Pith. sign in

REVIEW 2 major objections 5 minor 2 cited by

Far from equilibrium hydrodynamics of nonthermal fixed points

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nonthermal fixed points support hydrodynamic excitations, with a shear viscosity that grows linearly in time.

desk verdict A serious, transparent proposal for hydrodynamics around nonthermal fixed points; the untested 14-moment closure for inhomogeneous modes is the key soft spot. read the letter →

arxiv 2504.18754 v1 pith:ALKV27UG submitted 2025-04-26 hep-th cond-mat.quant-gas

Jürgen Berges , Gabriel S. Denicol , Michal P. Heller , Thimo Preis This is my paper · ORCID
classification hep-thcond-mat.quant-gas
keywords nonthermalfixedpointsfar-from-equilibriumhydrodynamicsshearviscosityrelaxationtime14-momentapproximationrelativistickinetictheorydirectenergycascadeQCD
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 proposes that nonthermal fixed points—the self-similar, far-from-equilibrium cascades seen in quark-gluon plasma, early-universe cosmology, and cold atomic gases—support hydrodynamic excitations. Working in relativistic kinetic theory for massless particles with binary collisions, the authors expand about an isotropic nonthermal fixed point rather than about local equilibrium and derive a closed, relaxation-type equation for shear-stress perturbations, with a shear viscosity $\eta(t)$ and relaxation time $\tau_\pi(t)$ that are intrinsically time dependent. For the direct energy cascade both coefficients grow linearly in time, $\eta(t),\tau_\pi(t)\propto (t-t_*)/t_{\rm ref}$, while the dimensionless combination $\eta/[\tau_\pi(E+P)]$ stays fixed at $1/5$. If correct, this gives nonthermal fixed points transport coefficients, predicts power-law rather than exponential relaxation of pressure anisotropy, and opens the way to measuring far-from-equilibrium viscosity in cold-atom experiments.

What carries the argument

The load-bearing object is the generalized 14-moment approximation taken around an isotropic nonthermal fixed point instead of local equilibrium: writing the one-particle distribution as $f_k=f_{0k}+\delta f_k$ with $\delta f_k = f_{0k}\,(2p^\mu p^\nu)/(15 N_4(t))\,\pi_{\mu\nu}(t)$ and $N_4(t)=\int dK\,|\mathbf{k}|^4 f_{0k}$ closes the exact but unclosed shear-stress equation of motion through a linearized collision kernel $\delta C(t)$. That kernel determines the time-dependent transport coefficients $\eta(t)=\frac{8}{15^2} E N_4(t)/\delta C(t)$ and $\tau_\pi(t)=\frac{2}{15} N_4(t)/\delta C(t)$. A scaling analysis of $\delta C(t)$ on the self-similar fixed-point distribution then yields the linear time growth $\eta(t),\tau_\pi(t)\propto(t-t_*)/t_{\rm ref}$ for the direct energy cascade.

What would settle it

A spatially inhomogeneous simulation of the full Boltzmann equation around the relativistic direct-energy-cascade fixed point, measuring the shear-channel response at fixed momentum $q$, would settle the claim: if the rescaled dispersion relation $\bar\Omega(\bar q)$ from Eqs. (31)–(34) is not reproduced, or if the long-time shear amplitude decays exponentially rather than as $\exp[-c q^2 (t-t_*)^2]$, the hydrodynamic description fails.

Watch

Extended reading notes

Core claim

The paper claims that spatial inhomogeneities near a nonthermal fixed point obey the same kind of hydrodynamic equations that describe small departures from thermal equilibrium, provided the background distribution used for the expansion is the self-similar nonthermal fixed point rather than local equilibrium. Concretely, the shear-stress perturbation satisfies $\tau_\pi(t)\dot{\pi}^{\langle\mu\nu\rangle}=-\pi^{\mu\nu}+2\eta(t)\sigma^{\mu\nu}+\tau_\pi(t)(-\tfrac{4}{3}\pi^{\mu\nu}\theta-\tfrac{10}{7}\pi^{\langle\mu}_{\ \ \lambda}\sigma^{\nu\rangle\lambda}-2\pi^{\langle\mu}_{\ \ \lambda}\omega^{\nu\rangle\lambda})$, with $\eta(t)$ and $\tau_\pi(t)$ intrinsically time dependent. On the relativistic direct energy cascade (scaling exponents $\alpha=-4/7$, $\beta=-1/7$), both coefficients grow linearly with time, $\eta(t),\tau_\pi(t)=\bar\eta,\bar\tau_\pi\,(t-t_*)/t_{\rm ref}$, so the ratio $\eta(t)/[\tau_\pi(t)(E+P)] = 1/5$ is time independent. The authors corroborate this with QCD kinetic-theory simulations showing the same scaling of the transport coefficients and a power-law, rather than exponential, relaxation of the pressure anisotropy after an anisotropic perturbation, and they show the same linear growth for the non-relativistic direct energy cascade.

Load-bearing premise

The derivation hinges on assuming that the 14-moment truncation—writing the deviation from the isotropic nonthermal distribution solely in terms of the shear-stress tensor—remains an accurate closure around a nonthermal fixed point, even though no small-parameter separation of scales has been demonstrated and only the homogeneous isotropization channel, not the inhomogeneous modes, has been tested directly.

Editorial extensions

If this is right

  • Pressure anisotropy near the fixed point decays as a power law, $\Delta P(t)\sim [(t_0-t_*)/(t-t_*)]^{t_{\rm ref}/\bar\tau_\pi}$, instead of the exponential decay seen near equilibrium.
  • Shear perturbations at small spatial momentum $q$ become arbitrarily long-lived as $q\to 0$, diffusing as $\exp[- (3\bar\eta/4E)\, q^2 ((t-t_*)/t_{\rm ref})^2]$ rather than with the usual $e^{-D q^2 t}$.
  • Sound waves still propagate at speed $1/\sqrt3$ around the fixed point, but their attenuation is governed by the same time-dependent transport coefficients.
  • The time-independent ratio $\eta/[\tau_\pi(E+P)] = 1/5$ provides a dimensionless transport benchmark that connects far-from-equilibrium behavior to the near-equilibrium and strong-coupling values for the same underlying theory.
  • In QCD kinetic theory, the extracted $\eta(t)\sim t$ and $\tau_\pi(t)\sim t$ scaling, together with power-law isotropization, are consistent with the fixed-point predictions even though inelastic $1\leftrightarrow 2$ processes are present.

Reading between the lines

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

  • If the 14-moment truncation is relaxed in the full Boltzmann equation, the exact dispersion relations may shift, but the noncommutativity of the large-time and small-momentum limits should persist, implying a finite time window of hydrodynamics at any fixed nonzero $q$; a large-volume cold-atom experiment could look for this breakdown.
  • The same construction should apply to the anisotropic nonthermal fixed point relevant to early-time heavy-ion collisions, potentially linking this far-from-equilibrium hydrodynamics to anisotropic-hydrodynamics descriptions used there.
  • A direct experimental test would be to imprint an anisotropic perturbation on a cold-atom system realizing the non-relativistic direct energy cascade and measure the pressure-anisotropy relaxation; a power-law rather than exponential decay with the predicted exponent would confirm the picture.
  • The time-independence of $\eta/[\tau_\pi(E+P)]$, in contrast to the time dependence of $\eta/s$, suggests that ratios of transport coefficients, rather than entropy-normalized viscosity, may be the more robust universal observables for comparing far-from-equilibrium systems across couplings.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proposes that long-wavelength spatial perturbations around isotropic nonthermal fixed points are governed by relativistic hydrodynamic equations with intrinsically time-dependent transport coefficients. Starting from the Boltzmann equation and taking the background f_0k to be an isotropic nonthermal fixed-point distribution rather than local equilibrium, the authors use the 14-moment approximation to close the moment hierarchy and derive a relaxation equation for the shear stress, Eq. (18), with shear viscosity eta(t) and relaxation time tau_pi(t) given in Eq. (19). For the direct energy cascade these coefficients grow linearly with time, Eqs. (26), while the ratio eta/(tau_pi(E+P)) equals 1/5, Eq. (20). The paper derives scaling forms for the transport coefficients, analyses shear and sound dispersion relations in the rescaled variables of Eq. (31), and compares the predicted homogeneous isotropization and the scaling of eta and tau_pi against QCD kinetic theory simulations. It also extends the scaling analysis to the nonrelativistic direct energy cascade relevant to cold atoms.

Significance. If the central claim holds, the paper establishes a genuinely new bridge between nonthermal fixed points and fluid dynamics, with falsifiable consequences for cold-atom experiments and possible implications for early-universe and heavy-ion physics. The derivation is self-contained in kinetic theory: the transport coefficients are evaluated from the linearized collision kernel, the ratio in Eq. (20) follows from the definitions without fitting, and the scaling analysis in Appendix C is explicit. The paper is also appropriately conservative in several places, noting that the QCD comparison is qualitative and that fully inhomogeneous ab initio tests remain future work. The main weakness is that the load-bearing 14-moment closure is assumed rather than derived for the nonthermal background, and the numerical support in Sec. IV is limited to homogeneous observables. The reported work is a plausible and interesting proposal whose central quantitative predictions require additional validation.

major comments (2)
  1. [§II.B, Eq. (15)/(B8), Appendix B] The central derivation closes the exact moment equation (B5) by assuming the 14-moment form delta f_k = f_0k (2 p^mu p^nu / (15 N_4(t))) pi_mu nu for perturbations around a nonthermal fixed point. Near thermal equilibrium this truncation is controlled by a separation between slow hydrodynamic modes and rapidly relaxing nonhydrodynamic modes; the manuscript's only stated justification for its nonthermal counterpart is the isotropy of f_0k, which is necessary but not sufficient. The background is itself time dependent, and Ref. [31] is cited as finding a tower of power-law decaying modes at q=0 rather than an exponential gap, so the dominance of the single pi^mu nu channel is not established. Because Eq. (18), the transport coefficients (19), and the dispersion relations (32)-(34) all rest on this closure, the paper should either supply a test of the truncation for inhomogeneous perturbations, for example a linearized full kinetic-theory computation along the lines suggested in Sec. VI, or explicitly frame the main claim as a conjecture pending such a test.
  2. [§IV, Figs. 1-2] The ab initio support in Sec. IV does not directly test the central claim about spatially inhomogeneous hydrodynamic modes. Fig. 2 compares homogeneous isotropization with Eq. (29), which is the q=0 sector of the approximated equations, and Fig. 1 checks only the scaling of the ingredients entering Eqs. (19); neither exercise tests the q-dependent shear and sound modes that carry the paper's main prediction. Moreover, the QCD simulation includes inelastic 1-to-2 processes and a different transition matrix element than the elastic 2-to-2 scalar collision kernel used in the derivation, a point the authors acknowledge, so the comparison is qualitative. A quantitative validation of Eqs. (32)-(34) with a broken-homogeneity kinetic-theory simulation would substantially strengthen the paper.
minor comments (5)
  1. [§III.B, after Eq. (32)] Please verify the exponent of the transient shear mode: from the homogeneous solution (29), the transient decay should scale as ((t-t*)/t_ref)^(-t_ref/tau_bar_pi), whereas the text appears to print the inverted exponent -tau_bar_pi/t_ref.
  2. [Eqs. (10) and (37)] The gain-loss terms in the collision integrals have unbalanced parentheses; for example Eq. (10) reads "(f_p f_p' (f_k + f_k') - f_k f_k' (f_p + f_p')" with an opening parenthesis that is never closed.
  3. [Fig. 1 caption] The sentence "delta C is evaluated using 2x10^9 MC samples at every 100th time step (Q dt = 0.1, or every t Q = 10 steps)" is ambiguous; please specify the time-step size and the sampling interval precisely.
  4. [Eq. (29) and Sec. III.A] The exponent t_ref/tau_bar_pi in Eq. (29) is dimensionless, but the notation invites confusion with the dimensionful bar_tau_pi; a brief definition of the dimensions of bar_tau_pi and bar_eta would improve readability.
  5. [Reference [22]] Reference [22] contains an "https://" URL inside the title field and would benefit from standard journal citation formatting.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hydrodynamic equations and transport coefficients are derived from the 14-moment closure and the linearized collision kernel; the QCD simulation is an external benchmark and the acknowledged 14-moment truncation is a validity concern, not a circular step.

full rationale

The derivation chain is self-contained in kinetic theory. Starting from the Boltzmann equation, Eq. (B5) is the exact, unclosed moment equation; it is closed by the explicit 14-moment ansatz δf_k = f_{0k} (2 p^μ p^ν)/(15 N_4(t)) π_{μν} (Eqs. (15)/(B8)), which the paper states is a truncation, not a derived theorem. The transport coefficients η(t) and τ_π(t) are then functionals of the background f_0 and the linearized collision kernel δC (Eqs. (19)/(B15) and (B13)); no parameter is fitted to the target dispersion relations. The ratio η/(τ_π(E+P)) = 1/5 follows algebraically from those definitions and isotropy, so it is an identity of the moment method rather than a fitted prediction. The QCD kinetic-theory comparison in Sec. IV is an external benchmark: δC is evaluated from the simulated gluon distribution using the QCD matrix element, and the only adjusted quantities are the non-universal t* and a reference amplitude, explicitly stated as such. The paper candidly flags the load-bearing nature of the 14-moment closure in Sec. VI ('The most pressing one would be going beyond the 14-moment approximation') and notes that inhomogeneous ab initio tests are future work; that is an uncontrolled-approximation caveat, not circularity. The self-citations [30,31] (overlapping authors) are used for motivation and consistency regarding stability and power-law relaxation, but the central derivation does not import their results: removing them would not change Eqs. (18)-(20) or the dispersion relations. No circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claim rests on kinetic theory in the overoccupied limit, isotropy of the background scaling distribution, and the unvalidated 14-moment closure; the linear time dependence ultimately inherits the previously computed scaling exponents α and β. No free parameters are introduced in the derivation itself, though t* and overall amplitudes are adjusted in the QCD comparison.

free parameters (2)
  • Nonuniversal time offset t* = t* Q ≈ 1.3 × 10^3 in QCD comparison
    Appears in the power-law time dependence (t - t*)^(-1) in Eq. (29) and is extracted from the QCD kinetic theory data at a reference time to account for prescaling effects; it does not enter the central derivation.
  • Overall amplitudes \bar η and \bar τπ = extracted from bulk observables at an arbitrary reference time
    Dimensionful prefactors in Eqs. (26) and (40) are set by evaluating the 14-moment formulas on the scaling function; they are normalizations, not used to derive the time dependence.
assumptions (6)
  • domain assumption The nonthermal fixed point is described by the overoccupied classical-wave Boltzmann equation with binary collisions and O(N) or QCD transition matrix elements (Eqs. (9)-(11)).
    Underlies the whole kinetic-theory calculation; the paper focuses on weakly coupled scale-invariant systems in the overoccupied limit f >> 1.
  • domain assumption The background f0k is isotropic in the local rest frame (near-isotropic scaling distribution).
    The 14-moment matching (B8) requires isotropy of f0k; the paper states it specializes to near-isotropic systems.
  • ad hoc to paper The 14-moment truncation closes the moment hierarchy: δfk = f0k 2 p^μ p^ν/(15 N4) πμν (Eq. (15)).
    No small-parameter justification is given for the truncation around a nonthermal fixed point; the authors call Eq. (18) an effective approximation.
  • domain assumption Scaling exponents α = -4/7 and β = -1/7 for the relativistic energy cascade (Eq. (25)).
    Taken from prior literature [47,63-65]; used to derive η(t) ~ t. They are inputs, not derived here.
  • domain assumption The nonthermal fixed point is an attractor whose stability and local conservation laws hold.
    Hydrodynamic relaxation to the fixed point presupposes attractiveness and stability, supported by simulations and [30,31].
  • domain assumption Bulk viscosity and heat flow are neglected (conformal system with no conserved charge).
    Stated in Sec. I; P = E/3 and charge dynamics are neglected.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Far from equilibrium hydrodynamics of nonthermal fixed points." pith.science (2026). https://pith.science/paper/ALKV27UG

@misc{pith2026250418754,
  author       = {Pith},
  title        = {Pith review of: Far from equilibrium hydrodynamics of nonthermal fixed points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ALKV27UG}},
  note         = {Machine review of arXiv:2504.18754}
}
read the original abstract

Nonthermal fixed points are paradigmatic far-from-equilibrium phenomena of relevance to high-energy physics, cosmology, and cold atomic gases. We propose that, despite their intrinsically nonequilibrium nature, nonthermal fixed points give rise to hydrodynamic excitations otherwise known in the vicinity of thermal equilibrium. As a result, nonthermal fixed points can also be characterized by transport coefficients, such as a far-from-equilibrium, and therefore manifestly time-dependent, incarnation of shear viscosity. We corroborate our proposal with explicit studies using relativistic kinetic theory with binary collisions of massless particles in the 14-moment approximation and comparisons to QCD kinetic theory simulations.

Figures

Figures reproduced from arXiv: 2504.18754 by the authors.

Figure 1
Figure 1. FIG. 1. Time evolution of shear viscosity ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Evolution of the pressure anisotropy after an [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Stochastic inflation from a non-equilibrium renormalization group

    hep-th 2026-05 unverdicted novelty 7.0 of 10

    Stochastic inflation is the leading infrared limit of a coarse-grained Schwinger–Keldysh effective theory, and the same Fokker–Planck dynamics follows from a Polchinski-type renormalization-group flow for the reduced ...

  2. Attractodynamics in 0+1D

    hep-ph 2026-07 conditional novelty 6.0 of 10

    Attractodynamics constructs a macroscopic closure around a nonthermal attractor; in the 0+1D BSY kinetic model, ideal and viscous truncations track the full kinetic evolution, with the viscous extension improving agre...

Reference graph

Works this paper leans on

92 extracted references · 4 canonical work pages · cited by 2 Pith papers

  1. [31]

    Martirosyan, M

    G. Martirosyan, M. Gazo, J. Etrych, S. M. Fischer, S. J. Morris, C. J. Ho, C. Eigen, and Z. Hadzibabic, (2024), arXiv:2410.08204 [cond-mat.quant-gas]

    arXiv 2024

  2. [1]

    Andersson and G

    N. Andersson and G. L. Comer, Living Rev. Rel. 10, 1 (2007), arXiv:gr-qc/0605010

    arXiv 2007

  3. [2]

    (8) The combination appearing on the left-hand side of the above inequality will play an important role later in the paper. The equations of motion for relativistic hydrodynam- ics of the type (6) can be derived from the Boltzmann equation [54], which provides a microscopic description of dilute gases in terms of the single particle momen- tum distributio...

  4. [3]

    Rezzolla and O

    L. Rezzolla and O. Zanotti, Relativistic Hydrodynamics (Oxford University Press, 2013)

    2013

  5. [4]

    Sch¨ afer and D

    T. Sch¨ afer and D. Teaney, Rept. Prog. Phys.72, 126001 (2009), arXiv:0904.3107 [hep-ph]

    arXiv 2009

  6. [5]

    The Many Faces of Relativistic Fluid Dynam- ics

    (20) Since the derivation behind (19) relied only on the isotropy of f0k, the expression (20) applies also to the standard near-equilibrium hydrodynamics. This indi- cates a degree of universality between the equilibrium and nonthermal fixed point hydrodynamics for the same underlying weakly-coupled quantum field theory. In holography at infinite coupling...

    2020

  7. [6]

    Busza, K

    W. Busza, K. Rajagopal, and W. van der Schee, Ann. Rev. Nucl. Part. Sci. 68, 339 (2018), arXiv:1802.04801 [hep-ph]

    arXiv 2018

  8. [7]

    L. D. Landau, E. M. Lifshitz, J. B. Sykes, and W. H. Reid, Fluid Mechanics (Pergamon Press Oxford, Eng- land, 1959)

    1959

Show all 92 references
  1. [8]

    M. P. Heller and M. Spalinski, Phys. Rev. Lett. 115, 072501 (2015), arXiv:1503.07514 [hep-th]

    2015 arXiv

  2. [9]

    Romatschke, Phys

    P. Romatschke, Phys. Rev. Lett. 120, 012301 (2018), arXiv:1704.08699 [hep-th]

    2018 arXiv

  3. [10]

    Fujii and T

    K. Fujii and T. Enss, Phys. Rev. Lett. 133, 173402 (2024), arXiv:2404.12921 [cond-mat.quant-gas]

    2024 arXiv

  4. [11]

    Mazeliauskas and T

    A. Mazeliauskas and T. Enss, (2025), arXiv:2501.19240 [cond-mat.quant-gas]

    2025

  5. [12]

    Soloviev, Eur

    A. Soloviev, Eur. Phys. J. C 82, 319 (2022), arXiv:2109.15081 [hep-th]

    2022 arXiv

  6. [13]

    Jankowski and M

    J. Jankowski and M. Spali´ nski, Prog. Part. Nucl. Phys. 132, 104048 (2023), arXiv:2303.09414 [nucl-th]

    2023 arXiv

  7. [14]

    Micha and I

    R. Micha and I. I. Tkachev, Phys. Rev. Lett. 90, 121301 (2003), arXiv:hep-ph/0210202

    2003 arXiv

  8. [15]

    Berges, A

    J. Berges, A. Rothkopf, and J. Schmidt, Phys. Rev. Lett. 101, 041603 (2008), arXiv:0803.0131 [hep-ph]

    2008 arXiv

  9. [16]

    Berges, K

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Phys. Rev. D 89, 074011 (2014), arXiv:1303.5650 [hep-ph]

    2014 arXiv

  10. [17]

    Kurkela and Y

    A. Kurkela and Y. Zhu, Phys. Rev. Lett. 115, 182301 (2015), arXiv:1506.06647 [hep-ph]

    2015 arXiv

  11. [18]

    Berges, M

    J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan, Rev. Mod. Phys. 93, 035003 (2021), arXiv:2005.12299 [hep-th]

    2021 arXiv

  12. [19]

    A. N. Mikheev, I. Siovitz, and T. Gasenzer, Eur. Phys. J. ST 232, 3393 (2023), arXiv:2304.12464 [cond-mat.quant- gas]

    2023 arXiv

  13. [20]

    Pr¨ ufer, P

    M. Pr¨ ufer, P. Kunkel, H. Strobel, S. Lannig, D. Linne- mann, C.-M. Schmied, J. Berges, T. Gasenzer, and M. K. Oberthaler, Nature 563, 217 (2018), arXiv:1805.11881 [cond-mat.quant-gas]

    2018 arXiv

  14. [21]

    S. Erne, R. B¨ ucker, T. Gasenzer, J. Berges, and J. Schmiedmayer, Nature 563, 225 (2018), arXiv:1805.12310 [cond-mat.quant-gas]

    2018 arXiv

  15. [22]

    J. A. P. Glidden, C. Eigen, L. H. Dogra, T. A. Hilker, R. P. Smith, and Z. Hadzibabic, Nature Phys. 17, 457 (2021), arXiv:2006.01118 [cond-mat.quant-gas]

    2021 arXiv

  16. [23]

    Navon, A

    N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadz- ibabic, Nature 539, 72 (2016), arXiv:1609.01271 [cond- mat.quant-gas]

    2016 arXiv

  17. [24]

    S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula, and K. Helmerson, Science 364, 1267 (2019), ”https://www.science.org/doi/pdf/10.1126/science.aat5793”

    2019 doi

  18. [25]

    Helmrich, A

    S. Helmrich, A. Arias, G. Lochead, T. Wintermantel, M. Buchhold, S. Diehl, and S. Whitlock, Nature 577, 481 (2020), arXiv:1806.09931 [cond-mat.quant-gas]

    2020 arXiv

  19. [26]

    A. D. Garc´ ıa-Orozco, L. Madeira, M. A. Moreno-Armijos, A. R. Fritsch, P. E. S. Tavares, P. C. M. Castilho, A. Cidrim, G. Roati, and V. S. Bagnato, Phys. Rev. A 106, 023314 (2022), arXiv:2107.07421 [cond-mat.quant- gas]

    2022 arXiv

  20. [27]

    Martirosyan, C

    G. Martirosyan, C. J. Ho, J. Etrych, Y. Zhang, A. Cao, Z. Hadzibabic, and C. Eigen, (2023), arXiv:2304.06697 [cond-mat.quant-gas]

    2023 arXiv

  21. [28]

    S. Huh, K. Mukherjee, K. Kwon, J. Seo, S. I. Mis- takidis, H. R. Sadeghpour, and J.-y. Choi, (2023), arXiv:2303.05230 [cond-mat.quant-gas]

    2023 arXiv

  22. [29]

    Lannig, M

    S. Lannig, M. Pr¨ ufer, Y. Deller, I. Siovitz, J. Dreher, T. Gasenzer, H. Strobel, and M. K. Oberthaler, (2023), arXiv:2306.16497 [cond-mat.quant-gas]

    2023 arXiv

  23. [30]

    M. Gazo, A. Karailiev, T. Satoor, C. Eigen, M. Ga lka, and Z. Hadzibabic, (2023), arXiv:2312.09248 [cond- mat.quant-gas]

    2023 arXiv

  24. [32]

    Preis, M

    T. Preis, M. P. Heller, and J. Berges, Phys. Rev. Lett. 130, 031602 (2023), arXiv:2209.14883 [hep-ph]

    2023 arXiv

  25. [33]

    De Lescluze and M

    M. De Lescluze and M. P. Heller, (2025), arXiv:2502.01622 [hep-th]

    2025 arXiv

  26. [34]

    P. K. Kovtun and A. O. Starinets, Phys. Rev. D 72, 086009 (2005), arXiv:hep-th/0506184

    2005 arXiv

  27. [35]

    S. A. Hartnoll and S. P. Kumar, JHEP 12, 036 (2005), arXiv:hep-th/0508092

    2005 arXiv

  28. [36]

    Romatschke, Eur

    P. Romatschke, Eur. Phys. J. C 76, 352 (2016), arXiv:1512.02641 [hep-th]

    2016 arXiv

  29. [37]

    Grozdanov, N

    S. Grozdanov, N. Kaplis, and A. O. Starinets, JHEP 07, 151 (2016), arXiv:1605.02173 [hep-th]

    2016 arXiv

  30. [38]

    Kurkela and U

    A. Kurkela and U. A. Wiedemann, Eur. Phys. J. C 79, 776 (2019), arXiv:1712.04376 [hep-ph]

    2019 arXiv

  31. [39]

    G. D. Moore, JHEP 05, 084 (2018), arXiv:1803.00736 [hep-ph]

    2018 arXiv

  32. [40]

    Ochsenfeld and S

    S. Ochsenfeld and S. Schlichting, JHEP 09, 186 (2023), arXiv:2308.04491 [hep-th]

    2023 arXiv

  33. [41]

    Brants, Phys

    R. Brants, Phys. Rev. D 110, 116027 (2024), arXiv:2409.09022 [hep-th]

    2024 arXiv

  34. [42]

    Kovtun, D

    P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. 10 Lett. 94, 111601 (2005), arXiv:hep-th/0405231

    2005 arXiv

  35. [43]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 01, 030 (2003), arXiv:hep-ph/0209353

    2003 arXiv

  36. [44]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe, JHEP 05, 051 (2003), arXiv:hep-ph/0302165

    2003 arXiv

  37. [45]

    Mazeliauskas and J

    A. Mazeliauskas and J. Berges, Phys. Rev. Lett. 122, 122301 (2019), arXiv:1810.10554 [hep-ph]

    2019 arXiv

  38. [46]

    G. S. Denicol and D. H. Rischke, Microscopic foundations of relativistic fluid dynamics , Lecture notes in physics, Vol. 990 (Springer, 2021)

    2021

  39. [47]

    G. S. Rocha, D. Wagner, G. S. Denicol, J. Noronha, and D. H. Rischke, Entropy26, 189 (2024), arXiv:2311.15063 [nucl-th]

    2024 arXiv

  40. [48]

    G. S. Denicol and J. Noronha, Nucl. Phys. A 1005, 121996 (2021)

    2021

  41. [49]

    M. P. Heller, A. Mazeliauskas, and T. Preis, Phys. Rev. Lett. 132, 071602 (2024), arXiv:2307.07545 [hep-th]

    2024 arXiv

  42. [50]

    In this work, we neglect the dynamics of any conserved charge

  43. [51]

    Landau and E

    L. Landau and E. Lifshitz, Press Oxford: London, UK (1959)

    1959

  44. [52]

    Israel and J

    W. Israel and J. M. Stewart, Annals of Physics 118, 341 (1979)

    1979

  45. [53]

    C. Gale, S. Jeon, and B. Schenke, Int. J. Mod. Phys. A 28, 1340011 (2013), arXiv:1301.5893 [nucl-th]

    2013 arXiv

  46. [54]

    Paquet, J

    J.-F. Paquet, J. Phys. G 51, 103001 (2024), arXiv:2310.17618 [nucl-th]

    2024 arXiv

  47. [55]

    S. Pu, T. Koide, and D. H. Rischke, Phys. Rev. D 81, 114039 (2010)

    2010

  48. [56]

    de Groot, W

    S. de Groot, W. van Leeuwen, and C. van Weert, Relativistic Kinetic Theory: Principles and Applications (North-Holland Publishing Co., 1980)

    1980

  49. [57]

    Pi˜ neiro Orioli, K

    A. Pi˜ neiro Orioli, K. Boguslavski, and J. Berges, Phys. Rev. D 92, 025041 (2015), arXiv:1503.02498 [hep-ph]

    2015 arXiv

  50. [58]

    G. S. Denicol, T. Koide, and D. H. Rischke, Phys. Rev. Lett. 105, 162501 (2010), arXiv:1004.5013 [nucl-th]

    2010 arXiv

  51. [59]

    Buchel, M

    A. Buchel, M. P. Heller, and R. C. Myers, Phys. Rev. Lett. 114, 251601 (2015), arXiv:1503.07114 [hep-th]

    2015 arXiv

  52. [60]

    R. A. Janik, G. Plewa, H. Soltanpanahi, and M. Spalin- ski, Phys. Rev. D 91, 126013 (2015), arXiv:1503.07149 [hep-th]

    2015 arXiv

  53. [61]

    M. P. Heller, A. Kurkela, M. Spali´ nski, and V. Svensson, Phys. Rev. D 97, 091503 (2018), arXiv:1609.04803 [nucl- th]

    2018 arXiv

  54. [62]

    X. Du, M. P. Heller, S. Schlichting, and V. Svensson, Phys. Rev. D106, 014016 (2022), arXiv:2203.16549 [hep- ph]

    2022 arXiv

  55. [63]

    Micha and I

    R. Micha and I. I. Tkachev, Phys. Rev. D 70, 043538 (2004), arXiv:hep-ph/0403101

    2004 arXiv

  56. [64]

    Kurkela and G

    A. Kurkela and G. D. Moore, Phys. Rev. D 86, 056008 (2012), arXiv:1207.1663 [hep-ph]

    2012 arXiv

  57. [65]

    M. C. Abraao York, A. Kurkela, E. Lu, and G. D. Moore, Phys. Rev. D 89, 074036 (2014), arXiv:1401.3751 [hep- ph]

    2014 arXiv

  58. [66]

    Schlichting, Phys

    S. Schlichting, Phys. Rev. D 86, 065008 (2012), arXiv:1207.1450 [hep-ph]

    2012 arXiv

  59. [67]

    Berges, K

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan, Phys. Rev. D 89, 114007 (2014), arXiv:1311.3005 [hep-ph]

    2014 arXiv

  60. [68]

    P. M. Chesler and L. G. Yaffe, Phys. Rev. Lett. 102, 211601 (2009), arXiv:0812.2053 [hep-th]

    2009 arXiv

  61. [69]

    M. P. Heller, D. Mateos, W. van der Schee, and D. Trancanelli, Phys. Rev. Lett. 108, 191601 (2012), arXiv:1202.0981 [hep-th]

    2012 arXiv

  62. [70]

    M. P. Heller, D. Mateos, W. van der Schee, and M. Tri- ana, JHEP 09, 026 (2013), arXiv:1304.5172 [hep-th]

    2013 arXiv

  63. [71]

    P. M. Chesler and L. G. Yaffe, JHEP 07, 086 (2014), arXiv:1309.1439 [hep-th]

    2014 arXiv

  64. [72]

    Florkowski, M

    W. Florkowski, M. P. Heller, and M. Spalinski, Rept. Prog. Phys. 81, 046001 (2018), arXiv:1707.02282 [hep- ph]

    2018 arXiv

  65. [73]

    Baier, P

    R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, and M. A. Stephanov, JHEP 04, 100 (2008), arXiv:0712.2451 [hep-th]

    2008 arXiv

  66. [74]

    M. A. Moreno-Armijos, A. R. Fritsch, A. D. Garc´ ıa- Orozco, S. Sab, G. Telles, Y. Zhu, L. Madeira, S. Nazarenko, V. I. Yukalov, and V. S. Bagnato, (2024), arXiv:2407.11237 [cond-mat.quant-gas]

    2024 arXiv

  67. [75]

    Adams, L

    A. Adams, L. D. Carr, T. Sch¨ afer, P. Steinberg, and J. E. Thomas, New J. Phys. 14, 115009 (2012), arXiv:1205.5180 [hep-th]

    2012 arXiv

  68. [76]

    Withers, JHEP 06, 059 (2018), arXiv:1803.08058 [hep- th]

    B. Withers, JHEP 06, 059 (2018), arXiv:1803.08058 [hep- th]

    2018 arXiv

  69. [77]

    Grozdanov, P

    S. Grozdanov, P. K. Kovtun, A. O. Starinets, and P. Tadi´ c, Phys. Rev. Lett. 122, 251601 (2019), arXiv:1904.01018 [hep-th]

    2019 arXiv

  70. [78]

    Grozdanov, P

    S. Grozdanov, P. K. Kovtun, A. O. Starinets, and P. Tadi´ c, JHEP11, 097 (2019), arXiv:1904.12862 [hep- th]

    2019 arXiv

  71. [79]

    Keegan, A

    L. Keegan, A. Kurkela, A. Mazeliauskas, and D. Teaney, JHEP 08, 171 (2016), arXiv:1605.04287 [hep-ph]

    2016 arXiv

  72. [80]

    Kurkela, A

    A. Kurkela, A. Mazeliauskas, J.-F. Paquet, S. Schlicht- ing, and D. Teaney, Phys. Rev. Lett.122, 122302 (2019)

    2019

  73. [81]

    Kurkela, A

    A. Kurkela, A. Mazeliauskas, J.-F. Paquet, S. Schlicht- ing, and D. Teaney, Phys. Rev. C 99, 034910 (2019)

    2019

  74. [82]

    M. E. Carrington, S. Mrowczynski, and J.-Y. Ollitrault, Phys. Rev. C 110, 054903 (2024), arXiv:2406.14463 [nucl-th]

    2024 arXiv

  75. [83]

    J. D. Bjorken, Phys. Rev. D 27, 140 (1983)

    1983

  76. [84]

    Baier, A

    R. Baier, A. H. Mueller, D. Schiff, and D. T. Son, Phys. Lett. B 502, 51 (2001), arXiv:hep-ph/0009237

    2001 arXiv

  77. [85]

    Florkowski and R

    W. Florkowski and R. Ryblewski, Phys. Rev. C 83, 034907 (2011), arXiv:1007.0130 [nucl-th]

    2011 arXiv

  78. [86]

    Martinez and M

    M. Martinez and M. Strickland, Nucl. Phys. A 848, 183 (2010), arXiv:1007.0889 [nucl-th]

    2010 arXiv

  79. [87]

    Alqahtani, M

    M. Alqahtani, M. Nopoush, and M. Strickland, Prog. Part. Nucl. Phys. 101, 204 (2018), arXiv:1712.03282 [nucl-th]

    2018 arXiv

  80. [88]

    G. S. Denicol, H. Niemi, E. Molnar, and D. H. Rischke, Phys. Rev. D 85, 114047 (2012), [Erratum: Phys.Rev.D 91, 039902 (2015)], arXiv:1202.4551 [nucl-th]

    2012 arXiv

  81. [89]

    G. S. Denicol and D. H. Rischke, Microscopic Founda- tions of Relativistic Fluid Dynamics (Springer, 2022)

    2022

  82. [90]

    Brewer, B

    J. Brewer, B. Scheihing-Hitschfeld, and Y. Yin, JHEP 05, 145 (2022), arXiv:2203.02427 [hep-ph]. 11 Appendix A: Notation We define the double symmetric traceless projection of a tensor as A⟨µν⟩≡ ∆µναβAαβ, (A1) with ∆µναβ = 1 2(∆µα∆νβ + ∆να∆µβ− 2 3∆µν∆αβ). (A2) We further define...

    2022 arXiv

  83. [91]

    IV in mind

    Relativistic energy cascade We consider first the relativistic energy cascade with the application of QCD kinetic theory in Sec. IV in mind. Starting from Eq. (B13), we evaluate the expressions as δCrel,UV(t) = Z d3k (2π)3ωk d3k′ (2π)3ωk′ d3p (2π)3ωp d3p′ (2π)3ωp′ 1 ωk k2Wkk′→...

  84. [92]

    (37) and Eq

    Non-relativistic energy cascade The kinetic description of a non-relativistic system differs to that of a relativistic one in the appearance of ωk factors in the Boltzmann equation and in the momentum integration measures, see Eq. (37) and Eq. (38). We can then obtain the non-...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.