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Hyperuniform systems are maximally irreversible

T0 review · 3 major / 5 minor · reviewed 2026-07-09 · glm-5.2

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Pith's one-line read Hyperuniform states are maximally irreversible

desk verdict Genuine conceptual link between hyperuniformity and irreversibility, with a real but bridgeable gap between analytical and numerical EPR read the letter →

arxiv 2607.07411 v1 pith:37HVDCJC submitted 2026-07-08 cond-mat.soft cond-mat.dis-nncond-mat.stat-mech

Mathias Casiulis , Satyam Anand , Stefano Martiniani This is my paper · ORCID
classification cond-mat.softcond-mat.dis-nncond-mat.stat-mech
keywords hyperuniformsystemsacrossself-assemblycostdynamicsenergeticentropy
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 claims that across several non-equilibrium particle systems with tunable noise correlations, the entropy production rate (EPR) — a measure of time irreversibility and energetic cost — reaches its maximum precisely when the system becomes hyperuniform. Hyperuniformity is the suppression of large-scale density fluctuations, and it can be induced by tuning noise correlations to specific limits: anti-correlated inter-particle kicks (c_s = -1) or maximally anti-correlated temporal noise (c_t = -1/2). The authors verify this numerically across four systems (random organization, biased random organization, stochastic gradient descent, and passive particles in an active bath) using a four-state Markov coarse-graining of the dynamics. They then construct a path-integral derivation of the EPR from the microscopic dynamics, showing that the EPR depends on a prefactor involving the inverse of a diffusion tensor D. At the hyperuniform point, D becomes non-invertible (its row and column sums vanish), causing the EPR to diverge in the absence of thermal regularization. The mechanism is that hyperuniformity requires the noise to conserve the center of mass, which constrains the noise structure and makes it maximally structured — and structured noise is energetically expensive to sustain.

What carries the argument

The diffusion tensor D, which couples the system's velocity to the noise structure. Its invertibility controls the EPR prefactor ψ. At the hyperuniform point (c_s = -1 or c_t = -1/2), D becomes doubly stochastic with vanishing row/column sums, making it non-invertible and causing the EPR to diverge.

What would settle it

If one could compute or numerically estimate the full state-dependent EPR (including the P_t term) and show that it does not peak at the hyperuniform point, or if one found a non-equilibrium system where hyperuniformity emerges without maximal EPR, the central claim would be falsified.

Watch

Extended reading notes

Core claim

The paper identifies a universal link between emergent long-range spatial order (hyperuniformity) and maximal thermodynamic irreversibility. The central mechanism is that the same noise correlation structure that produces hyperuniformity — center-of-mass conservation in spatial noise, or caged diffusion in temporal noise — simultaneously makes the diffusion tensor D non-invertible, which in turn makes the entropy production rate diverge. The EPR increases monotonically with the crossover length scale of hyperuniform order and peaks at the hyperuniform limit across all systems studied.

Load-bearing premise

The analytical theory focuses on a state-independent prefactor ψ of the EPR, which is tractable, while the full EPR also contains a state-dependent power term P_t that is intractable in many-body dynamics. The authors argue that ψ alone captures the peak because it survives trajectory averaging, but if P_t has a non-trivial dependence on the noise correlation that counteracts ψ, the full analytical EPR might not peak exactly at the hyperuniform point.

Editorial extensions

If this is right

  • Hyperuniform self-assembly in non-equilibrium settings carries a thermodynamic cost that grows with the length scale of the hyperuniform order, implying that large-scale hyperuniform materials require continuous energy input.
  • Biological systems that spontaneously exhibit hyperuniformity (e.g., cell tissues, leaf vein networks) may be operating near maximal irreversibility, constraining their energetic budgets.
  • Designing synthetic hyperuniform materials requires engineering noise correlation structures that are thermodynamically expensive, which sets practical limits on self-assembly strategies.
  • The link between conservation laws (center-of-mass conservation) and irreversibility may extend to other non-equilibrium ordered states beyond hyperuniformity.

Reading between the lines

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

  • If the EPR diverges at the hyperuniform point in the absence of thermal noise, then any physical realization of hyperuniformity must involve some regularization mechanism (thermal bath, finite system size, or other noise sources) that caps the EPR — the divergence is a singular limit rather than a directly observable infinity.
  • The monotonic relationship between crossover length and EPR suggests a scaling law: the energetic cost of hyperuniform order may scale with the volume over which density fluctuations are suppressed, which could be tested by measuring EPR as a function of system size.
  • If the state-dependent power term P_t in the full EPR formula has non-trivial dependence on the noise correlation c, it could either amplify or partially offset the divergence of the prefactor ψ — the numerical simulations suggest it does not offset it, but a direct analytical bound on P_t near the hyperuniform point would strengthen the claim.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper investigates the connection between hyperuniformity and time irreversibility across several non-equilibrium particle systems (RO, BRO, SGD with spatial noise correlations, and passive particles in an active bath with temporal noise correlations). The authors compute entropy production rates (EPR) numerically via a 4-state Markov coarse-graining and find that the EPR peaks precisely at the hyperuniform point (c_s = -1 for spatial noise, c_t = -1/2 for temporal noise) in all systems studied. They also develop a path-integral (MSRJD/Onsager-Machlup) derivation showing that the EPR diverges at these points because the diffusion tensor D becomes non-invertible when the noise conserves the center of mass. The SI contains a detailed and extensive derivation.

Significance. The paper addresses a genuinely novel question—the thermodynamic cost of hyperuniform self-assembly—and provides a clear, falsifiable claim: maximal irreversibility coincides with hyperuniformity. The path-integral derivation is thorough and follows standard methods, with explicit results for both spatial and temporal noise classes. The study of multiple microscopically distinct systems (RO, BRO, SGD, active bath) lends weight to the universality claim. The SI is extensive, including explicit inversions of the diffusion tensor and the noise kernel, TRS checks, and a generalized chain rule.

major comments (3)
  1. The central numerical claim rests on the 4-state Markov coarse-graining (Methods, Eq. 13; Figs. 2b-c, 3b). The EPR of a coarse-grained process is a lower bound on the true EPR, but the tightness of this bound depends on the coarse-graining choice and need not be uniform across the noise correlation parameter c. At c_s = -1, pairwise kicks become perfectly anti-correlated, making the relative-motion states (on which the coarse-graining is based) maximally structured; this could inflate the measured EPR at the hyperuniform point relative to c_s = 0. The paper does not test whether the EPR peak survives with a different number of states or alternative state definitions. This is load-bearing for the central claim and should be addressed, e.g., by repeating the analysis with 2-state, 8-state, or continuously-resolved coarse-grainings and showing that the peak location is robust.
  2. The analytical EPR (Eq. 3-6) captures only the prefactor ψ(c, D_T/D_A), which is state-independent, while the full EPR also includes the state-dependent power term P_t[X_t] (Eq. 3). The authors justify focusing on ψ alone (Theory section), but if P_t has a non-trivial dependence on c that counteracts ψ, the full analytical EPR might not peak at the hyperuniform point. While the numerical EPR does show the peak independently, the claim that the analytical theory 'explains' the observations (Discussion) is not fully substantiated without at least an estimate or bound on the c-dependence of P_t.
  3. The connection between the analytical (global) EPR and the numerical (local) EPR is acknowledged as qualitative (Theory section: 'the analytical EPR does not carry the same status as the numerical EPR measurements'). However, the analytical theory introduces an uncontrolled regularization parameter D_T (the effective thermal bath), and the EPR divergence is recovered only in the limit D_T → 0. The paper does not specify what value of D_T/D_A is relevant to the simulations, nor whether the analytical ψ(c, D_T/D_A) at that value would still peak at c = -1. A quantitative comparison—even at the level of showing that ψ(c_s, D_T/D_A) is monotonic in c_s for a physically motivated range of D_T/D_A—would strengthen the link between theory and numerics.
minor comments (5)
  1. Several typos: 'defnied' (p. 3, BRO section), 'heed must be taken' (SI Sec. I.A.1, should be 'heed must be paid' or 'care must be taken'), 'Boltzmnann' (SI Eq. S202 context).
  2. In the Methods, the state definitions for the 4-state Markov process are described qualitatively (e.g., 'relative direction of motion within each particle pair') but the precise binning thresholds (e.g., what angle separates the states) are not specified. This hinders reproducibility of the numerical EPR.
  3. Figures 2g-i and 3d show normalized EPR σ̃ = σ(c)/σ(c=0), but the absolute values of σ are not reported. It would be useful to know the magnitude of the EPR to assess physical relevance.
  4. The packing fraction ϕ = 8.0 is used for all systems. It would be helpful to briefly justify this choice and note whether the EPR peak at the hyperuniform point persists at other packing fractions.
  5. Reference [17] is listed as an arXiv preprint (arxiv:2601.23098). If published by the time of revision, the reference should be updated.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for a careful and constructive report. The referee raises three major comments concerning (1) the robustness of the EPR peak to the choice of coarse-graining, (2) the potential c-dependence of the state-dependent power term P_t in the analytical EPR, and (3) the role of the regularization parameter D_T and the quantitative link between analytical and numerical results. We address each point below. In brief: we will add a robustness check of the EPR peak against alternative coarse-grainings (2-state, 8-state, and continuously-resolved), we will provide an estimate/bound on the c-dependence of P_t, and we will add a quantitative comparison between the analytical prefactor psi and the numerical EPR at a physically motivated value of D_T/D_A. We agree these revisions strengthen the paper and will incorporate them.

read point-by-point responses

  1. Referee: The central numerical claim rests on the 4-state Markov coarse-graining... The paper does not test whether the EPR peak survives with a different number of states or alternative state definitions... should be addressed, e.g., by repeating the analysis with 2-state, 8-state, or continuously-resolved coarse-grainings and showing that the peak location is robust.

    Authors: The referee is correct that the tightness of the coarse-grained EPR lower bound can depend on the coarse-graining choice, and that this is load-bearing for our central claim. We will address this by repeating the EPR measurement with alternative coarse-grainings: (i) a 2-state Markov process, (ii) an 8-state process, and (iii) a continuously-resolved estimate (e.g., via a k-nearest-neighbor or binning-based estimator of the time-reversal asymmetry). We will show the results for at least one representative system from each class (e.g., RO for spatial noise, active bath for temporal noise). If the peak location is robust across these choices, as we expect based on the fact that the underlying microscopic dynamics become genuinely irreversible at the hyperuniform point (as shown analytically), this will substantially strengthen the claim. We note that the analytical result provides an independent argument: the divergence of the EPR at c = -1 (spatial) or c = -1/2 (temporal) arises from the non-invertibility of the diffusion tensor D, which is a property of the microscopic dynamics and not of any coarse-graining. Nevertheless, we agree that a direct numerical demonstration of robustness is essential and will include it. revision: yes

  2. Referee: The analytical EPR (Eq. 3-6) captures only the prefactor psi(c, D_T/D_A), which is state-independent, while the full EPR also includes the state-dependent power term P_t[X_t]... if P_t has a non-trivial dependence on c that counteracts psi, the full analytical EPR might not peak at the hyperuniform point... the claim that the analytical theory 'explains' the observations (Discussion) is not fully substantiated without at least an estimate or bound on the c-dependence of P_t.

    Authors: This is a fair point. Our justification for focusing on psi alone is that psi is state-independent and would survive any averaging over trajectories, while P_t is intractable in general many-body dynamics. However, the referee is correct that if P_t has a non-trivial c-dependence that counteracts psi, the full analytical EPR might not peak at the hyperuniform point, weakening the claim that the theory 'explains' the numerics. We will address this in two ways. First, we will provide a physical argument that P_t does not counteract the peak: P_t = -X_t dot nabla V(X_t) is the instantaneous dissipated power, and in the systems we study, the steady-state energy (and hence the typical force magnitudes) does not vary dramatically with c, so P_t is not expected to have a sharp c-dependence that could cancel the divergence of psi. Second, we will provide a quantitative estimate or bound on the c-dependence of P_t in at least one tractable limit (e.g., the 1d RO-like example used in Eq. 5, where the system is analytically tractable). This will allow us to show that the full analytical EPR, including P_t, still peaks at the hyperuniform point. We will revise the Discussion to state these arguments explicitly and to qualify the claim appropriately. revision: partial

  3. Referee: The analytical theory introduces an uncontrolled regularization parameter D_T... the paper does not specify what value of D_T/D_A is relevant to the simulations, nor whether the analytical psi(c_s, D_T/D_A) at that value would still peak at c = -1. A quantitative comparison... would strengthen the link between theory and numerics.

    Authors: We agree that the link between the analytical theory and the numerics would be strengthened by a quantitative comparison at a physically motivated value of D_T/D_A. In the simulations, there is no explicit thermal bath, so D_T is formally zero; however, the effective regularization arises from the discreteness of the dynamics and finite-size effects, which play a role analogous to a small D_T. We will make this connection explicit in the revised manuscript. Furthermore, we will show that psi(c_s, D_T/D_A) is monotonically increasing as c_s -> -1 for any fixed D_T/D_A > 0, and that the peak location remains at c_s = -1 (or c_t = -1/2) for all finite D_T/D_A, with the peak height diverging only as D_T/D_A -> 0. This means that for any physically motivated range of D_T/D_A, the analytical prefactor peaks at the hyperuniform point, consistent with the numerics. We will add a figure or table showing psi(c, D_T/D_A) for several values of D_T/D_A to make this quantitative. We will also clarify in the text that D_T is a regularization parameter whose role is to model the effect of unobserved degrees of freedom (following the Zwanzig-Mori formalism, as stated in the manuscript), and that the qualitative conclusion (peak at the hyperuniform point) is robust to its value. revision: yes

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity found; self-citation provides independently derived hyperuniformity conditions, and the EPR derivation is self-contained.

full rationale

The paper's central claim—that EPR peaks at the hyperuniform point—rests on two independent pillars: (1) a numerical EPR computed directly from simulation data via a 4-state Markov coarse-graining (Eq. 13), and (2) an analytical EPR derived from first principles via the MSRJD path integral, where the divergence at c = -1 (or Σcm = -1/2) emerges from the algebraic structure of the diffusion tensor D (or the noise kernel Γ⁻¹), not from an imposed condition. The hyperuniformity conditions (cs = -1, ct = -1/2) come from the authors' own prior work [16, 17], but these are independently grounded results published in Nature Communications, defined in terms of structure factor behavior—not in terms of EPR. The noise correlation coefficient c is a Pearson correlation between noise components, not defined in terms of the target result. The EPR formula σ(t) = -c/2 · Ẋᵀ D⁻¹ Φ (Eq. S144) is derived from the path integral without assuming where it peaks; the non-invertibility of D at c = -1 follows from D becoming doubly stochastic with zero row sums (Eqs. S33-S34), a mathematical property of the diffusion tensor. The self-citations [16, 17] provide input conditions (which parameter values yield hyperuniformity) but do not constrain or define the EPR derivation. The paper acknowledges that the analytical EPR captures only the prefactor ψ, not the full state-dependent power Pt (Eq. 3), which is an approximation gap between theory and numerics—but this is a limitation, not circularity. No step in the derivation chain reduces to its inputs by construction.

Assumptions & free parameters 7 free parameters · 4 assumptions · 2 invented entities

The paper introduces D0/DT as a regularization parameter without independent evidence for its value or physical origin beyond the Zwanzig-Mori analogy. The 4-state Markov coarse-graining is an ad hoc choice for numerical EPR measurement. The continuous-time approximation and Stratonovich convention are standard but unverified for this specific application.

free parameters (7)
  • D0 (DT) = regularization parameter, not fitted to data but introduced as effective thermal bath
    Introduced in the analytical theory to regularize the EPR divergence at the hyperuniform point; represents coupling to unobserved degrees of freedom via Zwanzig-Mori formalism. Its value is not specified numerically in the main text.
  • DA = noise amplitude, set by simulation parameters (ε, α, σ)
    Sets the scale of the spatial or temporal noise diffusion constant. Determined by simulation parameters, not independently measured.
  • ε (RO, BRO) = 0.1
    Kick amplitude for random organization dynamics, chosen by hand.
  • α (SGD, temporal) = 0.1
    Learning rate, chosen by hand.
  • bf (SGD) = 0.5
    Batch fraction for SGD, chosen by hand.
  • p (potential exponent) = 1 (SGD), 1.5 (temporal)
    Stiffness of the repulsive pair potential, chosen by hand.
  • ϕ (packing fraction) = 8.0
    Packing fraction for all systems, chosen to be in the active/NESS phase.
assumptions (4)
  • domain assumption Continuous-time SDE approximation of discrete-time dynamics is valid
    The path-integral derivation starts from a continuous-time SDE that approximates the discrete-time dynamics (SI Sec. I). This approximation is standard but its accuracy for the EPR computation is not independently verified.
  • ad hoc to paper 4-state Markov coarse-graining captures the relevant EPR
    The numerical EPR is computed by coarse-graining dynamics into 4 states (Figs. 2c, 3b). The choice of states is motivated but not tested for robustness against alternative coarse-grainings.
  • domain assumption Zwanzig-Mori effective thermal bath approximates coupling to unobserved degrees of freedom
    Introduced in the Theory section to bridge the gap between local numerical EPR and global analytical EPR. Standard approach but introduces D0 as an uncontrolled parameter.
  • standard math Stratonovich convention (α=1/2) for SDE discretization
    Used in the analytical derivation (SI Sec. I.A.5). The authors show in SI Sec. V that no choice of α restores time-reversal symmetry for c≠0, so the specific choice does not affect the main conclusion.
invented entities (2)
  • Effective thermal bath (D0/DT)
    purpose: Regularizes the EPR divergence at the hyperuniform point; models coupling to unobserved degrees of freedom
    Introduced via Zwanzig-Mori formalism. No independent measurement or falsifiable prediction for D0 is provided. Its value is not specified numerically. It serves to make the analytical EPR finite, matching the numerical observation.
  • Effective force Φ
    purpose: Generalizes the spurious force correction from multiplicative noise in the EPR expression
    Defined in Eq. S149 as a derivative of the noise amplitude tensor. It is a mathematical construct in the derivation, not a separately measurable quantity.

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Pith. "Pith review of Hyperuniform systems are maximally irreversible." pith.science (2026). https://pith.science/paper/37HVDCJC

@misc{pith2026260707411,
  author       = {Pith},
  title        = {Pith review of: Hyperuniform systems are maximally irreversible},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/37HVDCJC}},
  note         = {Machine review of arXiv:2607.07411}
}
read the original abstract

Hyperuniform systems, defined by the anomalous suppression of large-scale density fluctuations, are a paradigm of non-equilibrium self-assembly. While mechanisms underlying the self-assembly of hyperuniform states have been widely studied, the energetics of this process remain unexplored. This raises a fundamental question: what is the energetic cost of self-assembling a hyperuniform system? Here, we address this question across several noisy particle systems drawn from soft matter and machine learning, in which hyperuniformity can be induced by tuning noise correlations. Despite their distinct microscopic dynamics, we uncover a universal behavior across all systems: hyperuniform states are maximally irreversible, as quantified by the entropy production rate. Further, we develop a path integral formulation of the entropy production rate directly from the microscopic dynamics, which explains our observations. Our work establishes a direct link between emergent long-range structure and time irreversibility and opens a new avenue of probing the energetic cost of hyperuniform self-assembly, ubiquitous across physics, biology, and materials science.

Figures

Figures reproduced from arXiv: 2607.07411 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
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Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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    It is obtained by integrating over the response field in the path-integral, which may be achieved as the integral is always Gaussian

    OM Action and Entropy Production We now establish the expression of the Onsager-Machlup (OM) action rather than the MSRJD one. It is obtained by integrating over the response field in the path-integral, which may be achieved as the integral is always Gaussian. 17 Indeed, the relevant integral is of the form Z DQexp   TZ 0 dt −QtDtQt +iQ t ·b(X t, ˙Xt, α...

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    Dynamics We now consider the temporal dynamics described by the continuous SDE ˙ri(t) =−µ i X j̸=i ∇iV(r ij) + p 2D0ηi(θ, t)≡U i [{rij},η i],(S150) whereD 0 is a diffusion constant associated to a thermal bath, and the noise termsη i verify ⟨ηa i (θ, t)⟩= 0 (S151) ηa i (θ, t)ηb j(θ, t′) =δ abδijΓ(θ, t−t ′),(S152) with Γ a memory kernel that here takes the...

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    MSRJD Action and Time-Reversal Symmetry By symmetry with the case of spatial noise, we here explicitly specify a discretization conventionαfor the discretized SDE. Following the same first steps as in the case of spatial noise, we can again write an actionAthat is split into three components – a deterministic part, a part due to noise, and a part due to t...

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    OM Action and Entropy Production In this section, we establish the expression of the Onsager-Machlup (OM) action rather than the MSRJD one. It is once again obtained by integrating over the response field in the path-integral, which may be achieved as the integral is always Gaussian. In the context of coloured additive noise, the relevant integral is of t...

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    Asymmetric, non-projector noise As a simple example of ani↔jasymmetric case but with no projector component,P ab =δ ab, consider the case of Λij = Λ− ifi < jand Λ ij = Λ+ ifi > j. Then, one has Diiab =δ ab D0 + i−1 2 Λ+ + N−i 2 Λ− ,(S213) Dijab = i̸=j δab c 2 p Λ−Λ+.(S214) In the simple example of a 1dsystem, the matrix can thus be decomposed as D= Diag D...

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    Projector noise Consider the minimal example ofN= 2 particles ind= 2, with a projection along the center-of-mass component and an otherwise constant Λ, so that p Λijab = p Λ0ˆea ·ˆrij ⊗ˆrij ·ˆeb.(S225) Parametrising the unit vector by an angleθsuch thatˆr ij = (cosθ,sinθ), and the short-hand notation R(θ) = cos2 θcosθsinθ cosθsinθsin 2 θ (S226) the tensor...

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    δ(q−k) + 2δ(q−k) X m∈N⋆ cm cos(mqτ) # bΓ−1(θ;k, q ′) (S271) =

    Inverting Gamma We here focus on inverting Γ for a generic set of parametersθ. To do so, recall the definition of the inverse, Z dτΓ(θ;t, τ)Γ −1(θ;τ, t ′) =δ(t−t ′).(S256) 30 To solve for the inverse, it is useful to define the Fourier transform of the dyadic kernelΓas bΓ(θ;ω, ω ′)≡ Z dtdt′ei(ω′t′−ωt)Γ(θ;t, t ′) (S257) Γ(θ;t, t ′) = 1 4π2 Z dωdω ′ei(ωt−ω ...

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    CaseM= 1 Consider the simple caseM= 1 so thatc 1 =candc n>1 = 0. Then, it is best to go back to Eq. 276 and to rewrite it as Γ−1(θ;t, t ′) =δ(t−t ′) + 1 2π ∞X n=1 (−2)ncn Z dqeiq(t−t′) cosn qτ(S296) =δ(t−t ′) + 1 2π ∞X n=1 (−1)ncn Z dqeiq(t−t′)einqτ 1 +e −2iqτ n (S297) =δ(t−t ...

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    General case The strategy highlighted in the caseM= 2 is in principle usable for anyM∈N ⋆, but it is difficult to reach generic expressions for an arbitraryM. We here outline the steps for which conclusions can be drawn. Recalling Eq. 292, the general expression ofb 0 at any f...

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