspective these correlation also play a central role in extending the generalized Hydrodynamics beyond Euler scale see Ref. [1](Ref. 72 of modified manuscript) . In particular, their time evolution at, short-times gives Green-Kubo diffusive correction to the ballistic transport as done in supplimetary of Ref. [1].

3. Throughout the paper the authors stress that their approach applies to general integrable models. However, they only consider a theory with a single particle species and diagonal scattering. If the generalization to models with several particle species and scattering matrices as obvious as the authors' comments suggest, they should consider giving the relevant equations in an Appendix.

Response: The computation of FCS and density correlation can indeed be expanded to multiple species. Yet, the scope of this work is indeed to open the way for more complicated computations within a easier framework, the one of mapping to point particles. In multi-species models, this would require extra species labeling, and it will still map the GHD to free particle evolution as mentioned in Ref. [2] (Ref. 36 of the updated manuscript).



4. Regarding the observed non-Gaussianity in Fig 4: the saddle-point approximation holds only in the large-T limit, for which the data in Fig.4 is well described by a Gaussian. How can the authors be sure that the corrections to the saddle-point approximation for T=10 are negligible? I assume that the authors would claim that the numerical results show this, but the quality of the numerical data collapse for the values of $Q_1$ where there is data for a range of T's is quite difficult to judge without an estimate on the error bars.

Response: We have now replaced the plot of rate function at times shorter than $T=10$, which shows the non-Gaussian nature of the rate function clearly. These are robust across different times, indicating they are not numerical artifacts but clearly reflects the underlying non-gaussian nature. For the comparison of numerical data at late times (for which the referee is indeed correct that this would provide an exact numerical test of the theory), we have to perform importance sampling for hard rod gas, which is currently an open question, and that we leave it for future works. We stress however that it is quite interesting and remarkable that typical fluctuations at short times are indeed well approximated by our saddle point result, giving hope that the full FCS could be observed in real experimental settings.

5. What is the precise status of eqns (4) and (13)? Have they been proven (in a mathematical physics sense) or are they conjectures supported by convincing evidence? If they haven't been proven I think readers would benefit from a clear statement about what kinds of evidence supports them.

Response: The Eq. 4 describes the microscopic mapping to the point particles, which has been proven only for the case of classical Toda chain in Ref. [3] (Ref. 70 in modified manuscripts). For a quantum integrable models, heuristic derivation on the Lieb-Liniger gas is provided in Ref. [4] (Ref. 44 of modified manuscript) and can be easily extended to other integrable models. On the other hand Eq. 14 has been shown to map the GHD equation to point particle GHD, see Ref. [2].



[1] Hübner, Biagetti, De Nardis, Doyon, Diffusive hydrodynamics from long-range correlations. Phys. Rev. Letts. 134 (18) 2025, https://doi.org/10.1103/PhysRevLett.134.187101

[2] Doyon, Spohn, Yoshimura, A geometric viewpoint on generalized hydrodynamics. Nucl. Phys. B. 926 (1) 2018, https://doi.org/10.1016/j.nuclphysb.2017.12.002

[3] Aggarwal, Asymptotic scattering relation for the toda lattice, arXiv preprint, arXiv:2503.08018 2025, https://doi.org/10.48550/arXiv.2503.08018

[4] Doyon, Hübner, Ab initio derivation of generalised hydrodynamics from a gas of interacting wave packets. arXiv preprint arXiv:2307.09307. 2023, https://doi.org/10.48550/arXiv.2307.09307

The authors use a generalised hard rod mapping to express asymptotic quasi-particle trajectories in integrable systems to those of non-interacting particles. These facilitate a straightforward computation of large fluctuations and correlations after reversing the mapping from non-interacting to interacting particles.

Some comments and questions:

1. It is not clear whether the central equations 5 and 9 of the generalized hard rod transformation are simply postulated or whether they refer to previous work (specifically ref. 69, 70 being a detailed investigation of a particular example).

Response: Eq. 5 and 9 are the generlized hard rod lengths as described in the reference 37. In references 69 and 70, a particular example has been investigated in detail. We demonstrate this mapping leads to GHD equations in Appendix A.

2. The paper is at times difficult to follow due to extensive derivations and equations in the main text, e.g. BMFT of point particles in section 2, and two-point correlations on section 5, the majority of which could be moved to the appendix while preserving the main message of the paper.

Response: We have kept the BMFT construction for point particles in Section~2 in the main text, as it introduces the central framework and notation used throughout the paper. However, we have shortened Section-5 and relegated the details of the calculation of two-point correlations to Appendix~D in the main text.

3. What is a ``typical equilibrium profile" mentioned in 26? I could not find a statement as to how it is characterized. This also brings me to a conceptual problem - In eq. 21 the large deviations function is expressed as an average over a "typical" phase space density while large deviations invariable result from ``atypical" initial configurations. How is this reconciled?

Response: The term typical equilibrium profile refers to the average phase space density ($\bar{\rho}_0(x, \theta)$) of the system in the generalised Gibbs ensemble. This ensemble is characterized by the set of generalised temperatures (Lagrange multipliers associated with conserved charges), which uniquely determine the average phase space density via the thermodynamic Bethe ansatz (TBA). Then we use the point particle mapping given in Eq.~16 to get $\bar{\rho}_0(x, \theta) \to \bar{r}_0(x, \theta)$. So the typical profile is characterised as [see Fig. 2 of main text]

\[
\{\bar{\beta}_i(x, t=0)\}
\;\stackrel{\text{TBA Ref.~32}}{\longleftrightarrow}\;
\bar{\rho}_0(x, \theta)
\;\stackrel{\text{Mapping Eq.~16}}{\longleftrightarrow}\;
\bar{r}_0(X, \theta)
\]


Hence, providing a typical profile $\bar{r}_0(X, \theta)$ is equivalent to specifying a set of generalized temperatures defining the initial GGE state. This typical profile represents the most probable macroscopic state of the system. When a sample is prepared, the macroscopic state will fluctuate around the typical profile. The large deviation function then arises from rare fluctuations of the profile away from the typical profile. This is not a contradiction with Eq. 22 (21 in the old main text).

We added a discussion in the main text below Fig. 2 in page 4.

4. Section 3: ``... we have assumed that the statistical nature of the initial profile, when described in the point particle density ..., is governed by the probability density functional with the large deviation function...".

What precisely does this assumption encompass?

Response: In this paragraph, we are assuming the statistics of the initial state. We assume that when the initial state of a generic integrable model is prepared in a GGE, the probability density of observing an initial state mapped to the point particle coordinates $r_0(X, \theta)$ around typical profile $\bar{r}_0(X, \theta)$ is given in Eq.~ 28 (27 in the previous version of the main text). This large deviation form quantifies the statistics of the initial profile relative to the typical profile for the point particle system, as discussed in Appendix~ B.

A general proof of this large deviation form, for quantifying the initial state after mapping to point particles, is not available for generic integrable models. However, the assumption is well-motivated, as the Hamiltonian governing the mapped point particles is non-interacting (see Ref. [67]) and hence one expects that the large deviation functional is given by the free energy of the non-interacting system. In Appendix~B.2, we provide an explicit combinatorial calculation for the hard rods, showing that the equilibrium free energy, when expressed in terms of the point particle density, coincides with the large deviation form given in Eq.~(28) (27 in the previous version of the main text).

We added a discussion in the paragraph below Eq.~ 62 (61 in previous version of the text) in page 13 .


1. In Figure 4, the deviations from Gaussianity are appreciable only for the smallest time (T=10). Presumably, this is due to direct sampling of large values becoming exponentially hard with time. On the other hand, it is not clear from the presented data what part of this is due to finite-time effects. In principle, large deviations can be sampled efficiently by tilting the measure.

Response: We have now modified Figure 4 to show shorter times, which clearly shows that the rate function is non-Gaussian. The tilted sampling requires a substantial amount of numerics and we have left it for future works. Our current plot is already matching with theory and shows non-gaussian nature.


2. The identification of cumulants and correlators with derivatives of the asymptotic form of the their respective generating functions in eqs. 82, 95 123 requires additional regularity assumptions, see e.g. Ž Krajnik, J Schmidt, V Pasquier, E Ilievski, T Prosen Physical Review Letters 128 (16), 160601. This is more than a technical detail as the regularity condition is known to be violated in a number of integrable models.

Response: We now explicitly state that the identification of cumulants and correlation functions with derivatives of the scaled cumulant generating functions relies on the validity of Bryc’s regularity condition. This assumption ensures the differentiability of the asymptotic generating function and the applicability of a central limit theorem.

We also clarify that this the regularity condition is known to be violated in certain integrable models, as discussed in the above. Our results should therefore be understood as applying to regimes where the scaled cumulant generating function is sufficiently regular, and we have added a brief discussion of this limitation in the text.



3. In the Conclusions, the authors claim that the method can be applied to ``generic" and ``arbitrary" integrable models. At the moment this does not seem to be the case for classical integrable models. For example eq. 29 gives the free energy of classical particles which is correct but incomplete whereas 142 refers to classical systems which is wrong.

Specifically, it is known that description of certain classical integrable models involve radiative modes nor can solitons always be treated as classical particles, see e.g.:
A Bastianello, B Doyon, GMT Watts, T Yoshimura SciPost Physics 4 (6), 045
R Koch, JS Caux, A Bastianello Journal of Physics A: Mathematical and Theoretical 55 (13), 134001
A Bastianello, Ž Krajnik, E Ilievski Physical Review Letters 133 (10), 107102

To the best of my knowledge, a derivation of generalized hydrodynamics from kinetics of a generic classical integrable systems remains an open problem precisely because of this difficulty.

Response: We are indeed only focusing on the particle models and have explicitly mention in the manuscript. We have now specified it clearly in Eq. 128 (142 of the previous version) as well. However, extending these for radiative modes will involve accounting for the correct statistics in particular the Rayleigh-Jeans law, which we have not discussed in this article.


4. Minor comments/typos:

page 3: hard rods gas $\to$ hard rod gas page 5: in the express ... in the normal mode density (?) eqs 1, 14 and others: the order of integrals over x and theta changes sporadically Appendices: Hard rods $\to$ hard rods In 1.2 after eq. 14 the free coordinates $(X, k)$ are mentioned, but it is unclear what k is from the surrounding explanation.

Response: We have corrected the typos and accounted for the minor comments.

Overall, the work explores a natural idea to quantify dynamics of quasi-particles within a hydrodynamic framework and uses it to recover some previously obtained results. However, as noted above, some points require further clarification and it is not clear how different the method is from related hydrodynamic techniques used in the field.

Response: The main feature of our work is using the mapping to point particles for generic integrable models in which quasi-particles can be expressed as a function of the phase space density (Eq. 14). This in principle provides a single particle density level description complementary to existing BMFT, which as it has been previously formulated, is constructed through charge and currents densities and does not naturally provide access to particle-resolved observables. Hence, our approach provides a straightforward framework for investigating quantities like the tagged particle position distribution and the quasi-particle position distribution, whose computation using previously existing BMFT methods is not particularly clear.