Microscopic and hydrodynamic correlation in 1d hard rod gas

Anupam Kundu; Indranil Mukherjee; Seema Chahal

REVIEW 3 major objections 4 minor 27 references

Microscopic and hydrodynamic correlation in 1d hard rod gas

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper establishes that for a one-dimensional hard-rod gas, the exact microscopic density correlation, coarse-grained over fluid cells, equals the ballistic macroscopic fluctuation theory prediction, analytically validating hydrodynamic

desk verdict Exact finite-N hard-rod correlation formulas and their coarse-grained BMFT match are real, but the 'emergence/validation' framing overstates a correspondence built on a pre-correlated initial ensemble. read the letter →

arxiv 2601.04951 v2 pith:UYXBKUDV submitted 2026-01-08 cond-mat.stat-mech

Indranil Mukherjee , Seema Chahal , Anupam Kundu This is my paper · ORCID

classification cond-mat.stat-mech

keywords hardrodgasdensitycorrelationscoarse-grainingballisticmacroscopicfluctuationtheoryEulerhydrodynamicslong-rangeintegrablesystemsexactmicroscopicsolution

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a micro-to-macro bridge for density correlations in a one-dimensional gas of hard rods. For a specific initial ensemble, it computes the two-point correlation of the microscopic mass density exactly, using a mapping from rods to non-interacting point particles. It then coarse-grains that exact result over mesoscopic fluid cells and shows it agrees precisely with the correlation predicted by ballistic macroscopic fluctuation theory (BMFT), built from the Euler-scale solution of generalized hydrodynamics. If correct, this analytically validates the two core assumptions of hydrodynamic theory for hard rods: fluctuations of coarse-grained conserved densities are projected onto local equilibrium, and no independent dynamical noise emerges on the Euler scale. It also shows how long-range density correlations arise on the hydrodynamic scale from inhomogeneous initial conditions.

What carries the argument

The central object is the exact microscopic two-point correlation of the empirical mass density, obtained in Eqs. (19) and (23) as binomial order-statistics expressions. The key transformation is the hard-rod to hard-point mapping x_i = X_i − (i−1)a, which turns colliding rods into non-interacting point particles. The bridge to macroscopic scales is Eq. (8), which states that the coarse-grained correlation is exactly the fluid-cell average of the microscopic correlation. On the hydrodynamic side, BMFT supplies the path-integral saddle-point calculation whose free energy F[f] = ∫ f[g + ln f] encodes the same initial ensemble. The equality of the two sides is what carries the argument.

What would settle it

Simulate the same hydrodynamic setup starting from factorized non-overlapping hard rods (initial positions satisfying X_{i+1} ≥ X_i+a, velocities independent), coarse-grain the resulting micro-scale correlations over fluid cells, and compare the equal-time and two-time results with Eqs. (42) and (43); any deviation would show the validation is specific to the paper's correlated initial ensemble.

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Extended reading notes

Core claim

For an initial state in which point-particle positions are drawn independently from a smooth profile and velocities independently from h(v), then mapped to rods via X_i = x_i + (i−1)a, the authors obtain exact closed-form expressions for the connected two-point mass-density correlation, both equal-time and unequal-time. Coarse-graining these expressions through the fluid-cell average and taking the Euler scaling limit yields precisely the BMFT correlation formulas, including a delta-function local-equilibrium part and a long-range part built from derivatives of the Euler density. The agreement holds for both equal-time and two-time correlations and respects the scaling form C(ℓX,ℓt;ℓY,ℓt′) =

Load-bearing premise

The load-bearing premise is the initial ensemble: point-particle positions and velocities are sampled independently, and rods are then obtained by shifting the i-th point by (i−1)a; because this mapping already builds long-range correlations into the initial rod state, the exact formulas and the BMFT match are established only for this ensemble, not for a directly drawn non-overlapping rod state.

Editorial extensions

If this is right

Coarse-grained equal-time and two-time density correlations in hard rods are exactly the BMFT expressions, built only from the Euler solution and its derivatives.
Density fluctuations on the Euler scale thus evolve as deterministic transports of initial fluctuations, with no dynamically generated noise contributing to two-point correlations.
Long-range correlations in mass density emerge on the hydrodynamic scale for inhomogeneous initial conditions and vanish for homogeneous equilibrium, as BMFT predicts.
The scaling form C(ℓX,ℓt;ℓY,ℓt′) = (1/ℓ) C(X,t;Y,t′) is confirmed by data collapse across different particle numbers and fluid-cell sizes.
The exact finite-N microscopic expressions go beyond the Euler solution and match simulations, providing a controlled benchmark for hydrodynamic descriptions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A decisive open question is whether the agreement survives the alternative factorized non-overlapping rod ensemble, in which rods are drawn directly with no overlap. Since the paper's exact formulas rely on the point-particle i.i.d. ensemble, repeating the coarse-graining there would test the generality of BMFT rather than only one initial state.
The large-N Gaussian approximation for equal-time correlations suggests a way to probe higher-order effects: computing the third cumulant of the coarse-grained density microscopically could test whether the 'no dynamical noise' assumption holds beyond Gaussian order.
Extending the microscopic approach to the integrated current would allow a direct comparison with BMFT's prediction for current fluctuations, providing a stricter test of the hydrodynamic action than density correlations alone.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth reading if you work on hard rods or BMFT. For the i.i.d. point-particle ensemble mapped to rods, the authors obtain exact binomial order-statistics formulas for the microscopic density correlations (Eqs. 17, 19, 23), a finite-N Gaussian approximation (Eq. 30), and a non-equilibrium two-time BMFT expression (Eq. 43) that extends ref. [13]. The appendices are self-contained, the combinatorics checks out, and the simulations over 10^7 to 5×10^10 configurations agree visually. That is a serious piece of work. For anyone who wants benchmark formulas for finite-N hard-rod correlations, this paper is useful.

The soft spot is not the computation; it is the frame. The initial ensemble in Eq. (2) is i.i.d. point particles mapped to rods via X_i = x_i + a(i−1), and that shift already produces Euler-scale long-range correlations in the rod variables. The authors admit this in the last paragraph of Sec. 2. So the abstract's phrase "emergence of long-range correlations" is misleading, and the claim to "validate the underlying assumptions of hydrodynamic theory" is narrower than stated. What is actually shown is that, for this specially pre-correlated ensemble, coarse-graining the exact microscopic correlation reproduces the BMFT prediction derived from the same point-particle large-deviation functional. That is a consistency check between two calculations sharing the same ensemble, not an independent derivation of local relaxation from short-range initial data. The fact that the BMFT side is the same group's earlier work (ref. [13]) is not a flaw by itself, but it further reduces the weight of the agreement as external validation.

Two smaller issues. Eq. (42a) has a Dirac-delta term that does not look dimensionally consistent as written; that deserves a fix, not a scandal. And "agree precisely" appears without error bars or quantitative deviation metrics; given the simulation sizes, a simple L2 or relative-error plot would make the claim testable.

For the ensemble studied, I think the correspondence claim holds up. The exact finite-N formulas are new, the derivations are transparent, and the paper is honest about the initial-state correlations. The main fix is to revise the abstract and conclusion to say what was actually validated, add a quantitative comparison, and correct the typo. I would send this to a competent referee and expect a conditional accept.

Referee Report

3 major / 4 minor

Summary. The paper studies density correlations in a one-dimensional hard-rod gas at both microscopic and hydrodynamic scales. For a particular class of initial conditions—positions of point particles drawn i.i.d. from φ(x), velocities from h(v), ordered, and then mapped to rods via X_i = x_i + a(i−1)—the authors derive exact finite-N expressions for the two-time and equal-time microscopic density correlation (Eqs. (19) and (23)), and validate them against extensive numerical simulations. They then derive Euler-scale equal-time and space-time correlations from Ballistic Macroscopic Fluctuation Theory (Eqs. (42) and (43)) and compare them with the corresponding fluid-cell coarse-graining of the microscopic correlation via Eq. (8). The reported agreement is presented as a 'concrete validation of the underlying assumptions of hydrodynamic theory' and as evidence for the 'emergence of long-range correlations' on the Euler scale.

Significance. The exact combinatorial derivations in the main text and Appendices A–C are a genuine technical achievement: they provide closed-form finite-N correlation functions for hard rods that are verified by simulations with very large statistics, and the scaling collapse in Fig. 4 supports the Euler-scale scaling form (Eq. (40)). If the hydrodynamic claim were established for generic local-equilibrium initial states, this would be an important test of BMFT. However, as discussed below, the chosen initial ensemble already contains long-range correlations in the rod variables, and the BMFT calculation uses the same point-particle free-energy functional. The agreement therefore demonstrates internal consistency of two calculations based on the same pre-correlated ensemble more than it validates hydrodynamic projection or local relaxation for short-range initial states. This narrows the significance of the central claim considerably.

major comments (3)

[Sec. 2, final paragraph; Eqs. (1)–(2)] The initial rod ensemble is not a short-range local-equilibrium state. Because X_i = x_i + a(i−1), each rod position is coupled to the cumulative point-particle count over the entire interval, so the initial rod density already carries Euler-scale long-range correlations. The manuscript itself acknowledges this: 'In this case the hard-rod gas initially has only short-range correlation unlike our case where the gas already contains long-range correlation (produced by the transformation in Eq. (1)) to start with.' Since both the exact microscopic correlation (Eqs. (17), (19), (23)) and the BMFT saddle-point free energy (Appendix D, Eq. (D.5)) are built on the same i.i.d. point-particle ensemble, the micro-macro agreement may amount to a consistency check of two calculations based on the same pre-correlated ensemble, rather than a test of hydrodynamic projection / local relaxation for gener
[Abstract and Sec. 6] The abstract's phrase 'emergence of long-range correlations' and the conclusion's 'validating the assumptions of this theory in the context of one dimensional gas of hard rods' overstate what is demonstrated. Long-range correlations are present in the initial rod configuration by construction, so their presence at later times is not emergent from hydrodynamic evolution. To make the validation claim load-bearing, the authors would need either to (a) explicitly restrict all claims to the particular initial ensemble and show that BMFT reproduces the exact coarse-grained correlations for that ensemble, or (b) repeat the comparison for a factorized non-overlapping initial state with only short-range correlations (as used in Refs. [12,13]). As written, the central conclusion is narrower than the abstract claims.
[Sec. 5.1, Eq. (42a)] The BMFT equal-time correlation contains a singular term (1−ϱ)^2 ϱ δ(X_a−X_b), while the fluid-cell coarse-grained microscopic correlation (Eq. (8)) is obtained by integrating over cells of finite size ΔX and cannot produce a delta function. The comparison in Fig. 4 appears to be made for non-coincident points, which is reasonable, but the treatment of the singular part is not stated. Please clarify whether the singular term is meant to survive after coarse-graining (e.g., as a finite-cell contribution when X_a=X_b) or whether the comparison is only for the nonsingular long-range part. This is a presentational gap, but it matters for a precise statement of the micro-macro correspondence.

minor comments (4)

[Sec. 2, final paragraph] The sentence admitting the initial long-range correlations is important and should be placed prominently in the abstract or conclusion, not only in the initial-condition section, so that readers do not misinterpret 'emergence'.
[Throughout] Typographical errors: 'assumtions' in Sec. 1, 'compairing' in the Fig. 4 caption, 'symnbols' in the Fig. 3 caption, and 'equal-time correlation' in the Fig. 5 caption should be 'two-time' or 'space-time' correlation.
[Eq. (42c)] The formula for C_r would be easier to read with explicit parentheses; currently 'a(1−a¯ϱ(X_a,t))¯ϱ(X_a,t)(∂_{X_b}¯ϱ(X_b,t))' appears without a clear bracket structure.
[Appendix D, Eq. (D.9)] The notation g(z,u) is used both for −ln(φ(z)h(u)) and as the denominator weight; please use a different symbol (e.g., f_eq(z,u)) to avoid ambiguity in the saddle-point normalization.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the exact microscopic correlation is derived independently of BMFT, and the recorded initial long-range correlation is a stated scope limitation, not a circular step.

full rationale

The paper's central claim is that coarse-graining the exact microscopic hard-rod correlation reproduces the BMFT prediction. The microscopic formulas (Eqs. 17, 19, 23) are derived from exact binomial/order-statistics combinatorics of the stated initial point-particle ensemble and Jepsen mapping; they do not invoke BMFT or hydrodynamic projection. The BMFT side (Appendix D, Eqs. D.4-D.6) uses the large-deviation free energy Fp[f(0)] = ∫ f [g + ln f] with g determined by the same initial φ0 and h, together with the BMFT assumptions explicitly flagged in §1 and §5 as assumptions to be tested. Both calculations therefore share the same initial-condition input, but that is required for a controlled comparison and is not equivalent to fitting the output. The agreement is nontrivial because the exact microscopic computation includes the full finite-N combinatorial dynamics, while BMFT is a saddle-point/Euler-scale theory. Eq. (8) is an exact definitional relation between coarse-grained and microscopic correlations; using it to compare the two calculations is a method of verification, not a circular derivation. The paper itself notes in §2 that, because of the mapping X_i = x_i + a(i−1), the initial rod ensemble 'already contains long-range correlation...to start with.' This is an honest scope limitation: the claimed 'emergence' and the validation of hydrodynamic assumptions apply to this specially pre-correlated class of initial conditions, not to generic short-range local-equilibrium states. That caveat weakens the generality of the conclusion but does not make the derivation circular. No fitted parameter is relabeled as a prediction, and no load-bearing claim rests solely on a self-citation; refs. [12,13] supply the BMFT framework being tested, and ref. [20] supplies a previously derived mean-density formula that the present exact correlation calculation goes beyond.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the exact Jepsen mapping (standard), a specially chosen i.i.d. point-particle initial ensemble (ad hoc to this paper, and the source of pre-existing long-range correlations), the thermodynamic-limit Gaussian fluctuation structure, the BMFT axioms themselves (which the paper is testing), and saddle-point evaluation of the path integral. No parameters are fitted to data anywhere: inputs (N, a, σ, T, ℓ, ΔX/ℓ, profiles ϕ and h) are stated physical or simulation parameters, and ΔX is a probe scale constrained by a≪ΔX≪ℓ. The absence of fitted parameters is a genuine strength.

assumptions (5)

standard math Jepsen mapping: 1D hard-rod dynamics is exactly equivalent to labeled ballistic point particles under x_i = X_i − (i−1)a (Eq. 1), with label exchange reproducing collisions.
Foundation of all microscopic computations (Secs. 3-4, Appendices A-C); established result cited to [21-24].
ad hoc to paper Initial ensemble of Eqs. (1)-(2): i.i.d. point-particle positions and velocities, ordered, then mapped to rods; the rod gas therefore already contains long-range correlations at t=0.
Chosen for tractability — it turns all microscopic probabilities into binomial order-statistics products and fixes the BMFT free energy (D.5). Its long-range initial correlations limit the scope of the 'emergence' claim (admitted in §2).
domain assumption Thermodynamic/Euler limit: N→∞, system size→∞ with ¯ρ(x)=Nϕ(x) finite; then a→0 with Na/σ held fixed; Gaussian fluctuations of the empirical initial density (only terms up to O(k²) retained, Eq. C.3).
Yields the large-N formulas (12) and (30) and the BMFT saddle point; the Gaussian fluctuation structure is the same structure BMFT postulates through F_p (D.5).
domain assumption BMFT postulates: hydrodynamic projection and local relaxation — Euler-scale fluctuations are functionals of local conserved densities with no dynamically emergent noise (Eqs. 36-37).
These are the assumptions under test; Appendix D's saddle-point derivation requires them. The paper explicitly notes they are not rigorously derived (§2) and appeals to [26,27] for vanishing noise in integrable models.
standard math The BMFT path integral (Eqs. 38-41) is dominated by a single saddle point as ℓ→∞.
Standard large-deviation saddle-point evaluation used in Appendix D; subleading fluctuations are assumed negligible.

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Cite this review

Pith. "Pith review of Microscopic and hydrodynamic correlation in 1d hard rod gas." pith.science (2026). https://pith.science/paper/UYXBKUDV

@misc{pith2026260104951,
  author       = {Pith},
  title        = {Pith review of: Microscopic and hydrodynamic correlation in 1d hard rod gas},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UYXBKUDV}},
  note         = {Machine review of arXiv:2601.04951}
}

read the original abstract

We compute mass density correlations of a one-dimensional gas of hard rods at both microscopic and macroscopic scales. We provide exact analytical calculations of the microscopic correlation. For the correlation at macroscopic scale, we utilize Ballistic Macroscopic Fluctuation Theory (BMFT) to derive an explicit expression for the correlations of a coarse-grained mass density, which reveals the emergence of long-range correlations on the Euler space-time scale. By performing a systematic coarse-graining of our exact microscopic results, we establish a micro-macro correspondence and demonstrate that the resulting macroscopic correlations agree precisely with the predictions of BMFT. This analytical verification provides a concrete validation of the underlying assumptions of hydrodynamic theory in the context of hard rod gas.

Figures

Figures reproduced from arXiv: 2601.04951 by the authors.

**Figure 2.** The plot (a) shows the evolution of the two-time density correlation [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Plots comparing the mass density correlation [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Plots compairing the equal-time connected correlation function derived from [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: Comparison of the two-time connected correlation function obtained from [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [13]

A. Kundu. Ballistic macroscopic fluctuation theory of correlations in hard rod gas. Journal of Statistical Mechanics: Theory and Experiment, 2025(10):103203, Oct 2025

2025

[2] [1]

van Beijeren

H. van Beijeren. Exact results for anomalous transport in one-dimensional hamiltonian systems. Physical Review Letters, 108:180601, Apr 2012

2012

[3] [2]

H. Spohn. Nonlinear fluctuating hydrodynamics for anharmonic chains. Journal of Statistical Physics, 154:1191–1227, 2014

2014

[4] [3]

Narayan and S

O. Narayan and S. Ramaswamy. Anomalous heat conduction in one-dimensional momentum- conserving systems. Physical Review Letters, 89:200601, Oct 2002

2002

[5] [4]

B. Doyon. Lecture notes on Generalised Hydrodynamics. SciPost Physics Lecture Notes, page 18, 2020

2020

[6] [5]

L. D. Landau and E. M. Lifshitz. Fluid Mechanics, volume 6. Elsevier, 1987

1987

[7] [6]

J. M. O. De Zarate and J. V. Sengers. Hydrodynamic fluctuations in fluids and fluid mixtures. Elsevier, 2006

2006

[8] [7]

S. R. De Groot and P. Mazur. Non equilibrium thermodynamics. Courier Corporation, 2013

2013