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Relativistic dissipative fluids in the trace-fixed particle frame: Hyperbolicity, causality, and stability

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Pith's one-line read A single temperature-dependent inequality controls whether a new first-order relativistic fluid theory is hyperbolic and causal.

desk verdict A careful, worthwhile first-order dissipative fluid theory built on a new frame condition; the hard-sphere/disks verification has a real but fixable gap in the intermediate-temperature regime. read the letter →

arxiv 2412.03712 v2 pith:2SPSNX63 submitted 2024-12-04 gr-qc astro-ph.HEhep-th

J. Félix Salazar , Ana Laura García-Perciante , Olivier Sarbach This is my paper · ORCID
classification gr-qcastro-ph.HEhep-th MSC 76Y0583C5535L45
keywords relativisticdissipativefluidstrace-fixedparticleframefirst-orderhydrodynamicscausalityhyperbolicitystabilityhard-sphereandhard-diskgasestransportcoefficients
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 a first-order relativistic theory of dissipative fluids in the trace-fixed particle frame, a choice of state variables in which the temperature is fixed by the trace of the stress-energy tensor rather than by the internal energy. The central claim is that the theory is hyperbolic and causal—perturbations propagate at finite speed and the initial-value problem is well posed—whenever the single inequality $(k_B T/e)(1+2\eta/\kappa)\le 1$ holds. At long wavelengths the linearized equations reproduce the familiar damped shear, acoustic, and heat-diffusion modes, and with a particular choice of the free parameter $\Gamma_1$ the equilibrium state is stable against perturbations of all wave numbers. The authors verify the inequality and the stability conditions for a relativistic gas of hard spheres or disks at all temperatures. If correct, the theory offers a first-order dissipative fluid description that is simultaneously causal and stable, without the second-order terms usually invoked to secure those properties.

What carries the argument

The central object is the trace-fixed particle frame itself, in which the temperature is fixed by the trace of the stress-energy tensor, $T^\mu{}_\mu=-ne+dp$, and the constitutive relations for the bulk-viscous correction $\epsilon$, the heat flux $Q^\mu$, and the shear tensor $T^{\mu\nu}$ remain first order in derivatives. The argument then runs through the linearized mode equations: causality is read from the eigenvalues of a block matrix $M_\parallel$ whose nonzero eigenvalues are the roots of the $2\times 2$ matrix $RQ$, a structure summarized in Lemma 1. Under the choice $\Gamma_2=h/(k_B T)$, the conditions $0<D<1$ and $2\sqrt D<\mathrm{Tr}\le 1+D$ on $RQ$ collapse to the single inequality (30). Stability for all wave numbers is carried by an algebraic stability criterion applied to the fifth-order characteristic polynomial of the longitudinal system, with $\Gamma_1$ selected according to Eq. (36).

What would settle it

A concrete check would be to compute $(k_B T/e)(1+2\eta/\kappa)$ for a relativistic gas model other than hard spheres or disks, for instance a screened Coulomb interaction or a binary mixture, and search for a temperature at which it exceeds 1. If such a temperature exists, the trace-fixed first-order theory would predict superluminal characteristic speeds there, contradicting the claim that hyperbolicity and causality are controlled by this single inequality.

Watch

Extended reading notes

Core claim

The central claim is that the trace-fixed particle frame yields a first-order relativistic dissipative fluid theory whose hyperbolicity and causality reduce to one dimensionless condition, Eq. (30): $(k_B T/e)(1+2\eta/\kappa)\le 1$. Linearizing the evolution system for $(n,T,u^\mu,\epsilon,Q^\mu)$ around global equilibrium and applying the frozen-coefficient principle, the authors show that transverse and longitudinal modes have real characteristic speeds below the speed of light exactly when this inequality holds, with $\Gamma_2=h/(k_B T)$. The same linearized analysis recovers the standard damped shear, acoustic, and heat-diffusion modes at low wave numbers. Stability for all wave numbers is then established, under four structural assumptions on the equation of state and transport coefficients, by choosing $\Gamma_1$ as in Eq. (36) with the constant $\Lambda_0$ large enough. For a simple gas of hard spheres or hard disks, the inequality and the structural assumptions are argued to hold at all temperatures, giving explicit models in which the theory is hyperbolic, causal, and stable.

Load-bearing premise

The stability proof assumes that, at every temperature, the gas's internal energy, heat capacity, speed of sound, and viscosity ratios satisfy four structural bounds, and the paper verifies these for hard spheres and disks only through limiting formulas and a plot, not a complete analytic proof in the intermediate temperature range.

Editorial extensions

If this is right

  • For any fluid satisfying the inequality, the linearized first-order equations are hyperbolic and causal, so initial-value problems are well posed and signals do not outrun light.
  • In the hydrodynamic regime the theory reproduces the known shear damping $\eta k^2/(nh)$, heat diffusion with coefficient $\kappa k^2/(n c_p T)$, and sound waves with speed $v_s=\sqrt{(k_B T/h)(c_p/c_v)}$ and Stokes attenuation, independent of the free functions $\Gamma_1,\Gamma_2$.
  • With $\Gamma_2=h/(k_B T)$ and $\Gamma_1$ from Eq. (36) with large $\Lambda_0$, global equilibrium is stable against perturbations of all wave numbers whenever the equation of state and transport coefficients satisfy the four structural assumptions (i)-(iv).
  • The conditions are met at all temperatures for a dilute relativistic gas of hard spheres in three dimensions and hard disks in two dimensions, providing explicit examples where the theory is physically sound.
  • The formulation extends to curved spacetimes and background electromagnetic fields, though self-gravity is neglected; the authors state that coupling to the full Einstein equations is left to future work.

Reading between the lines

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

  • The inequality (30) can be read as a testable bound on transport data: any interaction model whose $(k_B T/e)(1+2\eta/\kappa)$ exceeds 1 at some temperature would force this first-order frame to lose causality, so measuring or computing $\eta/\kappa$ in other gases would directly probe the theory's domain.
  • Because the low-wave-number modes are independent of $\Gamma_1$ and $\Gamma_2$ while the high-frequency behavior depends on them, the free functions could be calibrated against kinetic theory or other microscopic models without changing the hydrodynamic predictions.
  • The mode crossing seen for hard disks at $T\sim m/k_B$, where two characteristic speeds become equal, may mark a transition in strict hyperbolicity; whether well-posedness persists at that crossing is an extension the paper does not settle.
  • If the theory is coupled to gravity, the trace-fixed frame's simple first-order structure could make it a practical starting point for simulations of viscous accretion flows and neutron star mergers, but that requires the Einstein-fluid Cauchy analysis the authors list as future work.
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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

1 major / 5 minor

Summary. This paper proposes a first-order relativistic dissipative fluid theory formulated in the trace-fixed particle frame, in which the temperature is fixed by the trace of the stress-energy tensor and the constitutive relations contain two free functions Γ1 and Γ2 of temperature. The authors linearize the evolution system around a homogeneous equilibrium in Minkowski spacetime and derive conditions for hyperbolicity and causality: for the choice Γ2 = h/(k_B T), the system is linearly hyperbolic and causal if the single inequality (k_B T/e)(1 + 2η/κ) ≤ 1 holds. They compute the characteristic speeds for hard spheres and disks, recover the expected damped shear, acoustic, and heat modes at low wave numbers, and prove, using the Routh-Hurwitz criterion, that for Γ1 chosen as in Eq. (36) with sufficiently large Λ0, all Fourier modes are stable provided certain structural assumptions (i)-(iv) on the equation of state and transport coefficients hold. The paper claims these assumptions are satisfied for a simple gas of hard spheres or disks.

Significance. If the claims are correct, the paper provides a first-order relativistic dissipative fluid theory with causal propagation and stable equilibria, with a concrete kinetic-theory example. The main strengths are the clear matrix analysis leading to the characteristic speeds, the detailed Routh-Hurwitz proof in Appendix B, and the recovery of the standard low-wave-number mode structure without adjustable parameters. The stability criterion is explicitly tied to a tunable parameter Λ0, which gives the theory flexibility. However, the universal verification for hard spheres or disks is not fully rigorous, as detailed in the major comments.

major comments (1)
  1. [Section IV and Appendix B.3] The verification of the fundamental inequality (30) and of the structural assumptions (i)-(iv) for hard spheres/disks is incomplete. The paper provides the asymptotic limits T→0 and T→∞ in Appendix A and the plots in Fig. 1, but no proof is given for intermediate temperatures. In particular, assumption (iii), ν ≤ v_s^2 ≤ 1/d, and the boundedness of νη/κ and νζ/κ over the whole temperature range are asserted as “automatically fulfilled” (Appendix B.3) without an analytic demonstration. Because the T→∞ limit of (k_B T/e)(1 + 2η/κ) for hard disks is 21/22 ≈ 0.9545, a small interior maximum could in principle exceed 1, so the plot alone does not establish inequality (30) for all T. Since the all-wave-number stability theorem depends on P ≥ δν with P defined in Eq. (B42) and on the boundedness arguments leading to Eq. (B44), the universal claim in the abstract and conclusions (“satisfied for a simple gas of hard spheres or disks”) is not rigorously supported. The authors should either supply a complete analytic proof using the explicit transport coefficients in Appendix A, provide a rigorous numerical verification with error bounds, or explicitly restrict the claim to the temperature range covered by Fig. 1 and mark the all-temperature statement as a conjecture.
minor comments (5)
  1. [Abstract and Section III] The statement that the theory is hyperbolic and causal should be qualified as “in the linearized regime” or accompanied by a reference to the companion paper [31] where the nonlinear result is established, to avoid overstating the scope of the present analysis.
  2. [Section III, after Eq. (26)] The phrase “trace T rand determinant D” should read “trace Tr and determinant D”.
  3. [Appendix B, Eq. (B9)] The notation ℓmd f appears to be a typo; it should presumably be ℓmfp (the mean free path).
  4. [Appendix B, text before Eq. (B1)] The word “paramters” should be “parameters”.
  5. [Figs. 1 and 2] The captions should state explicitly that the curves are computed from the analytic expressions in Appendix A, and that the all-temperature verification is numerical rather than analytic.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main derivation is self-contained algebra, while the hard-sphere/disks verification has a non-circular completeness gap.

full rationale

The paper's core derivation does not reduce to its inputs. Inequalities (20), (26), and (30) are obtained by direct linearization of the proposed evolution system (8)-(12) and by spectral analysis of the block matrices M⊥ and M∥; nothing in that path presupposes the inequality. Transport coefficients for hard spheres and disks are taken from independent kinetic-theory references [34,35], and the low-wave-number acoustic, shear, and heat modes are standard outputs, not fitted values. The stability theorem is an existence argument: Eq. (36) defines Γ1 through a positive constant Λ0, and Appendix B.3 proves that for sufficiently large Λ0 the Routh-Hurwitz conditions hold under assumptions (i)-(iv); this is a constructive sufficiency result rather than an input-output fit. The main weakness is not circularity but rigor: the universal statement that (30) and (i)-(iv) hold for hard spheres and disks rests on T→0 and T→∞ asymptotics plus Fig. 1, so an interior extremum violating (30) is not rigorously excluded. This is a missing proof, not a self-referential derivation. Self-citations [29] and [31] are used for the companion nonlinear well-posedness, entropy-production, and frame-transformation details; they are not the basis of the linearized causal inequality or the stability inequalities established in this paper, so they do not make the central claim circular.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The theory's claimed properties rest on a small set of physical inputs: an ideal gas equation of state, the trace-fixed frame definition, known transport coefficients from kinetic theory, and structural assumptions on the equation of state and transport coefficients used to prove stability. A free parameter Λ0 is introduced to guarantee the Routh-Hurwitz conditions, with no quantitative bound. No new particles or fields are postulated.

free parameters (3)
  • Λ0 = large enough (no explicit value)
    Dimensionless constant in the choice of Γ1, Eq. (36). The stability proof requires Λ0 to be sufficiently large to satisfy conditions (B26)-(B30); no quantitative lower bound is provided.
  • Γ2 = h/(kBT)
    Free function in the constitutive relation (5), fixed by hand in Eq. (21) to keep ηβ3 bounded in the nonrelativistic limit and to simplify the analysis.
  • Γ1 = 1 + (cv/kB)(e^2/(kBT h))((kB/cv - 1/d)^2)(κ/ζ)Λ0
    Free function in the constitutive relation (4), fixed in Eq. (36) as the inversion of Λ = Λ0/ν. It is introduced to guarantee the Routh-Hurwitz stability inequalities for large Λ0.
assumptions (7)
  • domain assumption Ideal gas equation of state p = n k_B T and internal energy e(T) only
    Used in Section II to define the constitutive relations and the trace-fixed temperature; restricts the theory to simple gases.
  • domain assumption Unique determination of T from trace requires cv < d kB
    Footnote 1 in Section II states this condition; it is satisfied for a simple relativistic gas.
  • standard math Linearization around homogeneous equilibrium and the principle of frozen coefficients
    Used in Section III to reduce the system to Fourier modes; standard for linear stability analysis.
  • domain assumption The constitutive relations (4)-(6) are the correct first-order transformations from the Eckart frame
    Stated in Section II with details deferred to Refs. [30,31]; the analysis treats these equations as given.
  • domain assumption Background electromagnetic field is set to zero for the linear analysis
    Section III states the electric field plays no essential role and sets E^mu = 0.
  • domain assumption Transport coefficients for hard spheres/disks from Refs. [34,35] are correct
    Appendix A uses formulas from the relativistic Boltzmann equation literature; the verification of inequality (30) depends on them.
  • ad hoc to paper Structural assumptions (i)-(iv) on e(T), cv, v_s, and η/κ are satisfied
    Appendix B.3 asserts these are fulfilled for hard spheres/disks based on limits only; they underpin the stability theorem.

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Cite this review

Pith. "Pith review of Relativistic dissipative fluids in the trace-fixed particle frame: Hyperbolicity, causality, and stability." pith.science (2026). https://pith.science/paper/2SPSNX63

@misc{pith2026241203712,
  author       = {Pith},
  title        = {Pith review of: Relativistic dissipative fluids in the trace-fixed particle frame: Hyperbolicity, causality, and stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SPSNX63}},
  note         = {Machine review of arXiv:2412.03712}
}
read the original abstract

We propose a first-order theory of relativistic dissipative fluids in the trace-fixed particle frame, which is similar to Eckart's frame except that the temperature is determined by fixing the trace of the stress-energy tensor. Our theory is hyperbolic and causal provided a single inequality holds. For low wave numbers, the expected damped modes in the shear, acoustic, and heat diffusion channels are recovered. Stability of global equilibria with respect to all wave numbers is also analyzed. The conditions for hyperbolicity, causality and stability are satisfied for a simple gas of hard spheres or disks.

Figures

Figures reproduced from arXiv: 2412.03712 by the authors.

Figure 1
Figure 1. FIG. 1. Left-hand side of Eq. (30) for hard spheres (black) and hard [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Characteristic speeds for hard spheres (black) and hard disks [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

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Works this paper leans on

43 extracted references · 39 canonical work pages

  1. [31]

    Gabarrete, A

    C. Gabarrete, A. L. Garc´ıa-Perciante, and O. Sarbach, In prepa- ration

    work page

  2. [1]

    The other three roots are obtained from the ansatzs = kry(k) with y a smooth 7 function of k which converges to y0 ̸= 0 as k → 0

    Behavior of the roots for small wave numbers For small |k| there are two roots of p(s) which are of the form s = −H0 + O(ℓmf pk2), s = −H1 + O(ℓmd fk2), (B9) and from now on we assume Γ1, Γ2 > 1 such that H0 and H1 are positive which implies that these roots describe rapidly damped modes, as explained in the main text. The other three roots are obtained f...

    work page

  3. [2]

    (B1) written as: p(s) = s5 + a1s4 + a2s3 + a3s2 + a4s + a5, (B18) with a1 = H0 + H1, a2 = H0H1 + c3k2, a3 = c2k2, a4 = c1k2 + d1k4, a5 = d0k4

    Mode stability for all wave numbers To analyze the stability with respect to all wave numbers k, the Routh-Hurwitz criterion (see, for instance, [38, 39] for elementary proofs) is applied to the fifth-order polynomial Eq. (B1) written as: p(s) = s5 + a1s4 + a2s3 + a3s2 + a4s + a5, (B18) with a1 = H0 + H1, a2 = H0H1 + c3k2, a3 = c2k2, a4 = c1k2 + d1k4, a5 ...

    work page

  4. [3]

    Maartens, Causal thermodynamics in relativity (1996), arXiv:astro-ph/9609119 [astro-ph]

    R. Maartens, Causal thermodynamics in relativity (1996), arXiv:astro-ph/9609119 [astro-ph]

    arXiv 1996

  5. [4]

    Brevik, O

    I. Brevik, O. Grøn, J. de Haro, S. D. Odintsov, and E. N. Sari- dakis, Viscous cosmology for early- and late-time universe, In- ternational Journal of Modern Physics D 26, 1730024 (2017)

    work page 2017

  6. [5]

    M. A. Abramowicz and P. C. Fragile, Foundations of black hole accretion disk theory, Living Reviews in Relativity16, 1 (2013)

    work page 2013

  7. [6]

    Chabanov, L

    M. Chabanov, L. Rezzolla, and D. H. Rischke, General- relativistic hydrodynamics of non-perfect fluids: 3+1 conserva- tive formulation and application to viscous black hole accretion, Monthly Notices of the Royal Astronomical Society 505, 5910 (2021)

    work page 2021

  8. [7]

    First Sagittarius A* Event Horizon Telescope results. I. The shadow of the supermassive black hole in the center of the Milky Way, The Astrophysical Journal Letters930, L12 (2022)

    work page 2022

Show all 43 references
  1. [8]

    Shen and L

    C. Shen and L. Yan, Recent development of hydrodynamic modeling in heavy-ion collisions, Nuclear Science and Tech- niques 31, 122 (2020)

    2020

  2. [9]

    Israel, Nonstationary irreversible thermodynamics: A causal relativistic theory, Annals of Physics 100, 310 (1976)

    W. Israel, Nonstationary irreversible thermodynamics: A causal relativistic theory, Annals of Physics 100, 310 (1976)

    1976

  3. [10]

    I.-S. Liu, I. M ¨uller, and T. Ruggeri, Relativistic thermodynam- ics of gases, Annals of Physics 169, 191 (1986)

    1986

  4. [11]

    Geroch and L

    R. Geroch and L. Lindblom, Dissipative relativistic fluid theo- ries of divergence type, Phys. Rev. D41, 1855 (1990)

    1990

  5. [12]

    Tsumura and T

    K. Tsumura and T. Kunihiro, Stable first-order particle-frame relativistic hydrodynamics for dissipative systems, Physics Let- ters B 668, 425 (2008)

    2008

  6. [13]

    Peralta-Ramos and E

    J. Peralta-Ramos and E. Calzetta, Divergence-type nonlinear conformal hydrodynamics, Phys. Rev. D 80, 126002 (2009)

    2009

  7. [14]

    V ´an and T

    P. V ´an and T. Bir´o, First order and stable relativistic dissipative hydrodynamics, Physics Letters B 709, 106 (2012)

    2012

  8. [15]

    Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, Journal of Physics A: Mathematical and Theoretical 45, 473001 (2012)

    P. Kovtun, Lectures on hydrodynamic fluctuations in relativistic theories, Journal of Physics A: Mathematical and Theoretical 45, 473001 (2012)

    2012

  9. [16]

    P. V ´an, Nonequilibrium thermodynamics: emergent and fun- damental, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, 20200066 (2020)

    2020

  10. [17]

    J. F. Salazar and T. Zannias, On extended thermodynamics: From classical to the relativistic regime, International Journal of Modern Physics D 29, 2030010 (2020)

    2020

  11. [18]

    Gavassino and M

    L. Gavassino and M. Antonelli, Unified extended irreversible thermodynamics and the stability of relativistic theories for dissipation, Frontiers in Astronomy and Space Sciences 8, 10.3389/fspas.2021.686344 (2021)

    2021

  12. [19]

    G. S. Rocha, D. Wagner, G. S. Denicol, J. Noronha, and D. H. Rischke, Theories of relativistic dissipative fluid dynamics, En- tropy 26, 10.3390/e26030189 (2024)

    2024 doi

  13. [20]

    Lehner, O

    L. Lehner, O. A. Reula, and M. E. Rubio, Hyperbolic theory of relativistic conformal dissipative fluids, Phys. Rev. D 97, 024013 (2018)

    2018

  14. [21]

    F. S. Bemfica, M. M. Disconzi, and J. Noronha, Causality and existence of solutions of relativistic viscous fluid dynamics with gravity, Phys. Rev. D98, 104064 (2018)

    2018

  15. [22]

    F. S. Bemfica, M. M. Disconzi, and J. Noronha, Nonlinear causality of general first-order relativistic viscous hydrodynam- ics, Phys. Rev. D 100, 104020 (2019)

    2019

  16. [23]

    Kovtun, First-order relativistic hydrodynamics is stable, JHEP 10, 034

    P. Kovtun, First-order relativistic hydrodynamics is stable, JHEP 10, 034

  17. [24]

    R. E. Hoult and P. Kovtun, Stable and causal relativistic Navier- Stokes equations, Journal of High Energy Physics 2020, 67 (2020)

    2020

  18. [25]

    F. S. Bemfica, M. M. Disconzi, and J. Noronha, First-order general-relativistic viscous fluid dynamics, Phys. Rev. X 12, 021044 (2022)

    2022

  19. [26]

    R. E. Hoult and P. Kovtun, Causal first-order hydrodynamics from kinetic theory and holography, Phys. Rev. D 106, 066023 (2022)

    2022

  20. [27]

    G. S. Rocha, G. S. Denicol, and J. Noronha, Perturbative ap- proaches in relativistic kinetic theory and the emergence of first- order hydrodynamics, Phys. Rev. D 106, 036010 (2022)

    2022

  21. [28]

    Disconzi, Recent developments in mathematical aspects of relativistic fluids, Living Reviews in Relativity27, 6 (2024)

    M. Disconzi, Recent developments in mathematical aspects of relativistic fluids, Living Reviews in Relativity27, 6 (2024)

    2024

  22. [29]

    Boillat and T

    G. Boillat and T. Ruggeri, Hyperbolic principal subsystems: Entropy convexity and subcharacteristic conditions, Archive for Rational Mechanics and Analysis 137, 305 (1997)

    1997

  23. [30]

    V . A. Cimmelli, D. Jou, T. Ruggeri, and P. V´an, Entropy princi- ple and recent results in non-equilibrium theories, Entropy 16, 1756 (2014)

    2014

  24. [32]

    Kovtun, First-order relativistic hydrodynamics is stable, Jour- nal of High Energy Physics 2019, 34 (2019)

    P. Kovtun, First-order relativistic hydrodynamics is stable, Jour- nal of High Energy Physics 2019, 34 (2019)

    2019

  25. [33]

    J. F. Salazar, A. L. Garc´ıa-Perciante, and O. Sarbach, Relativis- tic dissipative fluids in the trace-fixed particle frame: Strongly hyperbolic quasilinear first-order evolution equations, Phys. Rev. D 111, 084024 (2025)

    2025

  26. [34]

    H. O. Kreiss and J. Lorenz, Initial-boundary value problems and the Navier-Stokes equations (Academic Press, San Diego, 1989)

    1989

  27. [35]

    Sarbach, E

    O. Sarbach, E. Barausse, and J. Preciado-L ´opez, Well-posed Cauchy formulation for Einstein-æther theory, Class. Quant. Grav. 36, 165007 (2019)

    2019

  28. [36]

    Cercignani and G

    C. Cercignani and G. Kremer, The Relativistic Boltzmann Equation: Theory and Applications (Birkh¨auser, Basel, 2002)

    2002

  29. [37]

    A. L. Garc ´ıa-Perciante and A. R. M ´endez, Dissipative proper- ties of relativistic two-dimensional gases, Physica A: Statistical Mechanics and its Applications 530, 121559 (2019). 10

    2019

  30. [38]

    Weinberg, Entropy Generation and the Survival of Proto- galaxies in an Expanding Universe, Astrophys

    S. Weinberg, Entropy Generation and the Survival of Proto- galaxies in an Expanding Universe, Astrophys. J. 168, 175 (1971)

    1971

  31. [39]

    de Groot, W

    S. de Groot, W. van Leeuwen, and C. G. van Weert,Relativistic Kinetic Theory (North-Holland Publishing Company, Amster- dam, New York, Oxford, 1980)

    1980

  32. [40]

    J. J. Anagnost and C. A. Desoer, An elementary proof of the Routh-Hurwitz stability criterion, Circuits, Systems and Signal Processing 10, 101 (1991)

    1991

  33. [41]

    Meinsma, Elementary proof of the Routh-Hurwitz test, Sys- tems & Control Letters 25, 237 (1995)

    G. Meinsma, Elementary proof of the Routh-Hurwitz test, Sys- tems & Control Letters 25, 237 (1995)

    1995

  34. [42]

    Acu ˜na-C´ardenas, C

    R. Acu ˜na-C´ardenas, C. Gabarrete, and O. Sarbach, An intro- duction to the relativistic kinetic theory on curved spacetimes, General Relativity and Gravitation 54 (2022)

    2022

  35. [43]

    A. L. Garc ´ıa-Perciante, A. R. M ´endez, and O. Sarbach, Existence of the Chapman-Enskog solution and its relation with first-order dissipative fluid theories, Journal of Non- Equilibrium Thermodynamics 50, 295 (2025)

    2025

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