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Entropy from decoherence: a case study using glasma-based occupation numbers

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Pith's one-line read The entropy per particle produced by decoherence of the glasma coherent state falls below the two-dimensional thermal gluon-gas value, except for proton-nucleus collisions at small coupling.

desk verdict A clean application of phase-damping decoherence to Glasma occupation numbers whose central quantitative claim is underdetermined by an unspecified longitudinal box length Lz. read the letter →

arxiv 2507.04809 v2 pith:SZF43SA7 submitted 2025-07-07 hep-ph quant-ph

Gabriele Coci , Gabriele Parisi , Salvatore Plumari , Marco Ruggieri This is my paper · ORCID
classification hep-phquant-ph PACS 12.38.Aw12.38.Mh
keywords decoherenceentropyproductionopenquantumsystemsGlasmacoherentstatephasedampingvonNeumanncolorglasscondensate
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 asks whether quantum decoherence alone can create enough entropy to turn the coherent gluon state formed in the earliest moments of a high-energy nuclear collision into a thermalized gluon bath. It builds a coherent state whose occupation numbers are taken from the glasma fields of proton-nucleus and nucleus-nucleus collisions, couples that state to the vacuum through a phase-damping interaction that preserves particle number, and computes the asymptotic von Neumann entropy per particle $S_\infty/N$. The answer it finds is mostly no: apart from proton-nucleus collisions at small values of the coupling parameter $g\mu$, the decohered entropy per particle stays below the value $\simeq 2.19$ of a two-dimensional ultrarelativistic Bose gas of gluons. For nucleus-nucleus collisions at LHC-like values the ratio is only about one tenth of the thermal value. The paper concludes that phase decoherence from vacuum fluctuations is insufficient for thermalization, so additional entropy-generating mechanisms must operate.

What carries the argument

The central object is the phase-damping model of decoherence, defined by the system-reservoir Hamiltonian $H_{SR}=a^\dagger a(\Gamma^\dagger+\Gamma)$ and the vacuum Lindblad master equation. This interaction kills the off-diagonal elements of the density matrix exponentially while leaving occupation numbers intact, so each mode relaxes to the Poisson mixture $\rho_\infty = \sum_\ell e^{-n_k} n_k^\ell/\ell!\,|\ell\rangle\langle\ell|$. The map from glasma fields to occupation numbers is carried by the standard color-glass-condensate color-charge correlators, yielding $n_k^{AA}\propto L_z (g\mu_A)^2(g\mu_B)^2 I(k_T)/\omega_k$ and the analogous $pA$ expression, where $I(k_T)$ is a UV- and IR-finite transverse momentum integral. The comparison target is $S_{\rm SB}/N = 2.19$ for a two-dimensional ultrarelativistic Bose gas of gluons.

What would settle it

Compute $S_\infty/N$ using the full event distribution of $|a_k|^2$ instead of the single-coherent-state replacement $\beta_k=\sqrt{\langle |a_k|^2\rangle}$, and scan the longitudinal length $L_z$ in the occupation-number formulas: the central conclusion fails if the ratio crosses the thermal value $2.19$ or moves by more than a few percent as $L_z$ varies.

Watch

Extended reading notes

Core claim

The central discovery is that the asymptotic entropy of a decohered glasma coherent state is a sum over modes of the Shannon entropy of the Poisson distribution $\lambda_\ell = e^{-n_k} n_k^\ell/\ell!$, and that when the $n_k$ are taken from the glasma occupation numbers this entropy per particle falls short of the thermal benchmark. For nucleus-nucleus collisions, $S_\infty/N$ is independent of the transverse area and stays well below the two-dimensional ultrarelativistic gluon gas value $s_{SB}=2.19$; for proton-nucleus collisions it lies in the range roughly $0.35$--$0.75$ of $s_{SB}$ at LHC-relevant couplings, reaching or exceeding the thermal value only for the smallest $g\mu$ and largest transverse areas. As $g\mu$ grows the initial coherent state becomes denser and $S_\infty/N$ falls further behind the thermal value, because the total occupation number $N$ grows faster than the entropy $S_\infty$.

Load-bearing premise

The load-bearing premise is that the event-averaged glasma can be replaced, mode by mode, by one coherent state of amplitude $\beta_k=\sqrt{\langle |a_k|^2\rangle}$, and that the longitudinal box length $L_z$ cancels from the entropy per particle; because the von Neumann entropy is nonlinear in the occupation number, neither step is guaranteed, and $L_z$ is never fixed in the paper.

Editorial extensions

If this is right

  • For nucleus-nucleus collisions at $g\mu \simeq 2$ GeV, the decohered state reaches only about ten percent of the two-dimensional thermal gluon-gas entropy per particle, so most of the final entropy must come from later non-abelian dynamics.
  • As the glasma density parameter $g\mu$ increases, $S_\infty/N$ moves further below the thermal benchmark because $N$ grows faster than $S_\infty$; denser initial coherent states are therefore farther from decoherence-driven thermalization.
  • The two-dimensional thermal benchmark $S_{\rm SB}/N = 2.19$ is the appropriate comparison for a boost-invariant glasma; using the three-dimensional value $3.60$ would make the shortfall even larger.
  • In proton-nucleus collisions, decoherence brings the system closer to thermal equilibrium than in nucleus-nucleus collisions, with the gap closing only at small $g\mu$ and large transverse area.

Reading between the lines

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

  • Editorial extension: replacing the single coherent amplitude $\beta_k=\sqrt{\langle |a_k|^2\rangle}$ with the full distribution of $|a_k|^2$ would shift $S_\infty/N$, since entropy is nonlinear in $n_k$; a direct numerical test would settle the direction and size of the shift.
  • Editorial extension: because the environment is taken to be the vacuum, the computed entropy is a lower bound; coupling the glasma to a thermal reservoir through amplitude damping should add both particles and entropy, and the model predicts the $S/N$ gap would close as the reservoir temperature rises.
  • Editorial extension: the same Poisson-entropy map could be applied to anisotropic or overoccupied pre-equilibrium distributions to test whether subthermal decoherence entropy is generic for saturated gluon states.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper computes the asymptotic von Neumann entropy produced by phase-damping decoherence of a coherent state whose mode occupation numbers are taken from the McLerran-Venugopalan Glasma in AA and pA collisions. It compares the decoherence entropy per particle S∞/N with the entropy per particle of a thermal ultrarelativistic gluon gas and concludes that decoherence alone is insufficient to thermalize, except possibly for pA at small gμ and large transverse areas.

Significance. If fully established, the result would be a useful step in quantifying the role of decoherence in the early-stage thermalization problem. The single-mode decoherence result (Eqs. 23–28) is a standard and correct application of the phase-damping model, and the MV-model computation of the occupation numbers (Secs. IV) is internally consistent and clearly presented. The paper explicitly identifies the limitations of the phase-damping model. However, the central quantitative comparison is compromised by two load-bearing issues.

major comments (2)
  1. [Eqs. (69), (86)–(93), Figs. 4–5] The ratio S∞/N is not well defined because the occupation numbers scale with the longitudinal box length Lz. For AA, n_k ∝ Lz (Eq. 86); for pA, n_k ∝ Lz/AT (Eq. 87). In the mode sums of Eqs. (92)–(93), the integration is only over transverse momenta with dkx dky = (2π)^2/AT; there is no compensating dkz or factor 1/Lz. Since the Poisson entropy S∞(n_k) is a nonlinear function of n_k (Eq. 28), S∞ and N scale differently with Lz, so S∞/N depends on the arbitrary, never-specified Lz. The paper never states a value or physical justification for Lz, and Figs. 4–5 therefore do not follow from the equations. The central comparison with the thermal value sSB = 2.19 is underdetermined.
  2. [Eq. (46), Sec. III.A] The replacement of the event-averaged Glasma by a single coherent state with amplitude β_k = sqrt(⟨|a_k|^2⟩) is not justified. The MV model produces a distribution of color-charge configurations and hence a distribution of occupation numbers n_k per event. The decoherence entropy S(n_k) is a concave function of n_k, so by Jensen's inequality S(⟨n_k⟩) ≥ ⟨S(n_k)⟩. The calculation therefore overestimates the event-averaged entropy, and the reported S∞/N is not the entropy per particle of the decohered Glasma ensemble. The paper neither computes the distribution of n_k nor quantifies the resulting error.
minor comments (3)
  1. [Secs. II and III] The notation for the coherent-state amplitude changes from α (Sec. II) to β (Sec. III) without explanation; this is confusing when comparing Eqs. (10) and (40).
  2. [Figs. 4–5] The numerical value of Lz used for the curves is not stated anywhere; the captions and text should specify all inputs needed to reproduce the results.
  3. [Abstract] The abstract's qualifier 'except for proton-nucleus collisions at small values of gμ' is not precise, since the small-gμ range depends on the unstated Lz and on AT; the statement should be reformulated after the Lz issue is resolved.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the decoherence entropy is an un-fitted function of MV-model glasma occupation numbers, and the only self-citations are non-load-bearing; the Lz dependence is a determinacy concern, not a circularity.

full rationale

The central derivation chain is not circular. The phase-damping solution for a coherent state is independent of the target result: Eq. (28) gives S∞ as the von Neumann entropy of the Poisson distribution with mean n = |α|^2, and Eq. (29) is a standard consequence. The occupation numbers n_k are then computed from the McLerran-Venugopalan glasma, Eqs. (65)-(83), without fitting any parameter to the entropy or to the thermal benchmark sSB = 2.19. The comparison with a 2D/3D ultrarelativistic Bose gas is an external, independently defined reference, not an input to the calculation. The self-citations present, such as Ref. [56] for the pA parameters c = 1.25 and xg = 3.94, are not load-bearing because the paper states those values are taken from Refs. [55,57], which are external. Likewise, the coherent-state mapping follows Ref. [47], which is not by the present authors. A separate, non-circular weakness is that n_k in Eqs. (86)-(87) is proportional to the never-specified longitudinal length Lz, while the mode sums in Eqs. (92)-(93) are only over transverse momenta with dkx dky = (2π)^2/AT; since S∞(n_k) is nonlinear in n_k, S∞/N depends on Lz, so the numerical values in Fig. 5 are underdetermined. This is a correctness/robustness issue, not a self-referential reduction of the prediction to its inputs, and therefore does not raise the circularity score beyond the minor self-citation level.

Assumptions & free parameters 8 free parameters · 7 assumptions · 0 invented entities

The central calculation rests on standard open-quantum-system results plus the MV-model mapping from classical Glasma fields to coherent states. No new entities are introduced. The free parameters are mostly standard inputs from nuclear collision modeling; Lz, however, is an underspecified parameter that changes the final ratio.

free parameters (8)
  • m (IR regulator) = 0.2 GeV used for Fig. 1; stated range 0.1-0.4 GeV
    Chosen by hand to regularize the inverse Laplacian in the MV model; it enters I(kT) and thus all occupation numbers and the final entropy ratio.
  • g (QCD coupling) = g = 2 in Figs. 2-3; g mu scanned from 1 to 6 GeV
    Input coupling from the glasma description; the paper scans g mu rather than fitting it to the entropy result.
  • mu_A, mu_B (MV color charge density scales) = mu_A = mu_B = 0.5 GeV in Figs. 2-3; g mu scanned
    Standard MV model inputs setting the saturation momentum scale; not fitted to the target result.
  • c (hotspot normalization) = 1.25 from Refs. [55,57]
    Taken from prior literature for the pA hotspot model; affects pA occupation numbers and S/N.
  • xg (gluon distribution at fixed x) = 3.94 from Refs. [55,57]
    Input from global fits used in the pA hotspot model; affects pA occupation numbers.
  • AT (transverse quantization area) = 1, 4, 9 fm^2 in Figs. 4-5
    Varied to study dependence; for AA it cancels in S/N, for pA it leaves a residual dependence.
  • Lz (longitudinal box length) = not specified
    Appears in n_k through Eq. (69) and does not cancel in S_infinity/N because S_infinity is nonlinear in n_k; its value is never stated, leaving the numerical results underdetermined.
  • Bq, Bcq (proton thickness widths) = values not stated in text; taken from Refs. [55,56]
    Used in Eqs. (74)-(75) for the proton thickness function; not given in the paper, so pA results are not independently reproducible.
assumptions (7)
  • domain assumption Color charges of the colliding objects are Gaussian random variables with local correlations (MV model), with F(u) = delta^2(u).
    Sec. IV A, Eqs. (48) and (62); standard CGC/MV approximation. The entropy results inherit this assumption.
  • domain assumption Quantum field operators can be replaced by classical fields, with the coherent-state amplitude equal to the classical Fourier mode (following Ref. [47]).
    Sec. III A, text after Eq. (43): 'we assume that we can take the classical limit of Eq. (39), replacing the quantum operators with their classical counterparts.' This is load-bearing because it produces the occupation numbers.
  • domain assumption Decoherence of the Glasma against vacuum is described by the Born-Markov phase-damping master equation with a Markovian reservoir.
    Sec. II, Eqs. (14)-(19); the paper explicitly states the Markovian assumption is taken to simplify the problem.
  • domain assumption The Glasma is boost-invariant, so only kz = 0 modes contribute and the transverse gauge fields can be set to zero in Coulomb gauge.
    Sec. III A and Sec. IV A, around Eq. (59): 'as long as we work in the Coulomb gauge (59), we can neglect the fields (54).' This restricts the coherent state to longitudinal electric modes.
  • standard math Wick's theorem applies to the four-point correlator of color charges because fluctuations are Gaussian.
    Sec. IV B, footnote 2, used to obtain Eq. (60).
  • standard math The entropy of the total density matrix is the sum of single-mode von Neumann entropies because the modes are in a product state.
    Sec. V, Eqs. (84)-(85). Valid if modes are uncorrelated.
  • ad hoc to paper Each mode is represented by one coherent state with amplitude beta_k = sqrt(n_k), using the ensemble-averaged occupation number.
    Sec. III A, Eq. (46); this is an ad hoc mapping from the event-averaged classical Glasma to a pure coherent state, and it is not justified in the paper.

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Pith. "Pith review of Entropy from decoherence: a case study using glasma-based occupation numbers." pith.science (2026). https://pith.science/paper/SZF43SA7

@misc{pith2026250704809,
  author       = {Pith},
  title        = {Pith review of: Entropy from decoherence: a case study using glasma-based occupation numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZF43SA7}},
  note         = {Machine review of arXiv:2507.04809}
}
abstract

We compute the entropy-per-particle, $S/N$, produced by the decoherence of a coherent state interacting with an environment, using an analytical open quantum system approach. The coherent state considered is characterized by occupation numbers borrowed from the glasma fields produced in the early stages of high-energy nuclear collisions. The environment is modeled as the vacuum, and decoherence arises from the interaction of the state with vacuum fluctuations. We describe the system-environment interaction via a phase-damping model, which represents continuous measurements on the system without altering its energy or particle number. Starting from the occupation numbers typical of the Glasma in high-energy proton-nucleus and nucleus-nucleus collisions, we find that the final $S/N$ after decoherence is lower than that of a two-dimensional thermal bath of ultrarelativistic gluons, except for proton-nucleus collisions at small values of $g\mu$. Our results indicate that quantum decoherence alone does not generate sufficient entropy to transform the initial coherent state into a thermalized gluon bath.

Figures

Figures reproduced from arXiv: 2507.04809 by the authors.

Figure 1
Figure 1. FIG. 1. The function [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Total occupation number [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

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Forward citations

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