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Attractodynamics in 0+1D

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Near-attractor dynamics can be organized into a macroscopic theory, like hydrodynamics, built around a far-from-equilibrium attractor rather than local thermal equilibrium.

desk verdict A clean proof of principle for hydrodynamics-like theories around nonthermal attractors; the key truncation-completeness claim is not yet fully tested, and the paper partly says so. read the letter →

arxiv 2607.16393 v1 pith:GDS6QRS7 submitted 2026-07-17 hep-ph hep-thnucl-th

Jean F. Du Plessis , Bruno Scheihing-Hitschfeld , Rachel Steinhorst This is my paper · ORCID
classification hep-phhep-thnucl-th
keywords attractodynamicsfar-from-equilibriumattractorkinetictheoryanisotropichydrodynamicstransientviscousmodesmatchingconditionsboost-invariantdiffusionmodelmomenttruncation
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

Hydrodynamics is a closed macroscopic theory built around local thermal equilibrium. This paper asks whether the same kind of organization can be built around a far-from-equilibrium attractor — a lower-dimensional surface in the space of configurations that many initial conditions approach before thermalizing. The claim is yes: near-attractor dynamics can be captured by a finite set of macroscopic variables, generalized constitutive data, source terms, and, when needed, transient viscous modes. The paper proves this in a 0+1D boost-invariant kinetic model with an exactly known anisotropic Gaussian attractor, deriving an 'ideal' attractodynamic theory that evolves only the attractor chart variables and a 'viscous' extension that adds two transient off-attractor variables. Compared against the full kinetic evolution, the viscous extension agrees longer, until the finite truncation breaks down.

What carries the argument

The central object is the decomposition f = f_A + δf around the exact attractor manifold. The attractor chart variables A, B, C label the Gaussian attractor distribution; matching conditions (three moment conditions that set the δf corrections to number density and the longitudinal/transverse energy-weighted moments to zero) fix the chart and make δf perpendicular to the attractor's tangent directions. The residual δf is parametrized by a five-term ansatz in momentum variables (ζ = p/B, ν = p_z/p) whose coefficients are traded, via matching and the definitions of π and Π, for the two generalized viscous pressures. Projecting the kinetic equation onto the attractor tangent directions and the

What would settle it

Take the linearized kinetic equation around the attractor and compute the relaxation spectrum of modes orthogonal to the five retained moments; if any such mode has a relaxation rate comparable to or slower than the evolution of A, B, C over the window where agreement is claimed, the truncation is not closed. Alternatively, add a sixth moment to the ansatz and check whether the predictions for A, B, C, π, Π change visibly; a strong, slow coupling would falsify the claim that the hierarchy is under control.

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Extended reading notes

Core claim

The paper demonstrates that the attracting manifold itself — not local equilibrium — can serve as the organizing object for a macroscopic effective theory. In the model, the attractor is a Gaussian distribution parametrized by three chart variables A, B, C, with known evolution equations. The distribution is split as f = f_A + δf, with matching conditions fixing A, B, C so that δf carries no number density or attractor-tangent momentum moments. The leading 'ideal' attractodynamics evolves only A, B, C; the 'viscous' extension retains two residual moments, a generalized bulk pressure Π and shear pressure π, which evolve via projected kinetic equations with explicitly computed source integrals

Load-bearing premise

The construction works if the five retained moments (three chart variables plus two viscous variables) exhaust all dynamically slow directions near the attractor — omitted modes must relax fast or be only weakly sourced — and if the model's small energy non-conservation does not distort the benchmark.

Editorial extensions

If this is right

  • Ideal attractodynamics reproduces the full kinetic evolution for initial conditions on or close to the attractor, using only the three chart variables.
  • Adding the two transient viscous variables extends agreement for off-attractor initial data, in the same way transient viscous hydrodynamics extends fluid dynamics.
  • The residual variables behave as independent dynamical fields, not as constitutive corrections, so the hierarchy is improvable by retaining more moments.
  • The construction identifies the ingredients needed for a local 3+1D attractodynamics: chart variables, generalized equations of state in the currents, source terms, and residual relaxation data.
  • Because the additional moment-like variables are not protected by conservation laws, their source terms carry more microscopic information than ordinary hydrodynamics, but still in an organized form.

Reading between the lines

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

  • If this pattern generalizes, attractodynamics could become the pre-hydrodynamic stage in heavy-ion phenomenology, matched directly onto hydrodynamics at a switching time, with the attractor chart supplying the early-time initial conditions.
  • The same organizational principle could apply to any system with an identified nonthermal attractor, such as ultracold atomic gases or cosmological defect networks, once their attractor manifolds are characterized.
  • A sharper test would be to compute the full linearized mode spectrum around the attractor and verify that all modes beyond the retained five relax faster than the chart variables, which would make the truncation systematically improvable rather than empirically validated.
  • The energy non-conservation of the model's collision kernel is quantified in the paper and is small for the benchmark cases, but the comparison to the full solution inherits that kernel; repeating the test with an exactly energy-conserving kernel would isolate the attractodynamic closure error.
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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

3 major / 5 minor

Summary. The paper introduces attractodynamics, a macroscopic effective-theory construction organized around a far-from-equilibrium attractor manifold rather than around local thermal equilibrium, and demonstrates it in a 0+1D kinetic model. In the exactly solvable BSY model with anisotropic Gaussian attractor, the chart variables A, B, C define the 'ideal' attractodynamic theory, while a five-parameter δf ansatz, matching conditions (3.18), and the definitions (3.14)-(3.15) of generalized bulk (Π) and shear (π) pressures yield a viscous extension. The ideal and viscous reductions are compared with full numerical solutions of the kinetic equation for several couplings, occupancies, anisotropies, and initial deviations from the attractor; viscous attractodynamics improves on ideal and breaks down for sufficiently large off-attractor initial data. The paper also sketches how attractodynamics could be formulated macroscopically without an underlying kinetic description. Its central claim is that near-attractor dynamics can be organized in terms of a finite set of macroscopic variables plus a few transient modes, with the BSY model as a controlled proof of principle.

Significance. If the central claim holds, the paper provides a novel and potentially important template for building effective theories around nonthermal attractors, extending the logic of hydrodynamics to far-from-equilibrium settings relevant to heavy-ion pre-equilibrium dynamics and other systems with attractor behavior. The paper has several genuine strengths: the BSY attractor is an exact analytic solution, no constants are fitted to the kinetic theory benchmarks, the comparison with the full kinetic equation is direct, and the parameter scans in Appendices A-C are extensive. The linearized derivation and the explicit discussion of energy non-conservation in the BSY kernel are also transparent. The significance is conditional on the robustness of the 5-moment truncation, which is the main unresolved point in the manuscript.

major comments (3)
  1. [§3.3, Eqs. (3.35)–(3.42)] The viscous attractodynamic equations are the heart of the paper, but the text does not provide the reduced system in usable closed form. The functions g_i and operators O_1, O_2 are defined, but the coefficient vector c = (α, β0, w00, w33, v33) is stated to be fixed by the matching conditions and the definitions of π and Π without displaying the inversion, and the 'simple to compute' integrals I^(π)_ij and I^(Π)_ij are never evaluated. A reader cannot verify Eqs. (3.41)-(3.42) or reproduce Figs. 2-3 without redoing the entire computation. Please include the explicit c_i(π, Π, A, B, C; ξ) relations and the explicit integral matrices, or provide a computer-algebra supplement. This is a verifiability issue in the central derivation, not a cosmetic omission.
  2. [§2.1, §3.4, Appendix B, Sec. 5] The central claim is that a finite set of macroscopic variables plus two transient viscous modes organizes near-attractor dynamics. However, all benchmark initial conditions lie inside the 5-parameter ansatz (3.26). Appendix B scans over A0, B0, ξ0, π0, and Π0, but never over an unretained mode. Thus the condition stated in Sec. 2.1 — that omitted off-attractor modes must decay quickly or be weakly sourced — is never directly tested. A concrete test would initialize the full kinetic equation with f = f_A + ε δf_⊥, where δf_⊥ satisfies the matching conditions (3.18) but is orthogonal to the five ansatz functions, e.g. a term proportional to ν^4 ζ^4, and then compare the full solution with attractodynamics. Without such a test, the observed agreement may reflect only self-consistency of the chosen ansatz. The paper itself, in Sec. 5, identifies the systematic improvability of the truncatio
  3. [Appendix C, Figs. 11–12] The full kinetic benchmark solves the BSY equation, which does not conserve energy. The paper quantifies this violation but does not provide a criterion for when it is small enough to validate the late-time comparisons. For g_s = 1, the dimensionless violation shown in Fig. 12 reaches O(0.1) for some initial conditions, which is not obviously negligible for the agreement reported in Figs. 6-9. Please report the energy-violation measure along the specific trajectories used in Figs. 2-3 and state explicitly whether the quoted attractodynamic errors are affected by this artifact. This is not a demand to replace the kernel, but the benchmark should be interpreted with a quantitative caveat.
minor comments (5)
  1. [§3.4] The numerical solver used for the 'full kinetic evolution' is not described. Please add a few sentences on the discretization, convergence checks, and numerical error so the comparisons are reproducible.
  2. [§3.4] The sentence 'the exact and viscous solutions will agree exactly at the initial time' uses 'exact' for the full numerical solution. Rephrase to avoid confusion between the exact BSY attractor solution and the full kinetic solution.
  3. [§1, Eq. (1.1)] Eqs. (3.30) and (3.31) omit the measure; the reader must infer d^3p/(2π)^3 from the notation section. Write the measure explicitly at the first occurrence.
  4. [§2.2] The notation ζ is used in Sec. 3 before it is defined in Eq. (3.26); also the angle bracket in Eq. (2.7) should be explicitly linked to the definition (1.1).
  5. [§3.4, Fig. 1] The caption says 'different orange lines', but the figure appears to use a single color for all curves. Label or explain the line styles so the scan over initial π and Π is readable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the attractodynamic equations are derived by projection, not fitted, and all comparisons are genuine benchmarks against the full kinetic evolution.

full rationale

The paper's construction is self-contained in the required sense. The BSY attractor is taken as an exact analytic input from [71]; because it is an exact solution of the stated kinetic equation, it constitutes independent support rather than load-bearing self-citation. The chart variables A, B, C are defined by the matching conditions (3.18), and the residual viscous variables π and Π by the definitions (3.14)-(3.15); these are field-defining choices, not fitted parameters. The viscous equations (3.41)-(3.42) are obtained by projecting the kinetic equation onto the retained moments, and the ansatz (3.26) is an explicit truncation, not something inferred from the full solution. The numerical tests in Sec. 3.4 and Appendices A-B evolve the full kinetic equation independently and compare its output with the reduced attractodynamic evolution; no constants or initial data are adjusted to force agreement. The observation that ideal attractodynamics disagrees with the exact late-time solution when off-attractor initial data are used is itself a non-circular, falsifiable result. The main limitation—that initial conditions are restricted to lie inside the five-moment ansatz, so unretained orthogonal directions are not directly probed—is a scope limitation that the paper explicitly acknowledges, not a circularity. The paper never claims to have proven closure of the truncation, and its central claim is a proof-of-principle demonstration, which the benchmarks support. Self-citations appear, but they are either exact results or contextual references and are not used to forbid alternatives or supply the central derivation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The theory introduces no fitted constants; all parameters (g_s, N_c, σ0, ξ0, π0, Π0) are inputs of the underlying kinetic model or initial conditions. The construction itself rests on the exactness of the BSY attractor from Ref. [71] (self-cited), on the closure of the 5-moment ansatz (3.26), and on the hierarchy assumption that unretained modes relax or are weakly sourced. The model is acknowledged not to conserve energy (App. C), which is an explicit limitation of the benchmark rather than a hidden flaw.

free parameters (3)
  • Initial viscous pressure ratios π0/(PL)_i and Π0/(PL)_i (main comparisons) = 0.1 and 0.1 (also 0.5 in Fig. 3; scanned in Fig. 10)
    Hand-chosen magnitudes of the off-attractor residuals that define the benchmark; the comparison's qualitative conclusion (viscous improves over ideal) holds for 0.1 but degrades at 0.5.
  • Initial occupancy σ0 and anisotropy ξ0 (demo and scans) = σ0=4, ξ0=4 main; σ0∈{0.1,1,4}, ξ0∈{2,4,10} in App. B
    Hand-chosen initial conditions of the attractor chart; not fitted, but the regime of validity of the truncation is sensitive to them.
  • Coupling g_s = 0.1 main; 0.01 and 1 in App. B
    Model coupling of the underlying BSY kinetic theory; scans show the comparison quality varies with coupling (energy conservation violation grows with g_s).
assumptions (6)
  • domain assumption The BSY kinetic equation (3.1) with the specified q[f] possesses an exact Gaussian attractor f_A = A exp[-(p⊥²/B² + pz²/C²)/2] satisfying (3.3), as established in Ref. [71].
    The entire construction is built on this attractor; it is taken from the prior, self-cited work [71] and used as the chart for ideal attractodynamics.
  • ad hoc to paper The 5-parameter ansatz (3.26) spans the relevant off-attractor perturbation space, and the 3+2 matching/viscous definitions close the system.
    The ansatz is chosen by analogy with the 14-moment approximation; its completeness is not proven, only tested numerically.
  • ad hoc to paper Unretained off-attractor modes decay fast or are weakly sourced over the evolution (Sec. 2.1; probed numerically in App. B).
    This is the stability/closure assumption that justifies truncation; the paper states it as a requirement but does not prove it.
  • domain assumption The collision kernel's energy non-conservation is small enough that the comparison with the full kinetic theory remains a meaningful benchmark (quantified in App. C).
    The BSY model is known not to conserve energy; the paper argues the violation is mild, but this affects the interpretation of late-time comparisons.
  • standard math Tangency of the projection moments (3.30) follows from the Gaussian structure of f_A (Eq. 3.24).
    The projector moments ∫·, ∫p_z²·, ∫p_⊥²· correspond to the tangent directions 1, p_z², p_⊥² of the Gaussian attractor.
  • domain assumption Number density is conserved by the BSY kernel; energy is not (footnote 5).
    The model's conservation properties are background physics of the BSY collision kernel, stated in Sec. 3.2 and App. C.

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Cite this review

Pith. "Pith review of Attractodynamics in 0+1D." pith.science (2026). https://pith.science/paper/GDS6QRS7

@misc{pith2026260716393,
  author       = {Pith},
  title        = {Pith review of: Attractodynamics in 0+1D},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GDS6QRS7}},
  note         = {Machine review of arXiv:2607.16393}
}
read the original abstract

Hydrodynamics is a macroscopic theory of long-wavelength dynamics around local thermal equilibrium. We develop attractodynamics, the analogous construction around a far-from-equilibrium attractor. Dynamics near a far-from-equilibrium attractor retains some non-hydrodynamic microscopic information, which attractodynamics systematically organizes. We move towards this general structure by starting in 0+1D, and consider a model for which an anisotropic far-from-equilibrium attractor solution is exactly known. This setting provides a clean benchmark in which the ideal attractodynamic equations and a leading transient residual extension can be compared directly with the full kinetic evolution. The resulting hierarchy gives an improvable description of near-attractor dynamics: the ideal theory captures evolution close to the attracting manifold, while retaining the leading off-attractor moments extends the regime of agreement until the finite truncation breaks down. This example identifies the ingredients needed for local 3+1D attractodynamics and for macroscopic attractodynamic theories not derived from an underlying kinetic description.

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Works this paper leans on

82 extracted references · 77 linked inside Pith

  1. [1]

    U. W. Heinz and P. F. Kolb,Early thermalization at RHIC,Nucl. Phys. A702(2002) 269–280, [hep-ph/0111075]

    arXiv 2002

  2. [2]

    U. W. Heinz and P. F. Kolb,Two RHIC puzzles: Early thermalization and the HBT problem, in18th Winter Workshop on Nuclear Dynamics, 4, 2002.hep-ph/0204061

    arXiv 2002

  3. [3]

    P. F. Kolb and U. W. Heinz,Hydrodynamic description of ultrarelativistic heavy ion collisions,nucl-th/0305084

    arXiv

  4. [4]

    Romatschke and M

    P. Romatschke and M. Strickland,Collective modes of an anisotropic quark gluon plasma, Phys. Rev. D68(2003) 036004, [hep-ph/0304092]

    arXiv 2003

  5. [5]

    U. W. Heinz,Thermalization at RHIC,AIP Conf. Proc.739(2004), no. 1 163–180, [nucl-th/0407067]. – 25 – 5 10 50 100 500 0.8 0.9 1.0 1.1 1.2 1.3 1.4 τ/τ I (PT/PL) / (P T/PL)exact gs=1. σ0=4. ξ0=2. 0.8 0.9 1.0 1.1 1.2 1.3 1.4(PT/PL) / (P T/PL)exact gs=1. σ0=1. ξ0=2. 0.8 0.9 1.0 1.1 1.2 1.3 1.4(PT/PL) / (P T/PL)exact gs=1. σ0=0.1 ξ0=2. 5 10 50 100 500 τ/τ I gs...

    arXiv 2004

  6. [6]

    P. M. Chesler and L. G. Yaffe,Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma,Phys. Rev. Lett.102(2009) 211601, [arXiv:0812.2053]

    arXiv 2009

  7. [7]

    P. M. Chesler and L. G. Yaffe,Boost invariant flow, black hole formation, and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory,Phys. Rev. D82 (2010) 026006, [arXiv:0906.4426]

    arXiv 2010

  8. [8]

    P. M. Chesler and L. G. Yaffe,Holography and colliding gravitational shock waves in asymptotically AdS5 spacetime,Phys. Rev. Lett.106(2011) 021601, [arXiv:1011.3562]

    arXiv 2011

Show all 82 references
  1. [9]

    M. P. Heller, R. A. Janik, and P. Witaszczyk,The characteristics of thermalization of boost-invariant plasma from holography,Phys. Rev. Lett.108(2012) 201602, [arXiv:1103.3452]

    2012 arXiv

  2. [10]

    M. P. Heller, R. A. Janik, and P. Witaszczyk,A numerical relativity approach to the initial value problem in asymptotically Anti-de Sitter spacetime for plasma thermalization - an ADM formulation,Phys. Rev. D85(2012) 126002, [arXiv:1203.0755]

    2012 arXiv

  3. [11]

    M. P. Heller, D. Mateos, W. van der Schee, and D. Trancanelli,Strong Coupling Isotropization of Non-Abelian Plasmas Simplified,Phys. Rev. Lett.108(2012) 191601, [arXiv:1202.0981]. – 26 – 2 5 10 20 501 5 10 50 100 τ PT/PL π/PL,i=0.9 Π/PL,i=0 1 5 10 50 100PT/PL π/PL,i=0.5 Π/PL,i...

    2012 arXiv

  4. [12]

    van der Schee,Holographic thermalization with radial flow,Phys

    W. van der Schee,Holographic thermalization with radial flow,Phys. Rev. D87(2013), no. 6 061901, [arXiv:1211.2218]

    2013 arXiv

  5. [13]

    Schlichting,Turbulent thermalization of weakly coupled non-abelian plasmas,Phys

    S. Schlichting,Turbulent thermalization of weakly coupled non-abelian plasmas,Phys. Rev. D 86(2012) 065008, [arXiv:1207.1450]

    2012 arXiv

  6. [14]

    Kurkela and G

    A. Kurkela and G. D. Moore,UV Cascade in Classical Yang-Mills Theory,Phys. Rev. D86 (2012) 056008, [arXiv:1207.1663]

    2012 arXiv

  7. [15]

    Berges, K

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan,Turbulent thermalization process in heavy-ion collisions at ultrarelativistic energies,Phys. Rev. D89(2014), no. 7 074011, [arXiv:1303.5650]

    2014 arXiv

  8. [16]

    Berges, K

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan,Universal attractor in a highly occupied non-Abelian plasma,Phys. Rev. D89(2014), no. 11 114007, [arXiv:1311.3005]

    2014 arXiv

  9. [17]

    Berges, K

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan,Basin of attraction for turbulent thermalization and the range of validity of classical-statistical simulations,JHEP 05(2014) 054, [arXiv:1312.5216]

    2014 arXiv

  10. [18]

    M. C. Abraao York, A. Kurkela, E. Lu, and G. D. Moore,UV cascade in classical Yang-Mills theory via kinetic theory,Phys. Rev. D89(2014), no. 7 074036, [arXiv:1401.3751]. – 28 – Ideal Attractodynamics Viscous Attractodynamics 5 10 50 100 500 10-4 0.001 0.010 0.100 τ/τ I |τ∂ τΕ/...

    2014 arXiv

  11. [19]

    Kurkela and Y

    A. Kurkela and Y. Zhu,Isotropization and hydrodynamization in weakly coupled heavy-ion collisions,Phys. Rev. Lett.115(2015), no. 18 182301, [arXiv:1506.06647]

    2015 arXiv

  12. [20]

    M. P. Heller, A. Kurkela, M. Spali´ nski, and V. Svensson,Hydrodynamization in kinetic theory: Transient modes and the gradient expansion,Phys. Rev. D97(2018), no. 9 091503, [arXiv:1609.04803]

    2018 arXiv

  13. [21]

    Florkowski, R

    W. Florkowski, R. Ryblewski, and M. Spali´ nski,Gradient expansion for anisotropic hydrodynamics,Phys. Rev. D94(2016), no. 11 114025, [arXiv:1608.07558]

    2016 arXiv

  14. [22]

    Romatschke,Relativistic Fluid Dynamics Far From Local Equilibrium,Phys

    P. Romatschke,Relativistic Fluid Dynamics Far From Local Equilibrium,Phys. Rev. Lett. 120(2018), no. 1 012301, [arXiv:1704.08699]

    2018 arXiv

  15. [23]

    Alqahtani, M

    M. Alqahtani, M. Nopoush, and M. Strickland,Relativistic anisotropic hydrodynamics,Prog. Part. Nucl. Phys.101(2018) 204–248, [arXiv:1712.03282]

    2018 arXiv

  16. [24]

    Kurkela, A

    A. Kurkela, A. Mazeliauskas, J.-F. Paquet, S. Schlichting, and D. Teaney,Matching the Nonequilibrium Initial Stage of Heavy Ion Collisions to Hydrodynamics with QCD Kinetic Theory,Phys. Rev. Lett.122(2019), no. 12 122302, [arXiv:1805.01604]

    2019 arXiv

  17. [25]

    Schlichting and D

    S. Schlichting and D. Teaney,The First fm/c of Heavy-Ion Collisions,Ann. Rev. Nucl. Part. Sci.69(2019) 447–476, [arXiv:1908.02113]

    2019 arXiv

  18. [26]

    Berges, M

    J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan,QCD thermalization: Ab – 29 – initio approaches and interdisciplinary connections,Rev. Mod. Phys.93(2021), no. 3 035003, [arXiv:2005.12299]

    2021 arXiv

  19. [27]

    Mitra, S

    T. Mitra, S. Mondkar, A. Mukhopadhyay, A. Rebhan, and A. Soloviev,Hydrodynamic attractor of a hybrid viscous fluid in Bjorken flow,Phys. Rev. Res.2(2020), no. 4 043320, [arXiv:2006.09383]

    2020 arXiv

  20. [28]

    Pi˜ neiro Orioli, K

    A. Pi˜ neiro Orioli, K. Boguslavski, and J. Berges,Universal self-similar dynamics of relativistic and nonrelativistic field theories near nonthermal fixed points,Phys. Rev. D92 (2015), no. 2 025041, [arXiv:1503.02498]

    2015 arXiv

  21. [29]

    A. N. Mikheev, C.-M. Schmied, and T. Gasenzer,Low-energy effective theory of nonthermal fixed points in a multicomponent Bose gas,Phys. Rev. A99(2019), no. 6 063622, [arXiv:1807.10228]

    2019 arXiv

  22. [30]

    Pr¨ ufer, P

    M. Pr¨ ufer, P. Kunkel, H. Strobel, S. Lannig, D. Linnemann, C.-M. Schmied, J. Berges, T. Gasenzer, and M. K. Oberthaler,Observation of universal dynamics in a spinor Bose gas far from equilibrium,Nature563(2018), no. 7730 217–220, [arXiv:1805.11881]

    2018 arXiv

  23. [31]

    S. Erne, R. B¨ ucker, T. Gasenzer, J. Berges, and J. Schmiedmayer,Universal dynamics in an isolated one-dimensional Bose gas far from equilibrium,Nature563(2018), no. 7730 225–229, [arXiv:1805.12310]

    2018 arXiv

  24. [32]

    J. A. P. Glidden, C. Eigen, L. H. Dogra, T. A. Hilker, R. P. Smith, and Z. Hadzibabic, Bidirectional dynamic scaling in an isolated Bose gas far from equilibrium,Nature Phys.17 (2021), no. 4 457–461, [arXiv:2006.01118]

    2021 arXiv

  25. [33]

    S. Huh, K. Mukherjee, K. Kwon, J. Seo, J. Hur, S. I. Mistakidis, H. R. Sadeghpour, and J.-y. Choi,Universality class of a spinor Bose–Einstein condensate far from equilibrium,Nature Phys.20(2024), no. 3 402–408, [arXiv:2303.05230]

    2024 arXiv

  26. [34]

    M. Gazo, A. Karailiev, T. Satoor, C. Eigen, M. Ga lka, and Z. Hadzibabic,Universal coarsening in a homogeneous two-dimensional Bose gas,Science389(2025), no. 6762 ado3487, [arXiv:2312.09248]

    2025 arXiv

  27. [35]

    Lannig, M

    S. Lannig, M. Pr¨ ufer, Y. Deller, I. Siovitz, J. Dreher, T. Gasenzer, H. Strobel, and M. K. Oberthaler,Observation of two non-thermal fixed points for the same microscopic symmetry, arXiv:2306.16497

    arXiv

  28. [36]

    Martirosyan, C

    G. Martirosyan, C. J. Ho, J. Etrych, Y. Zhang, A. Cao, Z. Hadzibabic, and C. Eigen, Observation of Subdiffusive Dynamic Scaling in a Driven and Disordered Bose Gas,Phys. Rev. Lett.132(2024), no. 11 113401, [arXiv:2304.06697]

    2024 arXiv

  29. [37]

    Micha and I

    R. Micha and I. I. Tkachev,Relativistic turbulence: A Long way from preheating to equilibrium,Phys. Rev. Lett.90(2003) 121301, [hep-ph/0210202]

    2003 arXiv

  30. [38]

    Berges,Introduction to nonequilibrium quantum field theory,AIP Conf

    J. Berges,Introduction to nonequilibrium quantum field theory,AIP Conf. Proc.739(2004), no. 1 3–62, [hep-ph/0409233]

    2004 arXiv

  31. [39]

    Berges,Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology, arXiv:1503.02907

    J. Berges,Nonequilibrium Quantum Fields: From Cold Atoms to Cosmology, arXiv:1503.02907

    arXiv

  32. [40]

    Gorghetto, E

    M. Gorghetto, E. Hardy, and G. Villadoro,Axions from Strings: the Attractive Solution, JHEP07(2018) 151, [arXiv:1806.04677]

    2018 arXiv

  33. [41]

    Hindmarsh, J

    M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren, and J. Urrestilla,Approach to scaling in axion string networks,Phys. Rev. D103(2021), no. 10 103534, [arXiv:2102.07723]. – 30 –

    2021 arXiv

  34. [42]

    Kovtun,Lectures on hydrodynamic fluctuations in relativistic theories,J

    P. Kovtun,Lectures on hydrodynamic fluctuations in relativistic theories,J. Phys. A45 (2012) 473001, [arXiv:1205.5040]

    2012 arXiv

  35. [43]

    Crossley, P

    M. Crossley, P. Glorioso, and H. Liu,Effective field theory of dissipative fluids,JHEP09 (2017) 095, [arXiv:1511.03646]

    2017 arXiv

  36. [44]

    Liu and P

    H. Liu and P. Glorioso,Lectures on non-equilibrium effective field theories and fluctuating hydrodynamics,PoST ASI2017(2018) 008, [arXiv:1805.09331]

    2018 arXiv

  37. [45]

    A. H. Mueller,Toward equilibration in the early stages after a high-energy heavy ion collision,Nucl. Phys. B572(2000) 227–240, [hep-ph/9906322]

    2000 arXiv

  38. [46]

    A. H. Mueller,The Boltzmann equation for gluons at early times after a heavy ion collision, Phys. Lett. B475(2000) 220–224, [hep-ph/9909388]

    2000 arXiv

  39. [47]

    A. H. Mueller and D. T. Son,On the Equivalence between the Boltzmann equation and classical field theory at large occupation numbers,Phys. Lett. B582(2004) 279–287, [hep-ph/0212198]

    2004 arXiv

  40. [48]

    Baier, A

    R. Baier, A. H. Mueller, D. Schiff, and D. T. Son,’Bottom up’ thermalization in heavy ion collisions,Phys. Lett. B502(2001) 51–58, [hep-ph/0009237]

    2001 arXiv

  41. [49]

    P. B. Arnold, G. D. Moore, and L. G. Yaffe,Effective kinetic theory for high temperature gauge theories,JHEP01(2003) 030, [hep-ph/0209353]

    2003 arXiv

  42. [50]

    P. B. Arnold, J. Lenaghan, and G. D. Moore,QCD plasma instabilities and bottom up thermalization,JHEP08(2003) 002, [hep-ph/0307325]

    2003 arXiv

  43. [51]

    Kurkela and E

    A. Kurkela and E. Lu,Approach to Equilibrium in Weakly Coupled Non-Abelian Plasmas, Phys. Rev. Lett.113(2014), no. 18 182301, [arXiv:1405.6318]

    2014 arXiv

  44. [52]

    Kurkela, A

    A. Kurkela, A. Mazeliauskas, J.-F. Paquet, S. Schlichting, and D. Teaney,Effective kinetic description of event-by-event pre-equilibrium dynamics in high-energy heavy-ion collisions, Phys. Rev. C99(2019), no. 3 034910, [arXiv:1805.00961]

    2019 arXiv

  45. [53]

    Blaizot, F

    J.-P. Blaizot, F. Gelis, J.-F. Liao, L. McLerran, and R. Venugopalan,Bose–Einstein Condensation and Thermalization of the Quark Gluon Plasma,Nucl. Phys. A873(2012) 68–80, [arXiv:1107.5296]

    2012 arXiv

  46. [54]

    Blaizot, J

    J.-P. Blaizot, J. Liao, and L. McLerran,Gluon Transport Equation in the Small Angle Approximation and the Onset of Bose-Einstein Condensation,Nucl. Phys. A920(2013) 58–77, [arXiv:1305.2119]

    2013 arXiv

  47. [55]

    Tanji and R

    N. Tanji and R. Venugopalan,Effective kinetic description of the expanding overoccupied Glasma,Phys. Rev. D95(2017), no. 9 094009, [arXiv:1703.01372]

    2017 arXiv

  48. [56]

    Mazeliauskas and J

    A. Mazeliauskas and J. Berges,Prescaling and far-from-equilibrium hydrodynamics in the quark-gluon plasma,Phys. Rev. Lett.122(2019), no. 12 122301, [arXiv:1810.10554]

    2019 arXiv

  49. [57]

    Boguslavski, A

    K. Boguslavski, A. Kurkela, T. Lappi, F. Lindenbauer, and J. Peuron,Limiting attractors in heavy-ion collisions,Phys. Lett. B852(2024) 138623, [arXiv:2312.11252]

    2024 arXiv

  50. [58]

    M. P. Heller, D. Mateos, W. van der Schee, and M. Triana,Holographic isotropization linearized,JHEP09(2013) 026, [arXiv:1304.5172]

    2013 arXiv

  51. [59]

    Casalderrey-Solana, H

    J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal, and U. A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions. Cambridge University Press, 2014. – 31 –

    2014

  52. [60]

    P. M. Chesler and W. van der Schee,Early thermalization, hydrodynamics and energy loss in AdS/CFT,Int. J. Mod. Phys. E24(2015), no. 10 1530011, [arXiv:1501.04952]

    2015 arXiv

  53. [61]

    P. M. Chesler, N. Kilbertus, and W. van der Schee,Universal hydrodynamic flow in holographic planar shock collisions,JHEP11(2015) 135, [arXiv:1507.02548]

    2015 arXiv

  54. [62]

    M. P. Heller,Holography, Hydrodynamization and Heavy-Ion Collisions,Acta Phys. Polon. B 47(2016) 2581, [arXiv:1610.02023]

    2016 arXiv

  55. [63]

    P. M. Chesler,Colliding shock waves and hydrodynamics in small systems,Phys. Rev. Lett. 115(2015), no. 24 241602, [arXiv:1506.02209]

    2015 arXiv

  56. [64]

    P. M. Chesler,How big are the smallest drops of quark-gluon plasma?,JHEP03(2016) 146, [arXiv:1601.01583]

    2016 arXiv

  57. [65]

    Grozdanov and W

    S. Grozdanov and W. van der Schee,Coupling Constant Corrections in a Holographic Model of Heavy Ion Collisions,Phys. Rev. Lett.119(2017), no. 1 011601, [arXiv:1610.08976]

    2017 arXiv

  58. [66]

    Folkestad, S

    A. Folkestad, S. Grozdanov, K. Rajagopal, and W. van der Schee,Coupling Constant Corrections in a Holographic Model of Heavy Ion Collisions with Nonzero Baryon Number Density,JHEP12(2019) 093, [arXiv:1907.13134]

    2019 arXiv

  59. [67]

    G. S. Denicol and J. Noronha,Fluid dynamics far-from-equilibrium: a concrete example, Nucl. Phys. A1005(2021) 121996

    2021

  60. [68]

    Berges, G

    J. Berges, G. S. Denicol, M. P. Heller, and T. Preis,Far from equilibrium hydrodynamics of nonthermal fixed points,arXiv:2504.18754

    arXiv

  61. [69]

    Israel and J

    W. Israel and J. M. Stewart,Transient relativistic thermodynamics and kinetic theory, Annals Phys.118(1979) 341–372

    1979

  62. [70]

    Romatschke,Do nuclear collisions create a locally equilibrated quark–gluon plasma?,Eur

    P. Romatschke,Do nuclear collisions create a locally equilibrated quark–gluon plasma?,Eur. Phys. J. C77(2017), no. 1 21, [arXiv:1609.02820]

    2017 arXiv

  63. [71]

    Brewer, B

    J. Brewer, B. Scheihing-Hitschfeld, and Y. Yin,Scaling and adiabaticity in a rapidly expanding gluon plasma,JHEP05(2022) 145, [arXiv:2203.02427]

    2022 arXiv

  64. [72]

    Brewer, L

    J. Brewer, L. Yan, and Y. Yin,Adiabatic hydrodynamization in rapidly-expanding quark–gluon plasma,Phys. Lett. B816(2021) 136189, [arXiv:1910.00021]

    2021 arXiv

  65. [73]

    Rajagopal, B

    K. Rajagopal, B. Scheihing-Hitschfeld, and R. Steinhorst,Adiabatic Hydrodynamization and the emergence of attractors: a unified description of hydrodynamization in kinetic theory, JHEP04(2025) 028, [arXiv:2405.17545]

    2025 arXiv

  66. [74]

    Rajagopal, B

    K. Rajagopal, B. Scheihing-Hitschfeld, and R. Steinhorst,Attractors without scaling: adiabatic hydrodynamization with and without inelastic scattering,JHEP03(2026) 003, [arXiv:2507.21232]

    2026 arXiv

  67. [75]

    Grad,On the kinetic theory of rarefied gases,Commun

    H. Grad,On the kinetic theory of rarefied gases,Commun. Pure Appl. Math.2(1949), no. 4 331–407

    1949

  68. [76]

    De Lescluze, M

    M. De Lescluze, M. P. Heller, A. Mazeliauskas, B. Scheihing-Hitschfeld, and C. Werthmann, Adiabatic hydrodynamization and quasinormal modes of nonthermal attractors, arXiv:2510.15016

  69. [77]

    A. N. Mikheev, A. Mazeliauskas, and J. Berges,Stability analysis of nonthermal fixed points in longitudinally expanding kinetic theory,Phys. Rev. D105(2022), no. 11 116025, [arXiv:2203.02299]. – 32 –

    2022 arXiv

  70. [78]

    Florkowski and R

    W. Florkowski and R. Ryblewski,Highly-anisotropic and strongly-dissipative hydrodynamics for early stages of relativistic heavy-ion collisions,Phys. Rev. C83(2011) 034907, [arXiv:1007.0130]

    2011 arXiv

  71. [79]

    Martinez and M

    M. Martinez and M. Strickland,Dissipative Dynamics of Highly Anisotropic Systems,Nucl. Phys. A848(2010) 183–197, [arXiv:1007.0889]

    2010 arXiv

  72. [80]

    Martinez, R

    M. Martinez, R. Ryblewski, and M. Strickland,Boost-Invariant (2+1)-dimensional Anisotropic Hydrodynamics,Phys. Rev. C85(2012) 064913, [arXiv:1204.1473]

    2012 arXiv

  73. [81]

    Bazow, U

    D. Bazow, U. W. Heinz, and M. Strickland,Second-order (2+1)-dimensional anisotropic hydrodynamics,Phys. Rev. C90(2014), no. 5 054910, [arXiv:1311.6720]

    2014 arXiv

  74. [82]

    Strickland, J

    M. Strickland, J. Noronha, and G. Denicol,Anisotropic nonequilibrium hydrodynamic attractor,Phys. Rev. D97(2018), no. 3 036020, [arXiv:1709.06644]. – 33 –

    2018 arXiv

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