Hydro+ in Action: Understanding the Out-of-Equilibrium Dynamics Near a Critical Point in the QCD Phase Diagram

Gregory Ridgway; Krishna Rajagopal; Ryan Weller; Yi Yin

REVIEW 3 major objections 6 minor 2 cited by

Hydro+ in Action: Understanding the Out-of-Equilibrium Dynamics Near a Critical Point in the QCD Phase Diagram

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Hydro+ run captures out-of-equilibrium critical fluctuations in QGP

desk verdict A solid and honest first implementation of Hydro+ with back-reaction in an expanding fireball; the small-back-reaction takeaway is real but demonstrated only in a parameter corner the authors themselves flag as not the QCD-relevant one. read the letter →

arxiv 1908.08539 v2 pith:FDKWTA3W submitted 2019-08-22 hep-ph nucl-th

Krishna Rajagopal , Gregory Ridgway , Ryan Weller , Yi Yin This is my paper · ORCID

classification hep-phnucl-th PACS 25.75.-q64.60.Ht25.75.Nq

keywords QCDcriticalpointHydro+fluctuationsout-of-equilibriumdynamicsslowingdownheavy-ioncollisionsbulkviscositybeamenergyscan

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 reports the first explicit solution of the Hydro+ equations in a simplified but self-consistent model of a heavy-ion collision fireball: a boost-invariant, azimuthally symmetric droplet with a hypothetical critical point placed near zero baryon chemical potential. The authors show that the coupled equations for the hydrodynamic flow and the full spectrum of critical fluctuations can be solved together, and that the resulting dynamics reproduces the expected out-of-equilibrium effects quantitatively: fluctuations grow and decay with a lag induced by critical slowing down, preserving memory of earlier conditions, and are carried outward by radial flow. They further find that the back-reaction of these out-of-equilibrium fluctuations on the energy density and radial flow is small, typically at the one-to-a-few-percent level for relaxation rates $\Gamma_0$ in the range $0.5$ to $2~\mathrm{fm}^{-1}$. If that smallness persists in more realistic settings, future phenomenological modeling of the QCD critical-point search could ignore the feedback of fluctuations on the flow and still capture the essential physics.

What carries the argument

The central object is the Wigner transform of the equal-time two-point correlation function of the order-parameter fluctuation, $\phi_Q(t,x)$, which measures the width of the distribution of the fluctuating field at wave vector $Q$ in the local rest frame. It obeys a relaxation equation $D\phi_Q = -\Gamma_Q(\phi_Q - \bar{\phi}_Q)$, with $\Gamma_Q$ encoding critical slowing down through $\Gamma_Q \propto (\xi/\xi_0)^{-2}(1+(Q\xi)^2)$ in Model A dynamics and $\bar{\phi}_Q$ the equilibrium Ornstein-Zernike form. The feedback on the flow enters through a generalized entropy $s^{(+)} = s + \Delta s$, whose dependence on the out-of-equilibrium $\phi_Q$ modifies the pressure $p^{(+)}$ appearing in the stress-energy tensor; this is what produces the effective bulk viscosity and the modified sound velocity. The model is closed by a temperature-dependent correlation length $\xi(T)$ peaking at $\xi_{\max}/\xi_0 = 3$, an equation of state built from a $c_V(T)$ with a critical contribution proportional to $\xi$, and initial conditions with $\phi_Q$ set to equilibrium at $\tau_I = 1~\mathrm{fm}$.

What would settle it

Run the same Hydro+ setup with the Model H dynamical exponent $z \approx 3$ and a correlation-length enhancement $\xi_{\max}/\xi_0$ of ten or more; if the fractional shifts in $\varepsilon$ and $v_r$ then grow well beyond a few percent, the paper's simplification claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that Hydro+ can be exercised: the deterministic equations for the equal-time two-point function $\phi_Q$ of the critical order-parameter fluctuations can be solved alongside second-order viscous hydrodynamics for an expanding, cooling droplet, with each feeding back on the other. In this model the fluctuations rise and fall as the droplet crosses the critical regime, but always lag their equilibrium values because long-wavelength modes relax slowly; at fixed radius the lag is visible as a solid curve that trails the dashed equilibrium curve on both sides of the crossing. The simulation also produces a spatial, memory-dependent pattern of $\phi_Q$, and shows that radial flow advects a pre-existing peak in the fluctuations outward, a phenomenon not reported before. The quantitative finding is that the resulting modification of the entropy, pressure, energy density and radial velocity is modest: with $\Gamma_0$ in the range $0.5$--$2~\mathrm{fm}^{-1}$, the fractional changes in $\varepsilon$ and $v_r$ are at the few-percent level, adding only about $1.5$--$4.5\%$ to $\varepsilon$ relative to the no-critical-point evolution. The authors present this as evidence that the first half of Hydro+, evolving the fluctuations on a fixed hydrodynamic background, may be sufficient for many phenomenological purposes.

Load-bearing premise

The quantitative conclusion that the back-reaction is small rests on three modeling choices: a faster relaxation law ($z=2$) than the true QCD critical point likely has, a modest correlation-length growth (only a factor of three), and starting the fluctuations in equilibrium even in the shell initially near the critical temperature; if any of these is more extreme, the lag and back-reaction could be larger.

Editorial extensions

If this is right

If the central claim is right, out-of-equilibrium critical fluctuations in a heavy-ion fireball are computable with deterministic Hydro+ equations, not only with stochastic simulations.
The lag and memory effects seen in uniform cooling systems appear in a spatially inhomogeneous, radially expanding fireball, with a new advective transport of fluctuations by the flow.
Back-reaction on $\varepsilon$ and $v_r$ is small, often below one percent and up to a few percent, so direct fluctuation observables should matter more for the critical-point search than modifications of the bulk flow.
Modeling the QCD critical-point search may be simplified to evolving fluctuations on a hydrodynamic background, omitting the feedback loop, at least within the parameter range studied.
Future 3+1-dimensional Hydro+ with a realistic equation of state and freezeout will be needed before comparison to Beam Energy Scan data, but this paper maps the technical path for doing so.

Reading between the lines

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

If the real QCD critical point is in the Model H universality class with $z \approx 3$, as the authors note, its fluctuations relax more slowly than the $z=2$ law used here, so the lag and back-reaction in a comparable simulation could exceed the few-percent level reported.
The outward advection of a fluctuation peak implies that in event-by-event simulations with lumpy initial conditions, critical fluctuations born in one region could be transported into regions whose thermodynamic history never passed near the critical point, making the correlation between fluctuations and local temperature history a potentially useful diagnostic.
Because the feedback enters through the generalized pressure $p^{(+)}$, the same machinery can be used to isolate the effective bulk viscosity and modified sound speed as functions of $\xi$; quantifying how those scale with $\xi_{\max}/\xi_0$ would show whether the small back-reaction is a robust property or a consequence of the modest factor-of-three correlation-length enhancement.
A direct comparison of Hydro+ with stochastic simulations of the same model would test how much of the critical dynamics is carried by the deterministic two-point-function truncation and whether higher cumulants, which Hydro+ does not yet evolve, feed back appreciably on the hydrodynamic variables.

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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

Bottom line: this is the first explicit Hydro+ simulation in an inhomogeneous, radially expanding fireball, and it works. If you care about critical fluctuations at BES, this is worth a careful read. The central demonstration—lag and memory in phi(Q), feedback into epsilon and v_r, and advection—is credible and well documented. Appendix A gives enough detail to reproduce; the equations are complete; no fitting to a target result, so no circularity worry. The advection claim is appropriately nuanced: the big advected peak in Fig. 7 is seeded by their assumption phi(Q)=phi_eq(Q) initially in the critical shell, which they call unrealistic; they still argue advection will matter generically. That is fair.

The soft spot is the one the stress-test note hit. The paper's forward-looking simplification claim—that back-reaction is only a few percent and can be neglected—is tested for Gamma_0 = 0.5, 1, 2, 5 fm^-1 with Model A dynamics (z=2) and xi_max/xi_0=3. But the authors themselves note that a realistic QCD critical point at nonzero mu_B is Model H with z~3, with Gamma ∝ (xi/xi_0)^-3, and that within this model the appropriate Gamma_0 is about 0.25 fm^-1. That rate is used in Fig. 7, where it produces much larger out-of-equilibrium deviations in phi, but back-reaction on epsilon and v_r at 0.25 is never shown. And there is no sensitivity scan in xi_max/xi_0 even though all critical effects scale with it. So the 'small back-reaction' conclusion is conditionally true, not demonstrated for the parameter corner the authors identify as QCD-like. They are appropriately hedged in the abstract ('within our model', 'if it persists'), so it is not overclaiming—but the reader should not treat the simplification as established.

Minor point: the numerical regularization at r<0.12 fm is asserted to leave results unaffected but the evidence is not shown; a convergence plot would help. Also, no code is released, though the implementation description is detailed enough that a motivated group could reproduce it.

Who is this for? People building quantitative Hydro+ models for BES observables, and anyone who wants to see how the formalism actually behaves in a finite expanding system. It deserves a serious referee. I would send it out; for publication I would ask for at least a back-reaction estimate at Gamma_0=0.25 or a clear statement that the simplification claim is only for Model A parameters, plus a xi sensitivity point. I would cite it in any Hydro+ context.

Referee Report

3 major / 6 minor

Summary. The paper presents the first explicit numerical implementation of the Hydro+ formalism in a setting that resembles a heavy-ion collision: a boost-invariant, azimuthally symmetric fireball undergoing radial flow, with a model critical point placed near the μB = 0 axis. The authors fully specify the model ingredients: the equilibrium fluctuation spectrum φ(Q) (Ornstein-Zernike form), the relaxation rate Γ(Q) with Model A dynamics and dynamical exponent z = 2, a temperature-dependent correlation length ξ(T), an equation of state built from a parameterized c_V(T) with a critical contribution, and initial conditions at τ_I = 1 fm. They solve the coupled Hydro+ and viscous hydrodynamic equations and report three main results: (i) the out-of-equilibrium φ(Q) lags behind its equilibrium value, first below it while φbar rises and then above it while φbar falls, including momentum-dependent memory effects; (ii) in the radial profile, a peak in φ(Q) can be advected outward by the radial flow when Γ0 is small; and (iii) the back-reaction of the out-of-equilibrium fluctuations on the energy density and radial velocity is at the level of a few percent for Γ0 = 0.5–2 fm^-1, leading the authors to suggest that future phenomenological modeling may neglect this back-reaction.

Significance. If the results hold, this is a valuable proof-of-principle: it demonstrates that the full Hydro+ system—fluctuation evolution, hydrodynamic evolution, and their mutual feedback—can be solved consistently, and it identifies the momentum range Q ∼ 0.4–0.7 fm^-1 that dominates the feedback integral. The paper is unusually complete in its model specification, and Appendix A gives enough numerical detail for the calculation to be reproduced; the authors also test sensitivity to Γ0 and to box size. The implementation uses the published Hydro+ formalism of Ref. [26] with input parameters chosen a priori, so there is no circularity in the central demonstration. The main significance is the explicit, self-consistent exercise of the formalism and the qualitative guidance it provides. The forward-looking conclusion that back-reaction is negligible is, however, conditional on a parameter region that is not fully explored, and this limits the strength of the paper's main phenomenological suggestion.

major comments (3)

[§3.2, Fig. 9] The paper's forward-looking conclusion that the back-reaction is small and can be neglected rests on runs with Γ0 = 0.5, 1, 2, and 5 fm^-1 only. Because §3.1 argues that a real QCD critical point at nonzero μB is in the Model H class and should be mimicked in this z = 2 model by a much smaller Γ0 (the Γ0 = 0.25 fm^-1 case shown in Fig. 7 is the strongly out-of-equilibrium case), the parameter point most relevant to the QCD motivation is absent from the back-reaction plots. This is a load-bearing gap: Fig. 7 shows substantially larger deviations of φ(Q) from equilibrium at Γ0 = 0.25 fm^-1, and the feedback integrals in Eqs. (2.11)–(2.15) grow with those deviations. I request either back-reaction results at Γ0 = 0.25 fm^-1 (or at a Model-H-motivated rate) or a clear restriction of the 'considerably simplify future modelling' claim to the tested parameter range.
[§2.4, Eq. (2.42)] The smallness of the back-reaction is also not tested against the assumed correlation-length enhancement, ξmax/ξ0 = 3. Since c_V^crit is proportional to ξ/ξ0 and the generalized-pressure corrections in Eq. (2.15) depend on the critical contribution to the equation of state, a larger ξmax/ξ0 would increase both the equilibrium and the out-of-equilibrium feedback. The manuscript should either provide a scan in ξmax/ξ0 or explicitly state that the percent-level conclusion is contingent on this value.
[§2.5 and §3.1, Fig. 7] The advected peak in φ(Q) at Γ0 = 0.25 fm^-1 is, by the authors' own description, seeded by the initial condition φ = φbar in the outer shell where T is initially near Tc, an initialization they call 'almost certainly unrealistic.' Because this artifact is the basis for the new advection phenomenon highlighted in Fig. 7 and the bright band in Fig. 8, the claim that radial flow transports fluctuations outwards as a generic feature is not yet established. I would like to see a test with a less tuned initialization (for example, φ initialized small everywhere, or with a separate prescription for the shell) or an explicit statement that the advection illustration is conditional on that initial condition.

minor comments (6)

[Fig. 9 caption] The caption says the colored curves come from 'three different values' of Γ0, but the figure and the text present four values: Γ0 = 0.5, 1, 2, and 5 fm^-1.
[§3.1] The sentence 'We show φ(Q) at τ = 10.5 fm, when it is at its maximum value, in both panels of Fig. 7' appears to refer to Fig. 5, since Fig. 7 shows τ = 2, 3.5, and 5.5 fm.
[§2.5] There is a grammatical fragment in the sentence beginning 'at a relatively larger, near the edge of the fireball'; this should be rephrased.
[Eq. (2.34)] The equation has a typesetting artifact, 'sqrt√', which should be cleaned up.
[§3.2] The phrase 'the colored "no B.R." energy density curves' is inconsistent with Fig. 9, where the 'no B.R.' case is shown as black dashed curves; the colored curves are the back-reaction cases.
[Abstract and §4] The phrase 'quantitatively capture non-equilibrium phenomena' is stronger than what a single model study with acknowledged simplifications can establish; a phrasing such as 'explicitly compute' or 'self-consistently describe' would better match the caveated statements in the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper solves an externally published formalism with explicit model inputs and computes, rather than fits, its results.

full rationale

The paper's derivation chain is self-contained in the relevant sense. It imports the Hydro+ formalism from Stephanov and Yin (Ref. [26]) and then implements it with explicitly stated model inputs: the equilibrium fluctuation spectrum phi_Q = c_M xi^2/(1+(Q xi)^2) (Eq. 2.26), the relaxation rate Gamma_Q = Gamma_0 (xi/xi_0)^(-2)(1+(Q xi)^2) (Eqs. 2.32-2.33), the correlation-length ansatz (Eq. 2.34) with xi_max/xi_0 = 3, and an equation of state built from c_V(T) (Eqs. 2.42-2.46). The dynamical outputs in Figs. 5-9, including the lag, memory, advection, and the few-percent back-reaction on epsilon and v_r, are obtained by solving the coupled deterministic equations (2.16)-(2.17); they are not fitted to a target result. The central quantitative claim, that the back-reaction is small, is a computed consequence of these inputs and of the generalized-thermodynamics formulas from Ref. [26]; it is not imposed by construction. The self-citations (Refs. [23], [24], [26]) are contextual: Ref. [26] is the published formalism being exercised, and Refs. [23]-[24] are prior qualitative studies whose lagging/memory results the paper explicitly reproduces and extends. No uniqueness theorem from the authors is invoked to force a choice, and no ansatz is smuggled in via citation: the Model A z=2 dynamics and the OZ form are justified and stated openly as model choices. The acknowledged limitations, such as initializing phi_Q in equilibrium in a shell near T_c (Sec. 2.5) and not presenting back-reaction results for the Model-H-motivated Gamma_0 = 0.25 fm^-1, are parameter-sensitivity and realism concerns, not circularity. Accordingly, the paper receives a circularity score of 0.

Assumptions & free parameters 10 free parameters · 10 assumptions · 0 invented entities

The paper is an application of an existing formalism, so the main axioms are carried from Hydro+ [26]. The many free parameters are model choices needed to define the toy critical point and the background EoS; none are fitted to experimental data. The central quantitative conclusion (small back-reaction) is sensitive to several of these choices, especially xi_max/xi_0, Gamma_0, and the Model A dynamics.

free parameters (10)

xi_0 (microscopic correlation length) = 1 fm
Sets the overall scale of correlation lengths and enters the critical heat capacity ansatz; chosen by hand.
xi_max/xi_0 (maximum equilibrium correlation length enhancement) = 3
Caps the growth of the equilibrium correlation length at the imagined critical point; a small value is a key reason the back-reaction is small.
Tc (critical temperature of the imagined critical point) = 0.160 GeV
Location of the hypothetical critical point on the T axis; chosen to make the fireball trajectory pass through the critical region.
Delta_T (width of the critical regime) = 0.2 Tc = 0.032 GeV
Sets the temperature range over which xi and c_V are enhanced; controls the duration of critical slowing down along the trajectory.
Gamma_0 (microscopic relaxation rate) = varied: 0.25, 0.5, 1, 2, 5 fm^-1; central value 1 fm^-1
Controls how far phi_Q lags behind equilibrium. The paper scans this parameter and estimates that Gamma_0 of order 1 fm^-1 is plausible if a critical point existed near mu_B=0.
Prefactor in c_V^crit = 1/2
Nonuniversal constant mapping the Ising mean-field heat capacity to c_V^crit. The authors call it a guess, informed by (Tc/Delta_T)^2 about 25 and 25/(16 pi) about 1/2.
Matching coefficients c_0...c_5 in c_V(T) = {26.80, 7.29, 0.38, -0.27, -0.12, 0.01}
Fit to make c_V/T^3 and its first two derivatives continuous at T=TL and T=TH. No physics data are involved; they are smooth-connection parameters.
Non-critical heat capacity limits a_L, a_H = 0.1 a_QGP, 0.8 a_QGP
Low- and high-temperature asymptotics of c_V/T^3, chosen with guidance from lattice QCD behavior at high T and numerical simplicity at low T.
Shear viscosity ratio and relaxation time = eta/s = 1/(4 pi), T tau_Pi = 4 eta/s
Standard choices within the range used in heavy-ion hydrodynamics; tau_Pi is a causality-preserving regulator with claimed little effect.
Initial central temperature and radial profile = 330 MeV at tau_I=1 fm, Glauber Au-Au profile at sqrt(s)=200 GeV
Initial conditions for the bulk hydrodynamics, taken from Ref. [41]. Not realistic for BES energies; the authors state this.

assumptions (10)

domain assumption Hydro+ formalism, including the generalized entropy Eq. (2.11), pressure Eq. (2.15), and the relaxation equation (2.3), is taken as given from Stephanov and Yin, Phys. Rev. D98 (2018) 036006.
The paper does not re-derive the framework but implements it; it is central to every result. The framework is published independently, though one author (Yin) is a co-author of the formalism paper.
ad hoc to paper The equilibration rate is set to its equilibrium form, Gamma_Q = Gamma_Q, with Model A universality class, z=2 and f_Gamma(a)=1+a^2 (Eq. 2.32).
Chosen for simplicity; the authors state that the true QCD critical point is likely Model H with z about 3, so this assumption materially affects the quantitative lag and back-reaction.
ad hoc to paper Equilibrium fluctuation spectrum uses the Ornstein-Zernike form phi_Q_bar = c_M/(Q^2 + xi^-2) (Eq. 2.26).
Simple scaling form with unit normalization; more refined scaling functions exist but are not used.
ad hoc to paper Correlation length ansatz (xi/xi_0)^-2 = sqrt(tanh^2((T-Tc)/Delta_T)(1-(xi_max/xi_0)^-4) + (xi_max/xi_0)^-4) (Eq. 2.34), with xi_max/xi_0=3, xi_0=1 fm, Tc=0.160 GeV, Delta_T=0.2Tc.
Motivated by mean-field behavior, but the specific values and functional form are chosen by hand; the small xi_max/xi_0 caps the critical enhancement and directly limits the size of the back-reaction.
ad hoc to paper Critical contribution to the heat capacity is c_V^crit(T) = (1/2) xi_0^-3 xi(T)/xi_0 (Eq. 2.42), obtained by mapping the Ising mean-field result C_M,Is = (1/(16 pi)) xi_0^-3 xi/xi_0 with a guessed prefactor 1/2.
The prefactor is acknowledged to be a guess; it sets the scale of critical softening and thus the EoS effects.
ad hoc to paper The non-critical heat capacity c_V^(no C.P.)(T) uses a tanh crossover between a_L and a_H (Eq. 2.46), with matching polynomial coefficients c_0...c_5 fitted to continuity conditions at T_L and T_H.
These choices shape the background EoS and the sound speed; the coefficients are fixed by smoothness, not by data.
domain assumption Initial condition phi_Q(tau_I,r) = phi_Q_bar(T(tau_I,r)) everywhere, including the shell initially near T_c.
Acknowledged as unrealistic for the near-critical shell; it seeds the advected peak in Fig. 7 and therefore shapes part of the reported qualitative dynamics.
domain assumption The fluid is assumed to be in the Muller-Israel-Stewart causal viscous hydrodynamics regime, with tau_Pi T = 4 eta/s and eta/s=1/(4 pi).
Standard choices in heavy-ion phenomenology; tau_Pi acts as a causality regulator and is stated to have little effect on dynamics.
domain assumption The convective derivative D = u^mu partial_mu is used without the confluent derivative refinement discussed in Ref. [39].
Footnoted as a subtlety that is not considered; it could affect the advection terms in the phi_Q equation.
domain assumption Truncation at the two-point-function level, i.e., deterministic equations for phi_Q only, with no noise or higher cumulants.
This is intrinsic to Hydro+ as presented; the paper explicitly defers higher-point functions to future work.

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Pith. "Pith review of Hydro+ in Action: Understanding the Out-of-Equilibrium Dynamics Near a Critical Point in the QCD Phase Diagram." pith.science (2026). https://pith.science/paper/FDKWTA3W

@misc{pith2026190808539,
  author       = {Pith},
  title        = {Pith review of: Hydro+ in Action: Understanding the Out-of-Equilibrium Dynamics Near a Critical Point in the QCD Phase Diagram},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDKWTA3W}},
  note         = {Machine review of arXiv:1908.08539}
}

abstract

Upcoming experimental programs will look for signatures of a possible critical point in the QCD phase diagram in fluctuation observables. To understand and predict these signatures, one must account for the fact that the dynamics of any critical fluctuations must be out-of-equilibrium: because of critical slowing down, the fluctuations cannot stay in equilibrium as the droplet of QGP produced in a collision expands and cools. Furthermore, their out-of-equilibrium dynamics must also influence the hydrodynamic evolution of the cooling droplet. The recently developed Hydro+ formalism allows for a consistent description of both the hydrodynamics and the out-of-equilibrium fluctuations, including the feedback between them. We shall explicitly demonstrate how this works, setting up a Hydro+ simulation in a simplified setting: a rapidity-independent fireball undergoing radial flow with an equation of state in which we imagine a critical point close to the $\mu_B=0$ axis of the phase diagram. Within this setup, we show that we can quantitatively capture non-equilibrium phenomena, including critical fluctuations over a range of scales and memory effects. Furthermore, we illustrate the interplay between the dynamics of the fluctuations and the hydrodynamic flow of the fireball: as the fluid cools and flows, the dynamical fluctuations lag relative to how they would evolve if they stayed in equilibrium; there is then a backreaction on the flow itself due to the out-of-equilibrium fluctuations; and, in addition, the radial flow transports fluctuations outwards by advection. Within our model, we find that the backreaction from the out-of-equilibrium fluctuations does not yield dramatically large effects in the hydrodynamic variables. Further work will be needed in order to check this quantitative conclusion in other settings but, if it persists, this will considerably simplify future modelling.

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Works this paper leans on

55 extracted references · 13 canonical work pages · cited by 2 Pith papers

[26]

Stephanov and Y

M. Stephanov and Y. Yin,Hydrodynamics with parametric slowing down and ﬂuctuations near the critical point, Phys. Rev. D98 (2018) 036006 [1712.10305]

arXiv 2018
[1]

Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo,The Order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443 (2006) 675 [hep-lat/0611014]

arXiv 2006
[2]

Bazavov et al.,Equation of state and QCD transition at ﬁnite temperature, Phys

A. Bazavov et al.,Equation of state and QCD transition at ﬁnite temperature, Phys. Rev. D80 (2009) 014504 [0903.4379]

arXiv 2009
[3]

Borsanyi, G

S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg et al.,The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [1007.2580]

arXiv 2010
[4]

Bazavov et al.,The chiral and deconﬁnement aspects of the QCD transition, Phys

A. Bazavov et al.,The chiral and deconﬁnement aspects of the QCD transition, Phys. Rev. D85 (2012) 054503 [1111.1710]

arXiv 2012
[5]

Berges and K

J. Berges and K. Rajagopal,Color superconductivity and chiral symmetry restoration at nonzero baryon density and temperature, Nucl. Phys. B538 (1999) 215 [hep-ph/9804233]

arXiv 1999
[6]

A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov and J. J. M. Verbaarschot,On the phase diagram of QCD, Phys. Rev. D58 (1998) 096007 [hep-ph/9804290]

arXiv 1998
[7]

M. A. Stephanov, K. Rajagopal and E. V. Shuryak,Signatures of the tricritical point in QCD, Phys. Rev. Lett.81 (1998) 4816 [hep-ph/9806219]

arXiv 1998

Show all 55 references

[8]

M. A. Stephanov, K. Rajagopal and E. V. Shuryak,Event-by-event ﬂuctuations in heavy ion collisions and the QCD critical point, Phys. Rev. D60 (1999) 114028 [hep-ph/9903292]

1999 arXiv
[9]

Rajagopal and F

K. Rajagopal and F. Wilczek,The Condensed matter physics of QCD, inAt the frontier of particle physics. Handbook of QCD. Vol. 1-3, M. Shifman and B. Ioﬀe, eds., pp. 2061–2151, (2000), hep-ph/0011333, DOI

2000 arXiv
[10]

Stephanov,QCD phase diagram: An Overview, PoS LA T2006(2006) 024 [hep-lat/0701002]

M. Stephanov,QCD phase diagram: An Overview, PoS LA T2006(2006) 024 [hep-lat/0701002]

2006 arXiv
[11]

Fukushima and T

K. Fukushima and T. Hatsuda,The phase diagram of dense QCD, Rept. Prog. Phys.74 (2011) 014001 [1005.4814]

2011 arXiv
[12]

Luo and N

X. Luo and N. Xu,Search for the QCD Critical Point with Fluctuations of Conserved Quantities in Relativistic Heavy-Ion Collisions at RHIC : An Overview, Nucl. Sci. Tech.28 (2017) 112 [1701.02105]

2017 arXiv
[13]

Busza, K

W. Busza, K. Rajagopal and W. van der Schee,Heavy Ion Collisions: The Big Picture, and the Big Questions, Ann. Rev. Nucl. Part. Sci.68 (2018) 339 [1802.04801]

2018 arXiv
[14]

Bzdak, S

A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov and N. Xu,Mapping the Phases of Quantum Chromodynamics with Beam Energy Scan, 1906.00936

1906 arXiv
[15]

Hatta and M

Y. Hatta and M. A. Stephanov,Proton number ﬂuctuation as a signal of the QCD critical endpoint, Phys. Rev. Lett.91 (2003) 102003 [hep-ph/0302002]

2003 arXiv
[16]

M. A. Stephanov,Non-Gaussian ﬂuctuations near the QCD critical point, Phys. Rev. Lett. 102 (2009) 032301 [0809.3450]

2009 arXiv
[17]

Athanasiou, K

C. Athanasiou, K. Rajagopal and M. Stephanov,Using Higher Moments of Fluctuations and their Ratios in the Search for the QCD Critical Point, Phys.Rev. D82 (2010) 074008 [1006.4636]

2010 arXiv
[18]

Brewer, S

J. Brewer, S. Mukherjee, K. Rajagopal and Y. Yin,Searching for the QCD critical point via the rapidity dependence of cumulants, Phys. Rev. C98 (2018) 061901 [1804.10215]. – 38 –

2018 arXiv
[19]

Shen and B

C. Shen and B. Schenke,Dynamical initialization and hydrodynamic modeling of relativistic heavy-ion collisions, Nucl. Phys. A982 (2019) 411 [1807.05141]

2019 arXiv
[20]

Parotto, M

P. Parotto, M. Bluhm, D. Mroczek, M. Nahrgang, J. Noronha-Hostler, K. Rajagopal, C. Ratti, T. Schäfer, and M. Stephanov,Lattice-QCD-based equation of state with a critical point, 1805.05249

arXiv
[21]

Monnai, S

A. Monnai, S. Mukherjee and Y. Yin,Phenomenological Consequences of Enhanced Bulk Viscosity Near the QCD Critical Point, Phys. Rev. C95 (2017) 034902 [1606.00771]

2017 arXiv
[22]

and Schäfer, T

Martinez, M. and Schäfer, T. and Skokov, V.,Critical behavior of the bulk viscosity in QCD, 1906.11306

1906 arXiv
[23]

Berdnikov and K

B. Berdnikov and K. Rajagopal,Slowing out-of-equilibrium near the QCD critical point, Phys.Rev. D61 (2000) 105017 [hep-ph/9912274]

2000 arXiv
[24]

Mukherjee, R

S. Mukherjee, R. Venugopalan and Y. Yin,Real time evolution of non-Gaussian cumulants in the QCD critical regime, Phys. Rev. C92 (2015) 034912 [1506.00645]

2015 arXiv
[25]

Yin,The QCD critical point hunt: emergent new ideas and new dynamics, 1811.06519

Y. Yin,The QCD critical point hunt: emergent new ideas and new dynamics, 1811.06519

arXiv
[27]

J. I. Kapusta and J. M. Torres-Rincon,Thermal Conductivity and Chiral Critical Point in Heavy Ion Collisions, Phys. Rev. C86 (2012) 054911 [1209.0675]

2012 arXiv
[28]

J. I. Kapusta and C. Plumberg,Causal Electric Charge Diﬀusion and Balance Functions in Relativistic Heavy Ion Collisions, Phys. Rev. C97 (2018) 014906 [1710.03329]

2018 arXiv
[29]

Sakaida, M

M. Sakaida, M. Asakawa, H. Fujii and M. Kitazawa,Dynamical evolution of critical ﬂuctuations and its observation in heavy ion collisions, Phys. Rev. C95 (2017) 064905 [1703.08008]

2017 arXiv
[30]

Nahrgang, Marlene and Bluhm, Marcus and Schäfer, Thomas and Bass, Steﬀen A.,Diﬀusive dynamics of critical ﬂuctuations near the QCD critical point, Phys. Rev. D99 (2019) 116015 [1804.05728]

2019 arXiv
[31]

Akamatsu, A

Y. Akamatsu, A. Mazeliauskas and D. Teaney,A kinetic regime of hydrodynamic ﬂuctuations and long time tails for a Bjorken expansion, Phys. Rev. C95 (2017) 014909 [1606.07742]

2017 arXiv
[32]

Murase and T

K. Murase and T. Hirano,Hydrodynamic ﬂuctuations and dissipation in an integrated dynamical model, Nucl. Phys. A956 (2016) 276 [1601.02260]

2016 arXiv
[33]

Hirano, R

T. Hirano, R. Kurita and K. Murase,Hydrodynamic ﬂuctuations of entropy in one-dimensionally expanding system, Nucl. Phys. A984 (2019) 44 [1809.04773]

2019 arXiv
[34]

Singh, C

M. Singh, C. Shen, S. McDonald, S. Jeon and C. Gale,Hydrodynamic Fluctuations in Relativistic Heavy-Ion Collisions, Nucl. Phys. A982 (2019) 319 [1807.05451]

2019 arXiv
[35]

Nahrgang, S

M. Nahrgang, S. Leupold and M. Bleicher,Equilibration and relaxation times at the chiral phase transition including reheating, Phys. Lett. B711 (2012) 109 [1105.1396]

2012 arXiv
[36]

Nahrgang, S

M. Nahrgang, S. Leupold, C. Herold and M. Bleicher,Nonequilibrium chiral ﬂuid dynamics including dissipation and noise, Phys. Rev. C84 (2011) 024912 [1105.0622]

2011 arXiv
[37]

Herold, M

C. Herold, M. Nahrgang, Y. Yan and C. Kobdaj,Dynamical net-proton ﬂuctuations near a QCD critical point, Phys. Rev. C93 (2016) 021902 [1601.04839]. – 39 –

2016 arXiv
[38]

Herold, A

C. Herold, A. Kittiratpattana, C. Kobdaj, A. Limphirat, Y. Yan, M. Nahrgang et al., Entropy production and reheating at the chiral phase transition, Phys. Lett. B790 (2019) 557 [1810.02504]

2019 arXiv
[39]

X. An, G. Basar, M. Stephanov and H.-U. Yee,Relativistic Hydrodynamic Fluctuations, 1902.09517

1902 arXiv
[40]

Stephanov,On the sign of kurtosis near the QCD critical point, Phys.Rev.Lett

M. Stephanov,On the sign of kurtosis near the QCD critical point, Phys.Rev.Lett. 107 (2011) 052301 [1104.1627]

2011 arXiv
[41]

Baier and P

R. Baier and P. Romatschke,Causal viscous hydrodynamics for central heavy-ion collisions, Eur. Phys. J.C51 (2007) 677 [nucl-th/0610108]

2007 arXiv
[42]

Baier, P

R. Baier, P. Romatschke and U. A. Wiedemann,Dissipative hydrodynamics and heavy ion collisions, Phys. Rev. C73 (2006) 064903 [hep-ph/0602249]

2006 arXiv
[43]

Baier, P

R. Baier, P. Romatschke, D. T. Son, A. O. Starinets and M. A. Stephanov,Relativistic viscous hydrodynamics, conformal invariance, and holography, JHEP 04 (2008) 100 [0712.2451]

2008 arXiv
[44]

J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu and U. Heinz,Applying Bayesian parameter estimation to relativistic heavy-ion collisions: simultaneous characterization of the initial state and quark-gluon plasma medium, Phys. Rev. C94 (2016) 024907 [1605.03954]

2016 arXiv
[45]

Romatschke and U

P. Romatschke and U. Romatschke,Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2019, 10.1017/9781108651998, [1712.05815]

2019 arXiv
[46]

Romatschke,Causal viscous hydrodynamics for central heavy-ion collisions

P. Romatschke,Causal viscous hydrodynamics for central heavy-ion collisions. II. Meson spectra and HBT radii, Eur. Phys. J.C52 (2007) 203 [nucl-th/0701032]

2007 arXiv
[47]

Onuki,Phase Transition Dynamics

A. Onuki,Phase Transition Dynamics. Cambridge University Press, 2002

2002
[48]

Guida and J

R. Guida and J. Zinn-Justin,3-D Ising model: The Scaling equation of state, Nucl. Phys. B489 (1997) 626 [hep-th/9610223]

1997 arXiv
[49]

Son and M

D. Son and M. Stephanov,Dynamic universality class of the QCD critical point, Phys.Rev. D70 (2004) 056001 [hep-ph/0401052]

2004 arXiv
[50]

Fujii and M

H. Fujii and M. Ohtani,Soft modes at the critical end point in the chiral eﬀective models, Prog. Theor. Phys. Suppl.153 (2004) 157 [hep-ph/0401028]

2004 arXiv
[51]

P. C. Hohenberg and B. I. Halperin,Theory of dynamic critical phenomena, Rev. Mod. Phys. 49 (1977) 435

1977
[52]

Kardar,Statistical Physics of Particles

M. Kardar,Statistical Physics of Particles. Cambridge University Press, 2007

2007
[53]

G. S. Denicol, C. Gale, S. Jeon, A. Monnai, B. Schenke and C. Shen,Net baryon diﬀusion in ﬂuid dynamic simulations of relativistic heavy-ion collisions, Phys. Rev. C98 (2018) 034916 [1804.10557]

2018 arXiv
[54]

Du and U

L. Du and U. Heinz,(3+1)-dimensional dissipative relativistic ﬂuid dynamics at non-zero net baryon density, 1906.11181

1906 arXiv
[55]

Shen and B

C. Shen and B. Schenke,Dynamical initial state model for relativistic heavy-ion collisions, Phys. Rev. C97 (2018) 024907 [1710.00881]. – 40 –

2018 arXiv

Pith tools

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

[1] [26]

Stephanov and Y

M. Stephanov and Y. Yin,Hydrodynamics with parametric slowing down and ﬂuctuations near the critical point, Phys. Rev. D98 (2018) 036006 [1712.10305]

arXiv 2018

[2] [1]

Y. Aoki, G. Endrodi, Z. Fodor, S. D. Katz and K. K. Szabo,The Order of the quantum chromodynamics transition predicted by the standard model of particle physics, Nature 443 (2006) 675 [hep-lat/0611014]

arXiv 2006

[3] [2]

Bazavov et al.,Equation of state and QCD transition at ﬁnite temperature, Phys

A. Bazavov et al.,Equation of state and QCD transition at ﬁnite temperature, Phys. Rev. D80 (2009) 014504 [0903.4379]

arXiv 2009

[4] [3]

Borsanyi, G

S. Borsanyi, G. Endrodi, Z. Fodor, A. Jakovac, S. D. Katz, S. Krieg et al.,The QCD equation of state with dynamical quarks, JHEP 11 (2010) 077 [1007.2580]

arXiv 2010

[5] [4]

Bazavov et al.,The chiral and deconﬁnement aspects of the QCD transition, Phys

A. Bazavov et al.,The chiral and deconﬁnement aspects of the QCD transition, Phys. Rev. D85 (2012) 054503 [1111.1710]

arXiv 2012

[6] [5]

Berges and K

J. Berges and K. Rajagopal,Color superconductivity and chiral symmetry restoration at nonzero baryon density and temperature, Nucl. Phys. B538 (1999) 215 [hep-ph/9804233]

arXiv 1999

[7] [6]

A. M. Halasz, A. D. Jackson, R. E. Shrock, M. A. Stephanov and J. J. M. Verbaarschot,On the phase diagram of QCD, Phys. Rev. D58 (1998) 096007 [hep-ph/9804290]

arXiv 1998

[8] [7]

M. A. Stephanov, K. Rajagopal and E. V. Shuryak,Signatures of the tricritical point in QCD, Phys. Rev. Lett.81 (1998) 4816 [hep-ph/9806219]

arXiv 1998