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Entropy production in pp and Pb-Pb collisions at energies available at the CERN Large Hadron Collider

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Pith's one-line read Combining measured particle spectra with HBT source radii, the paper fixes the entropy per unit rapidity in pp and Pb-Pb collisions and concludes that in central Pb-Pb about 95% of it is already present by 1 fm/c.

desk verdict A careful Pal–Pratt entropy extraction for LHC pp and Pb–Pb that will likely become the reference set of values, with one reproducibility gap in the pp pion entropy polynomial and a model-dependent temperature reconstruction that needs clearer uncertainty labeling. read the letter →

arxiv 1908.02792 v2 pith:ZZ5QUIU7 submitted 2019-08-07 hep-ph nucl-exnucl-th

Patrick Hanus , Aleksas Mazeliauskas , Klaus Reygers This is my paper · ORCID
classification hep-phnucl-exnucl-th
keywords entropyproductionphase-spacedensityHanburyBrown-Twissradiiquark-gluonplasmaPb-Pbcollisionspppre-equilibriumstageviscoushydrodynamics
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 tries to establish how much entropy is produced in proton-proton and lead-lead collisions at the Large Hadron Collider, and when during the collision it is produced. It extracts the entropy per unit rapidity $dS/dy$ directly from measured hadron spectra and Hanbury Brown-Twiss source radii, without assuming thermal equilibrium: $dS/dy = 11\,335 \pm 1\,188$ for the 10% most central Pb-Pb collisions at $\sqrt{s_{NN}} = 2.76$ TeV, $dS/dy = 135.7 \pm 17.9$ for high-multiplicity pp collisions at $\sqrt{s} = 7$ TeV, and $dS/dy = 37.8 \pm 3.7$ for minimum-bias pp. The entropy per charged particle, $6.7 \pm 0.8$ in Pb-Pb, $5.4 \pm 0.7$ in high-multiplicity pp, and $5.2 \pm 0.5$ in minimum-bias pp, agrees with hadron-resonance-gas models within 1-2$\sigma$ for Pb-Pb but sits below the chemical-equilibrium prediction in pp, suggesting incomplete chemical equilibration in small systems. Feeding the final entropy into a kinetic-theory pre-equilibrium stage matched to viscous hydrodynamics, the paper finds that about 95% of the final entropy already exists by $\tau = 1$ fm/$c$ and that the central temperature of the Pb-Pb fireball at that time is about 400 MeV. If this is right, the measured final multiplicity is primarily a probe of the pre-equilibrium phase, not of the later hydrodynamic expansion.

What carries the argument

The key machinery is the Pal-Pratt phase-space density method. For each hadron species treated as stable, the measured Lorentz-invariant spectrum $E\,d^3N/d^3p$ and the measured one-dimensional source radius $R_{\rm inv}$ are combined through $R_{\rm inv}^3 \approx \gamma R_{\rm out} R_{\rm side} R_{\rm long}$ ($\gamma = m_T/m$) into the maximum phase-space density $F = (1/m)(2\pi)^{3/2}/(2J+1)\,R_{\rm inv}^{-3}\,E\,d^3N/d^3p$. That $F$ is inserted into the quantum entropy integral $S = \int d^3r\,d^3p/(2\pi)^3[-f\ln f \pm (1\pm f)\ln(1\pm f)]$, expanded in powers of $F$ up to fourth order for Pb-Pb and with an eighth-order polynomial for pp pions because their phase-space density exceeds unity. The conversion $R_{\rm inv}^3 \approx \gamma R_{\rm out} R_{\rm side} R_{\rm long}$ is what lets a single measured radius stand for the full three-dimensional source volume; the paper checks the alternative $h(\gamma) = \alpha\gamma^\beta$ in an appendix. For the time-resolved part, the same final entropy fixes the normalization of an initial entropy profile, which is evolved through a linearized QCD kinetic-theory pre-equilibrium stage and then viscous hydrodynamics with $\eta/s = 0.08$ to freeze-out.

What would settle it

Measure, in the same multiplicity classes, the full three-dimensional HBT radii $R_{\rm out}, R_{\rm side}, R_{\rm long}$ together with the one-dimensional $R_{\rm inv}$ for pions, kaons, and protons, and check whether $R_{\rm inv}^3/(R_{\rm out}R_{\rm side}R_{\rm long})$ equals $\gamma$ within uncertainties; the appendix's alternative fit already moves pp entropy by 10%, so a comparable measured deviation would propagate directly through $F \propto 1/(m R_{\rm inv}^3)$ into $dS/dy$. In pp, where the paper finds pion phase-space densities above unity, also evaluate the entropy integral with the exact Bose-Einstein expression instead of the truncated polynomial; if the difference exceeds the quoted systematic uncertainty, the pp result depends on the series truncation.

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Extended reading notes

Core claim

The central discovery, stated on the paper's own terms, is that the entropy of the final state at LHC energies can be measured rather than modeled: combining identified-particle transverse-momentum spectra with one-dimensional HBT radii $R_{\rm inv}$ gives the phase-space density of each hadron species, and integrating the standard quantum entropy functional over that density yields $dS/dy$ at kinetic freeze-out. The paper reports $dS/dy = 11\,335 \pm 1\,188$ for 0-10% central Pb-Pb at $\sqrt{s_{NN}} = 2.76$ TeV, $dS/dy = 135.7 \pm 17.9$ for high-multiplicity pp at $\sqrt{s} = 7$ TeV, and $dS/dy = 37.8 \pm 3.7$ for minimum-bias pp, with entropy per charged particle $6.7 \pm 0.8$, $5.4 \pm 0.7$, and $5.2 \pm 0.5$ respectively. These $S/N_{\rm ch}$ values are compared with hadron-resonance-gas calculations at $T_{\rm ch} = 156$ MeV; the Pb-Pb value agrees within 1-2$\sigma$, while the pp values are below the chemical-equilibrium predictions. When the Pb-Pb entropy is used to normalize initial conditions for a pre-equilibrium kinetic-theory stage followed by viscous hydrodynamics with $\eta/s = 0.08$, the simulation produces only about 5% of the final entropy after $\tau = 1$ fm/$c$, so the bulk of entropy is generated before that time, and the central temperature of the 0-10% Pb-Pb fireball at $\tau = 1$ fm/$c$ reaches about 400 MeV (about 250 MeV for high-multiplicity pp).

Load-bearing premise

The load-bearing premise is that the one-dimensional source radius $R_{\rm inv}$ measures the same three-dimensional Gaussian source volume as the product of HBT radii through $R_{\rm inv}^3 \approx \gamma R_{\rm out} R_{\rm side} R_{\rm long}$, and that this relation (with its $m_T$ scaling) can be applied to every hadron species, including those for which only spectra are measured; if this geometric conversion is biased, the reported $dS/dy$ shifts with it because the entropy is logarithmic in $F \propto 1/(m R_{\rm inv}^3)$.

Editorial extensions

If this is right

  • If the extracted final entropy is right, then the entropy per charged particle in pp collisions is below the chemically equilibrated hadron-resonance-gas value, indicating that the small system has not reached full chemical equilibrium at freeze-out.
  • The final-state entropy fixes the normalization of initial entropy profiles for hydrodynamic calculations; with $\eta/s = 0.08$, the central temperature of the 0-10% Pb-Pb fireball at $\tau = 1$ fm/$c$ is about 400 MeV.
  • Only about 5% of the final entropy is produced after $\tau = 1$ fm/$c$ in Pb-Pb at $\eta/s = 0.08$, so measurements of final multiplicity mainly constrain the pre-equilibrium stage rather than the viscous expansion.
  • The same spectra-plus-femtoscopy recipe yields a finite entropy in high-multiplicity pp, supporting the practice of modeling small collision systems hydrodynamically while showing that the pion phase-space density there exceeds unity, so quantum corrections matter.

Reading between the lines

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

  • Beyond the paper: applying the same entropy reconstruction to p-Pb collisions at LHC energies would directly test whether $S/N_{\rm ch}$ keeps falling with system size, which would sharpen the case that small systems freeze out away from chemical equilibrium.
  • Beyond the paper: because the pp pion phase-space density exceeds unity, a fully quantum (Bose-Einstein) evaluation of the entropy integral would likely raise the pp entropy estimate; the paper's eighth-order polynomial is already a partial correction, but an exact calculation would show whether the reported pp values are the right endpoint.
  • Beyond the paper: if only about 5% of entropy is produced after 1 fm/c, then the shear viscosity $\eta/s$ of the quark-gluon plasma is only weakly constrained by final multiplicities; multiplicity-based extractions of $\eta/s$ would need to be reconsidered, while pre-equilibrium entropy production becomes the quantity to constrain.
  • Beyond the paper: the 10% shift in pp entropy under the fitted $h(\gamma)$ relation suggests the pp numbers are more fragile than the Pb-Pb ones; measuring full three-dimensional HBT radii in high-multiplicity pp would settle that fragility empirically.
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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 / 4 minor

Summary. The manuscript applies the Pal–Pratt method to ALICE data to compute the phase-space density and entropy per unit rapidity dS/dy from measured identified-particle spectra and HBT radii for sqrt(s)=7 TeV pp and sqrt(s_NN)=2.76 TeV Pb–Pb collisions. The central results are dS/dy = 11335 ± 1188 (0–10% Pb–Pb), 135.7 ± 17.9 (high-multiplicity pp), and 37.8 ± 3.7 (minimum-bias pp), corresponding to S/Nch values of 6.7 ± 0.8, 5.4 ± 0.7, and 5.2 ± 0.5. These values are compared with hadron-resonance-gas predictions at T_ch ≈ 156 MeV. The second part of the paper uses the KøMPøST pre-equilibrium propagator and the FluiduM viscous hydrodynamics code, with initial entropy profiles normalized to the measured final entropy, to reconstruct the average temperature profile at τ0 ≈ 1 fm/c and to conclude that in Pb–Pb collisions about 95% of the final entropy is already present by that time, with most entropy produced in the pre-equilibrium phase.

Significance. If the extraction is correct, this is the first LHC-era Pal–Pratt determination of final-state entropy and provides useful data-driven constraints on initial conditions and on entropy-per-charged-particle in pp and Pb–Pb collisions. The analysis has several genuine strengths: the pT-to-0 extrapolations use multiple functional forms with the spread carried into the systematic uncertainty; the HBT radii are parametrized with two different functions; feed-down from η, η′, and Σ0 is simulated with Pythia and corrected; and the pT-integrated pion, kaon, and proton yields are checked against published ALICE results. The dependence on the 1D-to-3D HBT radius relation is also cross-checked in Appendix A. These strengths make the central extraction credible, provided the unvalidated polynomial approximation for the pion phase-space density in pp collisions, discussed in the major comments, is addressed.

major comments (2)
  1. [Section II, Eq. (8)] The order-8 polynomial replacement for the pion Bose-Einstein integrand in pp collisions is undocumented and unvalidated. The coefficients a_i, the fit range in F, and the maximum pion phase-space density F reached in the data are not given, and no comparison with the exact integrand h(f) = -f ln f + (1+f) ln(1+f) is shown. This is load-bearing because pions contribute 20.1 of 37.8 (minimum bias) and 71.4 of 135.7 (high multiplicity) to dS/dy in Tables II and III, so an error in this approximation propagates directly into the headline values and into the S/Nch comparison. Please provide the coefficients and an explicit validation (for example, the maximum relative error over the actual F range), or replace the polynomial with numerical integration of the exact integrand, which is straightforward in this one-dimensional integral.
  2. [Section III A and Table I] The contributions of unmeasured species (η, η′, n, Σ, and the neutral-kaon component) are estimated by assuming that the entropy per particle is similar to that of the nearest measured species, with yields determined from a Pythia simulation of a hadron gas at T = 156 MeV. No systematic uncertainty is assigned to this procedure. In Table I the η, η′, n, and Σ entries alone sum to 1345 out of 11335, about 12% of the Pb–Pb total, with a comparable fraction in the pp tables. Because the final dS/dy and S/Nch values depend on these estimates, please propagate an uncertainty for the unmeasured-species assumption, for example by varying the hadron-gas temperature or by testing alternative matching rules, and state the resulting contribution to the total systematic uncertainty.
minor comments (4)
  1. [Section II, Eq. (8)] The text calls this a polynomial of order 8, but the sum runs only to i = 7; please clarify whether an a_8 coefficient is intended or whether the polynomial is of order 7.
  2. [Appendix A, Eq. (A2)] There is a typo: 'In this section we use assume h(γ) = ...' should read 'we use' or 'we assume'.
  3. [Section V B] The pp simulation uses a single width σ = 0.6 fm and a fixed η/s = 0.08, with no sensitivity study for these choices; a brief statement of how the reconstructed temperature and the '5% after 1 fm/c' conclusion respond to varying σ and η/s would help the reader assess the model dependence.
  4. [Section IV] The comparison with hadron-resonance-gas models mixes a data-based S/Nch that sums a limited set of final-state species with model S/Nch values that sum all primary hadrons before decays; the text acknowledges this difference but does not quantify how much of the 1–2σ discrepancy it can explain. A numerical estimate would strengthen the comparison.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: dS/dy is computed directly from measured spectra and HBT radii, and the modeling half calibrates kappa_s to that entropy as a boundary condition rather than predicting it.

full rationale

The central entropy extraction in Secs. II and III is self-contained: dS/dy is evaluated from measured ALICE identified-particle spectra and HBT radii through the Pal-Pratt phase-space formula, Eq. (6), with no parameter fitted to the reported entropy values. The relation R_inv^3 ≈ gamma R_out R_side R_long, Eq. (5), is an explicit geometric assumption, and the paper cross-checks it in Appendix A by fitting h(gamma)=alpha gamma^beta to measured one- and three-dimensional HBT radii; that changes pp entropy by about 10% (1 sigma), so it is a evaluated systematic uncertainty rather than a circular reduction. The only input calibrated to the final entropy is the normalization kappa_s in Eqs. (17) and (21), where the paper states that kappa_s 'is adjusted to reproduce the final state entropy estimated in Sec. III A.' That calibration is the correct boundary condition for the backward-extrapolated temperature and entropy-production profiles, and the timing statement that only about 5% of the final entropy is produced after tau = 1 fm/c is a model output, not an input. The self-cited KøMPøST references [62,63,70] and [54] enter only in this modeling half, as a publicly released kinetic-theory pre-equilibrium propagator with stated assumptions, not as the source of the measured dS/dy values. The main non-circular weakness is Eq. (8): the order-8 polynomial for the pion Bose-Einstein entropy term in pp collisions is stated without its coefficients, fit range, or validation, and because pions dominate pp entropy this is a reproducibility and uncertainty concern, but it is not a reduction of the result to its own inputs. The comparison to hadron-resonance-gas models in Sec. IV is also an external benchmark rather than an input. Overall, the derivation of the headline entropy values is independent of the fitted modeling parameters, so the circularity score is low.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central dS/dy values rest on the measured spectra and HBT radii plus axioms 1-4; no newly invented entities appear. The temperature and entropy-production-timing claims of Sec. V additionally depend on the free parameters κ_s, η/s, α, σ, and T_fo, all of which are either calibrated to the final entropy or taken from prior literature, so the modeling section is an anchored reconstruction rather than a prediction.

free parameters (7)
  • κ_s (initial entropy profile normalization) = tuned: final dS/dy = 11335 (Pb-Pb) and 135.7 (pp)
    In Eqs. (17) and (21), κ_s is adjusted so that KøMPøST plus FluiduM evolution reproduces the measured final entropy; it is a boundary-condition calibration for the temperature profile reconstruction, not an independent measurement.
  • η/s (specific shear viscosity) = 0.08 (0.16 used for sensitivity)
    Assumed constant through the QGP phase; sets the viscous entropy production, hence the claim that 90-95% of the entropy exists by τ = 1 fm/c (Fig. 4(a)).
  • α (two-component Glauber mixing) = 0.128
    Taken from ALICE multiplicity centrality dependence [56]; controls the shape of the Pb-Pb initial entropy profile and thus the reconstructed central temperature.
  • σ (pp initial profile width) = 0.6 fm
    Gaussian width of the pp initial entropy density in Eq. (21), taken from Ref. [67]; directly sets the pp central entropy density and temperature.
  • T_fo (freeze-out temperature) = 156 MeV
    Used both for the hydro freeze-out surface and for the hadron-resonance-gas comparisons; taken from statistical model fits [17].
  • α, β (h(γ) = α γ^β fit) = (1.52, 0.51) Pb-Pb; (0.48, 1.18) MB pp; (0.54, 0.93) HM pp
    Fit to ALICE R_inv versus Rout Rside Rlong data in Appendix A; used only for the systematic cross-check, not for the standard results.
  • R_inv constant for minimum-bias pp = 1.1 ± 0.1 fm
    Assumed independent of m_T because the measured dependence is weak; enters the entropy through F ∝ 1/(m R_inv^3).
assumptions (6)
  • standard math Gibbs entropy formula for a non-interacting hadron gas, Eq. (1): S = (2J+1) ∫ [-f ln f ± (1 ± f) ln(1 ± f)] d^3r d^3p / (2π)^3
    Foundation of the extraction; assumes the hadrons are non-interacting at kinetic freeze-out and that resonance decays after freeze-out do not change the total entropy (footnote 2).
  • domain assumption Gaussian source parametrization with HBT radii and the 1D-to-3D radius relation R_inv^3 ≈ γ R_out R_side R_long, Eqs. (3)-(5)
    The Pal-Pratt geometric premise; the paper's own Appendix A shows the pp results shift by about 10% under the alternative h(γ) = αγ^β.
  • domain assumption m_T scaling of the scaled HBT radii R_inv / ((γ^{1/2}+2)/3)^{1/2} (Kisiel et al. [26])
    Used to extend R_inv(m_T) to particles without femtoscopic data (Λ, Ξ, Ω, η, η′) in Sec. III A.
  • domain assumption Unmeasured species' entropy from same-mass species; feed-down fractions and N/Nch = 1.115 ± 0.03 from Pythia 8.2 with hadron-gas densities at T = 156 MeV, μ = 0
    Secs. III A and IV; η, η′, Σ, n add about 12% of the Pb-Pb entropy, and the N/Nch value converts S/N model predictions to S/Nch.
  • ad hoc to paper KøMPøST linear response (pure-glue QCD kinetic theory) as the pre-equilibrium propagator, with a lattice EoS used to map entropy to energy at τ_EKT = 0.1 fm
    Sec. V A and footnote 4; the lattice EoS conversion is applied out of equilibrium and labeled a 'purely practical tool' by the authors.
  • domain assumption Boost-invariant, azimuthally symmetric averaged evolution
    Sec. V; the entropy and energy current contours of Figs. 3-8 rely on averages over azimuth and over the 0-10% centrality bin; event-by-event fluctuations are not modeled.

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Pith. "Pith review of Entropy production in pp and Pb-Pb collisions at energies available at the CERN Large Hadron Collider." pith.science (2026). https://pith.science/paper/ZZ5QUIU7

@misc{pith2026190802792,
  author       = {Pith},
  title        = {Pith review of: Entropy production in pp and Pb-Pb collisions at energies available at the CERN Large Hadron Collider},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZ5QUIU7}},
  note         = {Machine review of arXiv:1908.02792}
}
abstract

We use experimentally measured identified particle spectra and Hanbury Brown-Twiss radii to determine the entropy per unit rapidity $dS/dy$ produced in $\sqrt{s} = 7$ TeV pp and $\sqrt{s_{\rm NN}} = 2.76$ TeV Pb-Pb collisions. We find that $dS/dy = 11335 \pm 1188$ in 0-10% Pb-Pb, $dS/dy = 135.7 \pm 17.9$ in high-multiplicity pp, and $dS/dy = 37.8 \pm 3.7$ in minimum bias pp collisions and compare the corresponding entropy per charged particle $(dS/dy)/(dN_{\rm ch}/dy)$ to predictions of statistical models. Finally, we use the QCD kinetic theory pre-equilibrium and viscous hydrodynamics to model entropy production in the collision and reconstruct the average temperature profile at $\tau_0 = 1$ fm/$c$ for high multiplicity pp and Pb-Pb collisions.

Figures

Figures reproduced from arXiv: 1908.02792 by the authors.

Figure 1
Figure 1. FIG. 1. Transverse momentum spectrum of positive pions [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Entropy per primary hadron [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Temperature in hydrodynamic evolution of an av [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Entropy per rapidity in viscous hydrodynamic expansion with specific shear viscosity [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. (a) Entropy density profile (multiplied by [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Temperature in hydrodynamic evolution of an aver [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. (a) Entropy per rapidity in viscous hydrodynamic expansion with specific shear viscosity [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. (a) Entropy density profile (multiplied by [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Comparison of measured one-dimensional HBT [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.