Crystal to liquid cross-over for active particles with inverse-square power-law interaction

Abhishek Dhar; Anupam Kundu; Chandan Dasgupta; Gregory Schehr; Leo Touzo; Pierre Le Doussal; Prashant Singh; Saikat Santra; Suman Dutta

REVIEW 3 major objections 3 minor 61 references

Crystal to liquid cross-over for active particles with inverse-square power-law interaction

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Run-and-tumble particles with inverse-square repulsion melt from a crystal into a liquid when their activity reaches order one, and into a bell-shaped cloud when activity reaches order N, with exact large-N fluctuation formulas setting…

desk verdict Solid active-matter paper with clean Hessian calculations, but Eq. (34) is algebraically wrong and needs a fix before this is publishable as is. read the letter →

arxiv 2411.13478 v2 pith:6HRAZI4G submitted 2024-11-20 cond-mat.stat-mech cond-mat.soft

Saikat Santra , Leo Touzo , Chandan Dasgupta , Abhishek Dhar , Suman Dutta , Anupam Kundu , Pierre Le Doussal , Gregory Schehr

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Prashant Singh

This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.soft MSC 82C2260B2082C31 PACS 05.40.-a05.20.-y64.70.-n

keywords run-and-tumbleparticlesinverse-squareinteractionCalogero-MosermodelWignersemi-circleLindemannratioactivematterpositioncovariancedensitycrossover

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

This paper studies N run-and-tumble particles in one dimension, confined by a harmonic trap and repelling one another through an inverse-square power-law potential. It attempts to establish that the steady-state density crosses over through three regimes as activity grows: sharp crystalline peaks, a smooth Wigner semicircle, and a bell-shaped cloud. The analytical centrepiece is a closed-form expression for the steady-state covariance of particle positions in the weak-noise limit, from which the single-particle variance follows. Comparing fluctuations with the interparticle spacing and with the support size yields crossover thresholds at v0 ~ O(1) and v0 ~ O(N), which the paper verifies by simulation. If correct, this provides a rare exact large-N description of a genuinely nonequilibrium interacting active system.

What carries the argument

The load-bearing object is the Hessian matrix H_ISM of the inverse-square model, evaluated at the Hermite-polynomial ground state. The paper uses the identity H_ISM = H_L², where H_L is the Hessian of the logarithmic (Dyson) model, and diagonalizes H_L with Hermite-polynomial eigenvectors whose eigenvalues are the integers 1,…,N. This turns the covariance into a Chebyshev sum with a 1/k⁴ weight, which converges quickly enough for a large-N evaluation that is also valid for edge particles; the same structure yields Lindemann's ratio and the density-vs-Wigner distance measure χ used to locate the crossovers.

What would settle it

Integrate Eq. (2) numerically at small tumbling rate (e.g. γ = 0.01) and small activity (e.g. v0 = 0.1) for N = 128, extract the steady-state covariance matrix, and compare it element-by-element with the scaled formula v0²/N C(x_i/√(2N), x_j/√(2N)); a disagreement larger than statistical error for bulk pairs would refute the central claim.

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Extended reading notes

Core claim

The paper's central claim is that in the weak-noise, long-persistence limit the displacement covariance is ⟨δxiδxj⟩ = v0²/N C(w,z) with C(w,z) = c(arccos w, arccos z)/(√(1−w²)√(1−z²)), and the single-particle variance is ⟨δxi²⟩ = v0²/N V(w), V(w) = arccos² w (π − arccos w)²/[6(1−w²)], where w and z are equilibrium positions scaled by √(2N). These closed forms imply that fluctuations are of order v0/√N for bulk and edge particles alike, that the crystal-like multi-peaked density melts to a Wigner semicircle when v0 ~ O(1), and that the semicircle gives way to a bell-shaped profile when v0 ~ O(N). The same covariance yields the variance of interparticle gaps near the trap centre, and a finite-tumbling-rate extension, all corroborated by numerical simulations.

Load-bearing premise

The derivation assumes that in the γ → 0 limit every frozen realization of the run-and-tumble noise drives the linearized system to one unique fixed point, so the displacement is simply v0 times the inverse Hessian acting on the noise; if some realizations never relax, the variance formulas and the crossover thresholds do not follow.

Editorial extensions

If this is right

The density profile stays close to the Wigner semicircle across a broad intermediate range of activity, with deviations confined to the edges, and no extended 'wings' appear because the nearest-neighbour force at the edge is strong enough to balance the trap.
Position fluctuations decay as 1/N for both bulk and edge particles, so the small-displacement theory remains valid up to v0 ~ O(N), covering the entire crystal and liquid regimes.
The crystal-to-liquid crossover occurs at v0 ~ O(1), independent of N, while the liquid-to-bell crossover occurs at v0 ~ O(N).
Near the trap centre the variance of the gap between particles i and i+n grows as n² with coefficient π⁴/96 v0²/N³ in the long-persistence limit, and the finite-γ version matches simulations.
In the strongly active regime the density has a bell-shaped bulk and a 1/x³ tail that contains only O(1) particles, with strong sample-to-sample fluctuations.

Reading between the lines

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

Because the same covariance machinery works for the Dyson and inverse-square models with weights 1/k² and 1/k⁴ respectively, the method likely extends to harmonically confined Riesz gases with other power-law exponents; the edge behaviour and crossover thresholds should interpolate between these two cases.
The 1/x³ tail appears independent of the interaction potential provided particles repel infinitely on contact, which suggests a testable experimental signature: trapped active colloids should show a power-law edge decay whose exponent does not depend on the interaction law.
The Lindemann-ratio criterion fixed at 0.5 is a phenomenological choice; the analytic variance-to-gap ratio provides a parameter-free observable that could be used in experiments to map the melting line in the (v0, γ) plane.
If the weak-noise fixed-point assumption fails for some realizations at finite γ, the crossover thresholds would shift; a direct check would be to compare the simulated distribution of long-time displacements with the predicted form v0 H^{-1} σ for many frozen noise realizations.

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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

Punchline: this is a solid paper with one concrete fix needed. The printed closed-form covariance (Eq. 34) is algebraically wrong. It does not match the piecewise expression in Eq. (35) or direct summation; the final variance formula (Eq. 36) and the ordered covariance (Eq. 35) are correct, but Eq. (34) as written will send downstream users astray. Easy typo-level fix, but it sits in the center of the paper.

What's new: the active inverse-square model with run-and-tumble noise is studied for the first time, and the paper derives explicit 1/N covariance and gap-variance formulas that are not in the earlier active DBM papers. The Hessian/eigenfunction method from Refs. [30,49] is applied cleanly, and the numerics corroborate the analytic formulas for v0/N up to about 0.1 across the tumbling rates. I also like the edge analysis: no wings, uniform 1/N variance scaling at the edge, and the sharp contrast with the active DBM. The crossover scalings v0 = O(1) and v0 = O(N) are simple and convincingly supported.

Soft spots: first, the Eq. (34) error. I checked the algebra: the pi^2 uv term sits inside the pi/4 bracket, which cannot be right; the bracket needs (|u-v|^3 - (u+v)^3 + 4 pi uv) or alternatively the pi^2 uv term must move outside. For u=1.5, v=0.5, direct summation gives 0.4737 while Eq. (34) gives 0.2089. So the printed central formula is unreliable, even though Eq. (35) and Eq. (36) are correct. Second, the finite-gamma expressions are imported from the companion paper [51] without derivation; acceptable if that paper is solid, but the current paper should at least outline the derivation or clearly mark them as external. Third, the phase boundaries (Lindemann ratio 0.5, chi=0.1) are hand-picked; they work for the numerics shown but they're conventions rather than sharp transitions. That's fine for a crossover study, but the paper should say so plainly. The reader's worry about fixed-point convergence for frozen noise is not actually a weak point: H_ISM = H_L^2 is positive definite, so Eq. (19) follows immediately from the linearized dynamics; no assumption imported from the DBM case is needed.

Bottom line: this deserves a serious referee. The core derivations are standard, the numerics are honest, and the one real defect is a fixable algebraic typo. I'd recommend sending it to review with a request to correct Eq. (34) and to state more carefully what is imported from [51].

Referee Report

3 major / 3 minor

Summary. The paper studies a one-dimensional gas of N run-and-tumble particles in a harmonic trap with repulsive inverse-square interactions (the active inverse-square model, ISM). Numerically, the steady-state density profile shows three regimes as activity increases: a peaked crystal-like profile, a smooth Wigner semi-circle liquid-like profile, and a bell-shaped profile at very high activity. The authors quantify these regimes via Lindemann's ratio and a distance measure, and they construct a phase diagram in the (a,1/γ) plane, with v0 = aN. The analytical core is a small-noise (v0 → 0), long-persistence (γ → 0) computation of the position covariance using the Hessian of the ISM, which is related to the DBM Hessian through H_ISM = H_L^2. In the large-N limit the covariance is expressed in terms of a scaling function C(w,z) (Eqs. 31-34), the single-particle variance is given in closed form (Eq. 36), and the inter-particle gap variance is derived both for γ = 0 and for finite γ (Eqs. 47 and 52). The theoretical predictions are compared with numerical simulations, with good agreement reported for a ≲ 0.1.

Significance. If correct, the paper provides an exact large-N description of density crossovers in an interacting active system and identifies the crossover scalings v0 = O(1) (crystal-to-liquid) and v0 = O(N) (liquid-to-bell). It extends the method developed for the active Dyson Brownian motion to the inverse-square potential, exploiting the remarkable relation between the two Hessians. The derivation is largely transparent, the numerical verification is thorough in the stated regime, and the results are falsifiable predictions about variances and gap statistics. The paper should be of interest to the active-matter and exactly-solvable nonequilibrium communities. The main analytical result is, however, marred by an algebraic error in the printed closed form for the covariance (Eq. 34), which must be corrected before publication.

major comments (3)

[Sec. 4.2, Eq. (34)] The closed-form expression for c(u,v) is algebraically incorrect. As printed, c(u,v) = (1/6)[uv(u^2+v^2)/2 + (π/4)(|u−v|^3 − (u+v)^3 + π^2 uv)] does not agree with the piecewise form in Eq. (35) nor with the variance limit in Eq. (36). For example, at u=1.5, v=0.5, direct summation of sin(ku)sin(kv)/k^4 gives 0.4737, Eq. (35) gives 0.4737, but Eq. (34) gives 0.2089. On the diagonal, Eq. (34) does not reduce to Eq. (36), which is instead consistent with Eq. (35). The coefficient π^2uv inside the π/4 bracket should be 4πuv (equivalently, the π^2uv/6 term should appear outside the bracket). As written, a reader or downstream code using Eq. (34) will compute incorrect covariances, so this central formula must be corrected.
[Sec. 4, before Eq. (17), and Sec. 4.4] The linearization of the equation of motion is justified by the statement that the typical fluctuations δx_i are small compared with the equilibrium gap, which scales as 1/√N. In the liquid regime where the formula is applied, v0 = O(1), the typical displacement is v0/√N, i.e., of order the gap, so this stated premise fails. The a posteriori validity check in Eq. (53) instead uses the gap fluctuation g_{i,1} ~ v0/N^{3/2}, which is indeed the correct small parameter because the interaction force depends only on differences δx_i − δx_j. The paper should restate the linearization condition consistently in terms of gap fluctuations and explain why the individual-position variance formula (36) is nevertheless valid; as written, the derivation's explicit assumption is false in the regime of interest, even though the numerical results support the final formula.
[Sec. 4.2, Eq. (32)] The upper limit of the sum defining C(w,z) is written as N, but the subsequent closed-form evaluation and the finite-γ expression in Eq. (42) use an infinite sum. Since the approximation leading to Eq. (32) is valid only for k ≪ N, and the 1/k^4 tail is negligible, the upper limit should be ∞ (or the text should state that the sum is extended to infinity with negligible error). This is a typographical inconsistency in a displayed equation that could mislead a reader implementing the formula.

minor comments (3)

[Sec. 4.2, Eqs. (41)-(43)] The finite-γ results are imported from the companion paper [51] without derivation. A short derivation or a more explicit statement of the assumptions used in that derivation would improve self-containedness, since these formulas are used for quantitative comparison with simulations.
[Sec. 3.4, Fig. 8 caption] The caption notes that the first transition occurs at a values that depend on N, which is correct because a = v0/N and the crossover is at v0 = O(1). A one-sentence reminder that a = v0/N would help avoid apparent tension with the statement that the crossover is independent of N in terms of v0.
[Sec. 4.3, Eq. (46)] The expansion D(w,z) = (π^2/24)(w−z)^2 + O(w^3,z^3) would benefit from a brief derivation or at least a statement that the linear terms cancel by symmetry, since the reader cannot easily verify the coefficient from the preceding expressions as written.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: covariance and variance are derived parameter-free from linear response and checked against independent simulations; overlapping-author citations supply supporting lemmas, not forced conclusions.

full rationale

Walking the derivation chain, there is no step where a claimed prediction reduces to an input by construction. The steady-state covariance (Eq. 21) follows directly from the linearized equation (17): since H_ISM = H_L^2 has positive eigenvalues 1,...,N (Eqs. 18-24), the frozen-noise fixed point (Eq. 19) is guaranteed by linear algebra, so the citation to Ref. [30] for relaxation is not load-bearing. Averaging over ±1 noises gives Eq. (21) with no parameters. The large-N evaluation (Eqs. 28-30) uses standard Chebyshev asymptotics of Hermite zeros as a mathematical approximation, and the resulting formulas (31)-(36) are parameter-free; they are compared with independent numerical simulations (Figs. 9-10), not fitted. The crossover statements (39)-(40) are scaling comparisons of the derived variance with geometric length scales, not redefinitions. Refs. [30,49,51] with overlapping authorship are cited for method/lemmas: Ref. [49] for the Hessian identity, Ref. [30] for the linear-response method, Ref. [51] for the finite-gamma extension, all externally checkable published results that do not assume the present conclusion. The algebraic misprint in Eq. (34) relative to Eq. (35) is a correctness concern, not a circularity. Therefore no circular step is present.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No invented entities. The only hand-chosen numbers are the two phase-diagram thresholds. The analytic results use established Calogero and log-gas identities plus one unproven dynamical fixed-point assumption inherited from the active DBM analysis; the latter is the main liability in the ledger.

free parameters (2)

Lindemann ratio threshold eta_c = 0.5
Hand-chosen reference line in Sec. 3.4: the system is called liquid when eta_i at the trap center exceeds 0.5. This is a classification convention, not fitted to data, and it does not enter the covariance formulas.
Distance-measure threshold chi_c = 0.1
Hand-chosen criterion in Sec. 3.4 and Appendix B: the strongly active regime is defined by chi > 0.1, described as a 10% deviation from the Wigner semicircle. Again a convention used to draw the phase diagram.

assumptions (4)

standard math At v0=0, equilibrium positions are the zeros of the Hermite polynomial HN, and the inverse-square Hessian factorizes as H_ISM = H_L^2, where H_L is the log-gas Hessian.
Used at the start of Sec. 4. Established in Ref. [49] and diagonalized in Ref. [53]; the paper relies on these identities without reproving them.
domain assumption In the gamma -> 0 limit, for each fixed realization of the dichotomous noises sigma, the linearized dynamics converges to a unique fixed point so that delta x = v0 H_ISM^{-1} sigma.
Stated in Sec. 4 before Eq. (19) as 'Anticipating this to be true for the active ISM also'. This is carry-over from active DBM numerics in Ref. [30] and is not proven for the inverse-square model.
standard math For large N and small mode index k, the eigenvectors of H_L are accurately given by Chebyshev polynomials (Eq. 28), and the normalization sum in Eq. (30) equals N; high-k modes are negligible because the spectral sum decays as 1/k^4.
Asymptotic approximations taken from Ref. [30] and used to derive the scaling functions C(w,z) and V(w) in Eqs. (31)-(36). No rigorous error bound is provided.
domain assumption The noise variables sigma_i are independent symmetric dichotomous processes with correlation exp(-2 gamma |t-t'|), and the particles stay ordered due to infinite repulsion.
Model definition in Sec. 2, Eq. (2); the ordering is used in the definition of gaps and in the linearization around an ordered equilibrium.

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Pith. "Pith review of Crystal to liquid cross-over for active particles with inverse-square power-law interaction." pith.science (2026). https://pith.science/paper/6HRAZI4G

@misc{pith2026241113478,
  author       = {Pith},
  title        = {Pith review of: Crystal to liquid cross-over for active particles with inverse-square power-law interaction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HRAZI4G}},
  note         = {Machine review of arXiv:2411.13478}
}

abstract

We consider a one-dimensional system comprising of $N$ run-and-tumble particles confined in a harmonic trap interacting via a repulsive inverse-square power-law interaction. We numerically compute the global density profile in the steady state which shows interesting crossovers between three different regimes: as the activity increases, we observe a change from a density with sharp peaks characteristic of a crystal region to a smooth bell-shaped density profile, passing through the intermediate stage of a smooth Wigner semi-circle characteristic of a liquid phase. We also investigate analytically the crossover between the crystal and the liquid regions by computing the covariance of the positions of these particles in the steady state in the weak noise limit. It is achieved by using the method introduced in Touzo {\it et al.} [Phys. Rev. E {\bf 109}, 014136 (2024)] to study the active Dyson Brownian motion. Our analytical results are corroborated by thorough numerical simulations.

Figures

Figures reproduced from arXiv: 2411.13478 by the authors.

**Figure 1.** The total (scaled) density profiles ˜ρ(z) for different (small) system sizes with several values of the activity, v0 = aN with a = 0.01 in (a), 0.1 in (b) and 1.0 in (c) show the crossover: from sharply peaked density profile in (a) at very small activity to bell shaped density profile in (c) at very large activity via an intermediary WSc profile in (b). For all the panels, the tumbling rate γ is kept fixed at 1. 3.… view at source ↗

**Figure 2.** The transition in the (scaled) density profile ˜ρ(z) is shown once again with the same set of parameters as in [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** Showing the total support of the density profile ρ(x) approximates to [−aN, aN] in the large activity limit. The steady-state density profiles are drawn as a function of x/N for three different system sizes which seem to converge and show a sharp drop at x = aN. Near the edges, the density profiles (dashed lines) decay as ∼ 1/x3 (red solid line). increasing v0 or equivalently by decreasing the interaction strength g… view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: Plot of the density ¯ρ(z) vs z showing the melting of the ‘crystal’ to a ‘liquid’ in a system of 64 particles because of reducing the tumbling rate γ keeping the activity parameter a fixed at a = 0.025. For the system size N = 128 with a = 0.025, the system has already…

**Figure 5.** Figure 5: Demonstration of the emergence of wing-like structures in the density profile within the strongly active region (a = 0.25) as a result of the reduced tumbling rate, γ. 3.4. Discussion of crossovers Thus, based on values of the parameters v0 and γ, the steady-state dens…

**Figure 6.** Figure 6: Numerically computed Lindemann’s ratio ηi (points) as a function of i/N for different values of the tumbling rate γ within a system comprising N = 64 particles. Three panels correspond to three different values of the speed v0, 0.32, 0.64 and 1.6 in (a),(b) and (c) res…

**Figure 7.** Figure 7: Lindemann’s ratio ηi as a function of i/N for different values of the tumbling rate γ is shown in [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Lower diagram depicts three distinct regions in the active ISM within the parameter space represented by a and 1/γ, with red, blue, and black points representing weakly active, intermediate, and strongly active regimes, respectively. Red points signify scenarios where …

**Figure 9.** Figure 9: Numerically computed variance of particle positions s 2 i ≡ ⟨x 2 i ⟩ − ⟨xi⟩ 2 (points) in the steady-state of active ISM for different values of the tumbling rate γ keeping the activity parameter a fixed at 0.05 in (a), 0.1 in (b) and 0.5 in (c) for the system size N =…

**Figure 10.** Figure 10: Variance of the gap g 2 n as defined in Eq. (47), is plotted as a function of n for two different values of the activity parameters, a = 0.01, 0.05, and 0.1, in panels (a), (b) and (c) respectively, within a system of 128 particles for three different choices of the t…

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Pith tools

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