Stochastic dynamics of quasiparticles in the hard rod gas

• simple, extensive calculations, and a phenomenological model, that clarify and generalise a recent result on correlations in hard rods
• easy to follow

• some open questions are not quite resolved

In this paper, the authors study the dynamics of quasiparticles (which I call particles below) in the hard rod gas. They study in particular correlations of tagged particles, and recover in a more general setting including inhomogeneous initial conditions (but perhaps under strong constraints, see below), the recent result [1] that was proven rigorously. They provide exact microscopic calculations, calculations based on BMFT and the recent understanding that Euler transport of fluctuations holds beyond the Euler scale in integrable systems, and calculations based on a nice phenomenological model of Brownian particle that extracts the physics of the correlations.

The paper is well explained overall and the results are interesting - they generalise recent rigorous results, the calculations themselves show how these recent results are in fact relatively straightforward and natural, and the phenomenological model really provides a clear understanding. All these results are strongly confirmed by numerics. There are some aspects which are still unclear about how these results connect with fluctuating hydrodynamics, these are discussed in the conclusion, and certainly are interesting open problems.

My comments, with changes to be made, are as follows:

1. P3: Description of the results, points 2,3: here it is not clear what $X_{v_0}(t)$ means, and the result expressed in point 3 is not entirely accurate.

a) My understanding is that the label $v_0$ does not completely specifies which particle we are taking (otherwise the results on correlations would not be meaningful). But then this means in the ensemble constructed, there are many particles for any given label $v_0$, which is slightly not so natural as typically an event where two different rods are drawn from the ensemble with the same velocity would be of measure zero. It would be good to specify what is meant here; see also the more general comments below on the specification of the distribution and variables (here, the most natural would be to choose two different particles, of velocities ``near to’’ $v_0$ and $u_0$, and the result would be smooth in the distances to $v_0,u_0$).

b) In point 3, my understanding is that the particles cannot in fact be arbitrarily far from each other we also need its initial position. One should specify in $X_{v_0}, X_{v_0’}$ not only the velocity, but also $x_0, x_0’$ the initial positions of the two particles. Then the result holds for $t\sim,> (x_0-x_0’)^2$, i.e. within the diffusive scale. For $t\sim x_0-x_0’$, for instance, then the statement does not hold.

1. P6: The distribution is Gaussian at large times in the sense of the CLT. Maybe mention the more accurate statement is simply that all cumulants scale linearly in $p_{rl}(t), p_{lr}(t) \sim t$ (and there is a large-deviation statement)?

2. P10: Results 26-28: it would be good to know the correction terms. Also, reference to the work [1] is made, but there, as I understand it, the initial separation was allowed to be of order $\sqrt{t}$ (diffusive scale). In a sense, the results here are an "easier version’’, with fixed initial distance. Is this correct? Can the diffusive scaling be considered here too?

3. P12: “noted previously in Eq. (27)” I believe it is Eq (28).

4. Formulas in section 3 are, as far as I understand, established here rigorously - the derivations are elementary and it seems without any real “hole”. Is this correct? Perhaps the authors could comment on this.

5. I have a number of comments about the chosen distribution and variables.

a) Choosing i.i.d. positions is fine for the homogeneous case, but in the inhomogeneous case I believe it introduces, on the hard rods themselves, special long-range correlations. Thus, the initial state is not the “physically sensible” local GGE that is typically taken in many-body system. The local GGE would have the measure $[dx dv] \Theta(x_{i+1}-x_i-a) e^{-\sum_i \beta(x_i,v_i)}$ for some function $\beta(x,v)$ that is slowly varying in $x$, and it is this measure that is expected to have exponentially decaying correlations. In this measure, however, it is much more difficult to evaluate correlation functions of tag particles. It would be good for the authors to comment on this.

b) Fixing the initial positions of tagged particles, and fixing $\bar N$, is convenient for calculations, but in principle it affects their distribution and could affect the results. A more ``natural’’ way would be to sample a hard rod configuration, and then choose particles, e.g. those nearest to $X_0$ and $Y_0$. I think that the calculations of section 5, using Euler transport of fluctuations, essentially does that. Is this correct? If so, then it indicates that the same results are obtained, at least in the homogeneous case. Could the author comment on this?

This area of study can be very confusing, with some apparently inconsistent results. Often I believe this comes from subtleties in the choice of distributions and random variables, and this may relate to the questions raised in the conclusion. It would be good for the author to specify, already from the beginning, the particularities (points a and b above) of their distributions and how they define their random variables (from which microscopically derive the results), and to mention if their conclusions are expected (or argued for, such as in section 5 I believe) to hold in more "natural’’ settings.

1. P13 Citations [11-14]: as here it is the diffusive scale that is being discussed, maybe it is worth citing the refs where it is proposed that Euler-scale transport is a good description for fluctuations / correlations beyond the Euler scale. Possible refs are [14,21,22,24] as well as

S. Gopalakrishnan, A. Morningstar, R. Vasseur, and V. Khemani, Phys. Rev. B 109, 024417 (2024).

Z. Krajnik, J. Schmidt, V. Pasquier, E. Ilievski, and T. Prosen, Phys. Rev. Lett. 128, 160601 (2022).

T. Yoshimura and Z. Krajnik, Phys. Rev. E 111, 024141 (2025).

S. Gopalakrishnan, E. McCulloch, and R. Vasseur, Proc. Natl. Acad. Sci. U.S.A. 121, e2403327121 (2024).

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