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Foundations of magnetohydrodynamics

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper derives closed magnetohrodynamic equations from non-equilibrium thermodynamics, valid beyond the ideal gas limit, with the standard dilute-gas kinetic-theory closure recovered as the ideal gas case.

desk verdict A useful tutorial synthesis, but the single-fluid reduction at Eq. (35) is flawed as written and needs correction before this can be relied on. read the letter →

arxiv 2504.13352 v1 pith:XPNORGYM submitted 2025-04-17 physics.plasm-ph

Jarett LeVan , Scott Baalrud This is my paper · ORCID
classification physics.plasm-ph
keywords magnetohydrodynamicsnon-equilibriumthermodynamicsgeneralizedOhm'slawFourier'sNewton'selectronchemicalpotentialGreen-Kuborelationstransportcoefficients
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper sets out to show that a closed single-fluid set of magnetohydrodynamic equations can be derived from non-equilibrium thermodynamics alone, without assuming an ideal gas equation of state. It claims that for non-ideal plasmas the constitutive relations take structurally different forms: Ohm's law and Fourier's law contain a gradient of the electron chemical potential over temperature, and Newton's law contains bulk and cross viscosity terms that vanish for a monatomic ideal gas. If correct, this means dense, strongly coupled, or degenerate plasmas cannot be modeled by inserting non-ideal transport coefficients into the standard ideal-gas MHD structure; the equations themselves must change. The derivation also supplies a microscopic closure, connecting every transport coefficient to equilibrium time correlations and the excess equation of state to pair correlations.

What carries the argument

The load-bearing object is the single-fluid entropy production rate after the electron-mass-ratio expansion, Eq. (35): $\sigma_S = -(1/T^2)\, q\cdot\nabla T + j\cdot[E'/T - (1/z_e)\nabla(\mu_e/T)] - (1/T)\hat{P}:\nabla V$. This bilinear form identifies conjugate force-flux pairs, fixes the diffusive entropy flux, and dictates which thermodynamic forces enter the linear constitutive relations. Inversion and rotational symmetry about the magnetic field eliminate vector-tensor coupling and reduce each second-rank transport tensor to three coefficients, while time-reversal symmetry relates the electrothermal and thermoelectric tensors. The viscosity tensor is reduced to seven independent coefficients by symmetry and written in components parallel, perpendicular, and cross to the magnetic field. The macroscopic fluxes are expressed in terms of particle phase-space averages by the Irving-Kirkwood procedure, and their equilibrium autocorrelation functions yield the transport coefficients through Green-Kubo relations, closing the system.

What would settle it

Compute, from a molecular dynamics simulation of a dense plasma in a steady state with a fixed density gradient, uniform temperature, and no applied electric field in the flowing frame, the current density that appears. The generalized Ohm's law predicts a current proportional to $\nabla(\mu_e/T)$; the standard ideal-gas MHD prediction is zero. If the measured current vanishes or lacks the predicted chemical-potential dependence, the central claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that the entropy production rate of a quasi-neutral electron-ion plasma, after expanding in the small electron-to-ion mass ratio, can be written in single-fluid form with heat flux, electrical current, and viscous stress as the only dissipative fluxes. From this bilinear form, linear constitutive relations follow, and spatial symmetry decouples vector and tensor forces while time-reversal symmetry reduces the coupling between current and heat flux. The resulting generalized Ohm's law is $j = \sigma \cdot [E' - (T/z_e)\nabla(\mu_e/T)] + \phi \cdot \nabla T$, Fourier's law is $q = -\lambda \cdot \nabla T + \varphi \cdot [E' - (T/z_e)\nabla(\mu_e/T)]$, and the viscous stress tensor has seven independent coefficients including bulk viscosity and a cross term. The paper shows explicitly that substituting the ideal-gas expression for $\nabla(\mu_e/T)$ reproduces the standard kinetic-theory forms of Ohm's and Fourier's laws, and that dropping bulk and cross viscosity reproduces the conventional magnetized-plasma form of Newton's law. Thus the traditional MHD equations appear as the ideal gas limit of a more general non-ideal closure.

Load-bearing premise

The single-fluid closure assumes that the electron diffusion flux and the electron specific chemical potential per unit mass are large enough to dominate the ion contributions, so that the multi-species entropy production can be rewritten with electron quantities alone.

Editorial extensions

If this is right

  • In dense, strongly coupled, or degenerate plasmas, Ohm's law and Fourier's law contain a term proportional to $\nabla(\mu_e/T)$; using the pressure-gradient form of ideal-gas MHD would misrepresent the driving force for current and heat flow.
  • Newton's law for a non-ideal plasma includes bulk viscosity and a cross-viscosity coefficient that vanish for a monatomic ideal gas, so these terms should appear whenever internal degrees of freedom such as electronic excitation or molecular states are active.
  • Because every transport coefficient is expressed as a Green-Kubo autocorrelation of phase-space-averaged fluxes, the full MHD closure for a given plasma can in principle be obtained from equilibrium molecular dynamics trajectories without a kinetic-theory solution.
  • The standard dilute-gas MHD equations are exactly recovered when the ideal-gas equation of state is inserted, including an explicit mapping between the generalized electrothermal and thermoelectric coefficients and the conventional ones.
  • Phenomena often labeled two-fluid effects, including the Hall effect, appear in the single-fluid description through the off-diagonal components of the resistivity tensor; only the inertial correction is genuinely second order and omitted.

Reading between the lines

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

  • The single-fluid closure rests on electron-dominated diffusion, so in partially ionized or molecular plasmas with comparable species masses, the reduction to Eq. (35) would need explicit multi-species diffusion terms and the generalized Ohm's law would likely require modification.
  • If the chemical-potential gradient term is real, then a dense plasma in a steady state with a density gradient, uniform temperature, and no applied electric field should exhibit a diffusive current; a molecular dynamics simulation could isolate this term from the usual resistive response.
  • The appearance of bulk and cross viscosity suggests that MHD closures for plasmas with electronic excitation or molecular degrees of freedom need at least these additional coefficients, and Green-Kubo evaluation from molecular dynamics would give their magnitude and dynamical importance.
  • The derivation implies that dense-plasma MHD codes using the ideal-gas form of Ohm's law are missing a term of the same order as the non-ideal part of the electron chemical potential; estimates from equation-of-state data could show where this correction matters.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper presents a tutorial derivation of single-fluid magnetohydrodynamic (MHD) equations from non-equilibrium thermodynamics, aiming to be valid beyond the ideal-gas approximation. It derives conservation laws, constructs the entropy production rate, obtains linear constitutive relations for the current density, heat flux, and viscous stress tensor (generalized Ohm, Fourier, and Newton laws), reduces the number of independent transport coefficients using symmetry arguments and Onsager-Casimir relations, and connects the transport coefficients and equation-of-state quantities to microscopic dynamics through the Irving-Kirkwood procedure and Green-Kubo relations. The authors show that the Chapman-Enskog form of MHD is recovered in the ideal-gas limit, and they discuss connections to extended and resistive MHD models.

Significance. If correct, the derivation would provide a useful structural framework for dense-plasma MHD, showing that non-ideal equations of state modify Ohm's and Fourier's laws through chemical-potential gradients and introduce bulk and cross viscosity terms. The symmetry analysis is a genuine strength: the reduction from 225 phenomenological coefficients to 16 independent ones via inversion symmetry, rotational symmetry about the magnetic field, and Onsager-Casimir relations is explicit and checkable. The Green-Kubo and Irving-Kirkwood connections also give a principled route to compute the coefficients from particle trajectories. However, the central single-fluid reduction of the multi-species entropy production contains an internally inconsistent inequality, and the resulting generalized Ohm's and Fourier's laws are derived under unquantified assumptions about ion chemical-potential gradients. This affects the main advertised claim that the equations are valid beyond the usual ideal-gas approximation.

major comments (2)
  1. [Section II.C, Eq. (35)] The reduction from Eq. (34) to Eq. (35) rests on the statement that 'd_e ≫ Σ_i d_i'. This is inconsistent with the definition d_α = ρ_α(V_α − V) in Eq. (7), because Σ_α d_α = 0 identically, so for a single ion species Σ_i d_i = −d_e and the magnitudes are exactly equal. The current is electron-dominated because |z_e| ≫ |z_i|, not because the electron diffusion flux is larger than the ion diffusion flux. The reduction should use Σ_α d_α = 0 to eliminate the ion fluxes; the exact two-species entropy production contains the force (1/(z_e − z_i))∇[(μ_e − μ_i)/T] conjugate to j. The simplified form in Eq. (35) follows only after additionally assuming |z_i| ≪ |z_e| and |∇(μ_i/T)| ≪ |∇(μ_e/T)|, neither of which is stated or quantified.
  2. [Eqs. (49) and (53)] Because the reduction in Eq. (35) is not generally valid, the generalized Ohm's law and Fourier's law as written depend only on ∇(μ_e/T), whereas the correct single-fluid reduction of the species sum involves ∇[(μ_e − μ_i)/T]/(z_e − z_i). For an electron–proton plasma this reduces to the paper's form, but in plasmas with heavier multiply charged ions, multiple ion species of comparable mass, or strong ion–ion coupling, the ion chemical-potential gradient need not be negligible. The paper should either adopt the exact two-species form or give explicit, quantitative conditions under which the simplified form applies. This qualification bears directly on the abstract's claim that the derivation is 'valid beyond the usual ideal gas approximation'.
minor comments (4)
  1. [Section IV.B] There is a typographical error: 'Champman-Enskog' should be 'Chapman-Enskog'.
  2. [Section II.D.5] The text following Eq. (65) lists 'η1, η2, η3, η4 and η5', but only η0 through η4 are defined; the list should refer to η0, η1, η2, η3, η4. Also, in Eq. (68c) the symbol 'ηB_o' should be 'ηB_0'.
  3. [Section III.C] The paper itself states that Eq. (97) for the excess electron chemical potential is only approximate because derivatives of pex depend on higher-order correlation functions, and that classical descriptions of dense electron-ion systems suffer Coulomb collapse. These are important limitations of the 'closed and self-consistent' description advertised in the introduction, and they should be reflected more prominently in the abstract or conclusions.
  4. [Section III.A] The sentence 'the average may be dropped so that instantaneous representations of the fluxes are obtained' is imprecise: what is needed is the instantaneous phase-space expression from which the ensemble average is then taken, not the dropping of the average from an averaged quantity. Rephrasing would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constitutive relations are assembled from entropy-production forces with transport coefficients left as external or Green-Kubo inputs; the only self-citations are peripheral.

full rationale

The paper's derivation chain is self-contained rather than circular. Starting from conservation laws and the Gibbs relation, it identifies an entropy-production bilinear form (Eq. 34), reduces it to single-fluid form under an explicit ordering assumption (Eq. 35), and then writes linear constitutive relations whose coefficients are not fitted but are left as material inputs or computed independently from Green-Kubo correlation functions (Eqs. 89-90). The generalized Ohm's, Fourier's, and Newton's laws follow from these force-flux identifications plus symmetry/Onsager-Casimir arguments; they are not re-imported from the microscopic results. The comparison with the Chapman-Enskog/ideal-gas limit is a consistency check using thermodynamic identities, not a derivation of the general laws from the ideal-gas result. The authors' own prior work is cited only for peripheral context, such as bulk-viscosity behavior in molecular species and the Braginskii-type viscosity notation, and the central derivation does not rest on those citations. The most substantive concern in the manuscript is the Sec. II.C reduction of Eq. (34) to Eq. (35), which assumes electron diffusion dominance; indeed, since sum_alpha d_alpha = 0, the stated inequality d_e >> sum_i d_i is internally inconsistent for a two-species plasma as written. That is a correctness/validity issue about an ordering assumption rather than a circularity: the single-fluid constitutive laws are conditional on that assumption, not constructed to equal it. The paper also explicitly flags its own approximation in deriving the excess chemical potential from pex (Sec. III.C), which again limits accuracy but does not make the derivation circular. Overall, no prediction reduces by construction to inputs, and no load-bearing chain depends on unverified self-citation.

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

The derivation rests on standard thermodynamic and statistical-mechanical assumptions; the paper introduces no new physical entities and fits no data. The transport coefficients and equation of state are external inputs.

free parameters (3)
  • Transport tensors sigma, phi, lambda
    The generalized Ohm's and Fourier's laws (Eqs. 49 and 53) contain phenomenological conductivity, electrothermal, and thermal conductivity tensors that are not derived in this paper; they must be supplied by Green-Kubo integration, experiment, or kinetic theory.
  • Viscosity coefficients eta0-eta4, eta_v, zeta
    Newton's law (Eqs. 65-68) introduces seven viscosity coefficients that depend on the fourth-rank phenomenological tensor L_PP; no values are computed, the paper only gives their definitions.
  • Equation of state (u, p, mu_e)
    Closure requires the internal energy, pressure, and electron chemical potential as functions of state; for dense plasmas these are not provided because classical EOS suffers Coulomb collapse (Sec. III.C).
assumptions (7)
  • domain assumption Local thermodynamic equilibrium at the fluid element scale (Sec. II).
    Required for the Gibbs relation (Eq. 29) to hold locally; the entire entropy-production derivation depends on this.
  • domain assumption Linear constitutive relations between fluxes and forces near equilibrium (Eq. 20).
    Assumes small thermodynamic forces so that flux-force relations are linear; the validity for strongly coupled dense plasmas is not established.
  • domain assumption Quasi-neutrality z=0 and non-relativistic pre-Maxwell equations (Eq. 19).
    Standard MHD assumptions; removes Gauss's law and displacement current.
  • domain assumption Onsager-Casimir reciprocal relations (Eq. 45).
    Used to reduce the number of independent transport coefficients; assumes time-reversal symmetry of microscopic dynamics.
  • domain assumption Electron-dominance ordering de >> sum_i di and mu_e >> sum_i mu_i (Eq. 35).
    Single-fluid reduction of the entropy production; may fail in multi-ion or molecular plasmas.
  • domain assumption Green-Kubo relations and Onsager regression hypothesis (Eq. 88).
    Connects transport coefficients to equilibrium fluctuations; assumes the decay of fluctuations follows macroscopic transport laws.
  • domain assumption Classical statistical mechanics for the equation of state (Eqs. 92-95).
    The excess EOS expressions rely on pair correlations; the paper notes classical Coulomb collapse prevents dense plasma application without quantum corrections.

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Pith. "Pith review of Foundations of magnetohydrodynamics." pith.science (2026). https://pith.science/paper/XPNORGYM

@misc{pith2026250413352,
  author       = {Pith},
  title        = {Pith review of: Foundations of magnetohydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XPNORGYM}},
  note         = {Machine review of arXiv:2504.13352}
}
read the original abstract

In this tutorial, a derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented. Non-equilibrium thermodynamics is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell's equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. These properties are connected to microscopic dynamics using the Irving-Kirkwood procedure and Green-Kubo relations. Symmetry arguments and the Onsager-Casimir relations allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm's law, Fourier's law, and Newton's law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman-Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations.

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Cited by 1 Pith paper

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Works this paper leans on

57 extracted references · 37 canonical work pages · cited by 1 Pith paper

  1. [1]

    merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aapmrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  2. [2]

    merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipauth4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translat...

    2010

  3. [3]

    merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs aipnum4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  4. [4]

    + command, where the argument is the citation key mentioned above. +

  5. [5]

    The AIP styles for REV 4 include Bib \ style files +aipnum.bst+ and +aipauth.bst+, appropriate for numbered and author-year bibliographies, respectively

    + commands may be crafted by hand or, preferably, generated by using Bib . The AIP styles for REV 4 include Bib \ style files +aipnum.bst+ and +aipauth.bst+, appropriate for numbered and author-year bibliographies, respectively. REV 4 will automatically choose the style appropriate for the document's selected class options: the default is numerical, and y...

  6. [6]

    merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  7. [7]

    merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

    FUNCTION id.bst "merlin.mbs apsrmp4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

    2010

  8. [8]

    author author J. P. \ Freidberg ,\ 10.1103/RevModPhys.54.801 journal journal Rev. Mod. Phys. \ volume 54 ,\ pages 801 ( year 1982 ) NoStop

    work page doi:10.1103/revmodphys.54.801 1982

Show all 57 references
  1. [9]

    Alfv \'e n ,\ @noop journal journal Nature \ volume 150 ,\ pages 405 ( year 1942 ) NoStop

    author author H. Alfv \'e n ,\ @noop journal journal Nature \ volume 150 ,\ pages 405 ( year 1942 ) NoStop

    1942

  2. [10]

    author author D. I. \ Pontin \ and\ author E. R. \ Priest ,\ @noop journal journal Living Reviews in Solar Physics \ volume 19 ,\ pages 1 ( year 2022 ) NoStop

    2022

  3. [11]

    Beresnyak ,\ @noop journal journal Living Reviews in Computational Astrophysics \ volume 5 ,\ pages 2 ( year 2019 ) NoStop

    author author A. Beresnyak ,\ @noop journal journal Living Reviews in Computational Astrophysics \ volume 5 ,\ pages 2 ( year 2019 ) NoStop

    2019

  4. [12]

    author author J. F. \ Hawley ,\ 10.1063/1.1542885 journal journal Physics of Plasmas \ volume 10 ,\ pages 1946 ( year 2003 ) NoStop

    1946 doi

  5. [13]

    Burlaga ,\ @noop journal journal Space science reviews \ volume 39 ,\ pages 255 ( year 1984 ) NoStop

    author author L. Burlaga ,\ @noop journal journal Space science reviews \ volume 39 ,\ pages 255 ( year 1984 ) NoStop

    1984

  6. [14]

    Chapman \ and\ author T

    author author S. Chapman \ and\ author T. G. \ Cowling ,\ @noop title The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases \ ( publisher Cambridge university press ,\ year 1990 ) NoStop

    1990

  7. [15]

    Braginskii ,\ @noop journal journal Reviews of plasma physics \ volume 1 ,\ pages 205 ( year 1965 ) NoStop

    author author S. Braginskii ,\ @noop journal journal Reviews of plasma physics \ volume 1 ,\ pages 205 ( year 1965 ) NoStop

    1965

  8. [16]

    author author S. R. \ De Groot \ and\ author P. Mazur ,\ @noop title Non-equilibrium thermodynamics \ ( publisher Courier Corporation ,\ year 2013 ) NoStop

    2013

  9. [17]

    Onsager ,\ 10.1103/PhysRev.37.405 journal journal Phys

    author author L. Onsager ,\ 10.1103/PhysRev.37.405 journal journal Phys. Rev. \ volume 37 ,\ pages 405 ( year 1931 ) NoStop

    1931 doi

  10. [18]

    author author H. B. G. \ Casimir ,\ 10.1103/RevModPhys.17.343 journal journal Rev. Mod. Phys. \ volume 17 ,\ pages 343 ( year 1945 ) NoStop

    1945 doi

  11. [19]

    author author J. H. \ Irving \ and\ author J. G. \ Kirkwood ,\ 10.1063/1.1747782 journal journal The Journal of Chemical Physics \ volume 18 ,\ pages 817 ( year 1950 ) NoStop

    1950 doi

  12. [20]

    author author M. S. \ Green ,\ 10.1063/1.1740082 journal journal The Journal of Chemical Physics \ volume 22 ,\ pages 398 ( year 1954 ) NoStop

    1954 doi

  13. [21]

    Kubo , author M

    author author R. Kubo , author M. Yokota , \ and\ author S. Nakajima ,\ @noop journal journal Journal of the Physical Society of Japan \ volume 12 ,\ pages 1203 ( year 1957 ) NoStop

    1957

  14. [22]

    author author L. J. \ Stanek , author A. Kononov , author S. B. \ Hansen , author B. M. \ Haines , author S. X. \ Hu , author P. F. \ Knapp , author M. S. \ Murillo , author L. G. \ Stanton , author H. D. \ Whitley , author S. D. \ Baalrud , author L. J. \ Babati , author A. D...

  15. [23]

    LeVan , author M

    author author J. LeVan , author M. D. \ Acciarri , \ and\ author S. D. \ Baalrud ,\ 10.1103/PhysRevE.110.015208 journal journal Phys. Rev. E \ volume 110 ,\ pages 015208 ( year 2024 ) NoStop

    2024 doi

  16. [24]

    LeVan \ and\ author S

    author author J. LeVan \ and\ author S. D. \ Baalrud ,\ 10.1103/PhysRevE.111.015202 journal journal Phys. Rev. E \ volume 111 ,\ pages 015202 ( year 2025 ) NoStop

    2025 doi

  17. [25]

    author author R. J. \ Bearman \ and\ author J. G. \ Kirkwood ,\ 10.1063/1.1744056 journal journal The Journal of Chemical Physics \ volume 28 ,\ pages 136 ( year 1958 ) NoStop

    1958 doi

  18. [26]

    author author G. A. \ Maugin ,\ title Non-equilibrium thermodynamics of electromagnetic solids , \ in\ 10.1007/978-3-7091-4321-6_4 booktitle Non-Equilibrium Thermodynamics with Application to Solids: Dedicated to the Memory of Professor Theodor Lehmann ,\ editor edited by\ edi...

    doi

  19. [27]

    Wolff \ and\ author A

    author author P. Wolff \ and\ author A. Albano ,\ https://doi.org/10.1016/0378-4371(79)90149-3 journal journal Physica A: Statistical Mechanics and its Applications \ volume 98 ,\ pages 491 ( year 1979 ) NoStop

    1979 doi

  20. [28]

    Kluitenberg ,\ https://doi.org/10.1016/0031-8914(73)90131-6 journal journal Physica \ volume 68 ,\ pages 75 ( year 1973 ) NoStop

    author author G. Kluitenberg ,\ https://doi.org/10.1016/0031-8914(73)90131-6 journal journal Physica \ volume 68 ,\ pages 75 ( year 1973 ) NoStop

    1973 doi

  21. [29]

    Curie ,\ @noop title Works of Pierre Curie: published by the French Physical Society \ ( publisher Gauthier-Villars ,\ year 1908 ) NoStop

    author author P. Curie ,\ @noop title Works of Pierre Curie: published by the French Physical Society \ ( publisher Gauthier-Villars ,\ year 1908 ) NoStop

    1908

  22. [30]

    author author G. B. \ Arfken , author H.-J. \ Weber , \ and\ author F. E. \ Harris ,\ @noop title Mathematical Methods for Physicists \ ( publisher Academic ,\ address Oxford ,\ year 2012 ) NoStop

    2012

  23. [31]

    Hooyman , author P

    author author G. Hooyman , author P. Mazur , \ and\ author S. de Groot ,\ https://doi.org/10.1016/S0031-8914(54)91998-9 journal journal Physica \ volume 21 ,\ pages 355 ( year 1954 ) NoStop

    1954 doi

  24. [32]

    Scheiner \ and\ author S

    author author B. Scheiner \ and\ author S. D. \ Baalrud ,\ 10.1103/PhysRevE.102.063202 journal journal Phys. Rev. E \ volume 102 ,\ pages 063202 ( year 2020 ) NoStop

    2020 doi

  25. [33]

    author author V. A. \ Istomin \ and\ author E. V. \ Kustova ,\ 10.1063/1.4975315 journal journal Physics of Plasmas \ volume 24 ,\ pages 022109 ( year 2017 ) NoStop

    2017 doi

  26. [34]

    Ferziger \ and\ author H

    author author J. Ferziger \ and\ author H. Kaper ,\ https://books.google.com/books?id=mxS2AAAAIAAJ title Mathematical Theory of Transport Processes in Gases \ ( publisher North-Holland Publishing Company ,\ year 1972 ) NoStop

    1972

  27. [35]

    \ Hansen \ and\ author I

    author author J.-P. \ Hansen \ and\ author I. R. \ McDonald ,\ @noop title Theory of simple liquids: with applications to soft matter \ ( publisher Academic press ,\ year 2013 ) NoStop

    2013

  28. [36]

    Reif ,\ @noop title Fundamentals of Statistical and Thermal Physics \ ( publisher McGraw Hill ,\ address Tokyo ,\ year 1965 ) NoStop

    author author F. Reif ,\ @noop title Fundamentals of Statistical and Thermal Physics \ ( publisher McGraw Hill ,\ address Tokyo ,\ year 1965 ) NoStop

    1965

  29. [37]

    author author R. K. \ Pathria \ and\ author P. D. \ Beale ,\ @noop title Statistical Mechanics ,\ edition 3rd \ ed.\ ( publisher Elsevier/Academic Press ,\ address Amsterdam ; Boston ,\ year 2011 ) NoStop

    2011

  30. [38]

    author author Ichimaru ,\ @noop title Statistical Plasma Physics Volume I \ ( publisher CRC Press, ,\ year 1991 ) NoStop

    1991

  31. [39]

    J Evans \ and\ author G

    author author D. J Evans \ and\ author G. P Morriss ,\ @noop title Statistical mechanics of nonequilbrium liquids \ ( publisher ANU Press ,\ year 2007 ) NoStop

    2007

  32. [40]

    Vieillefosse \ and\ author J

    author author P. Vieillefosse \ and\ author J. P. \ Hansen ,\ 10.1103/PhysRevA.12.1106 journal journal Phys. Rev. A \ volume 12 ,\ pages 1106 ( year 1975 ) NoStop

    1975 doi

  33. [41]

    Ott , author M

    author author T. Ott , author M. Bonitz , \ and\ author Z. Donk\'o ,\ 10.1103/PhysRevE.92.063105 journal journal Phys. Rev. E \ volume 92 ,\ pages 063105 ( year 2015 ) NoStop

    2015 doi

  34. [42]

    Donk\'o , author B

    author author Z. Donk\'o , author B. Ny\' ri , author L. Szalai , \ and\ author S. Holl\'o ,\ 10.1103/PhysRevLett.81.1622 journal journal Phys. Rev. Lett. \ volume 81 ,\ pages 1622 ( year 1998 ) NoStop

    1998 doi

  35. [43]

    Daligault , author K

    author author J. Daligault , author K. O. \ Rasmussen , \ and\ author S. D. \ Baalrud ,\ 10.1103/PhysRevE.90.033105 journal journal Phys. Rev. E \ volume 90 ,\ pages 033105 ( year 2014 ) NoStop

    2014 doi

  36. [44]

    Zwanzig ,\ 10.1063/1.1725558 journal journal The Journal of Chemical Physics \ volume 40 ,\ pages 2527 ( year 1964 ) NoStop

    author author R. Zwanzig ,\ 10.1063/1.1725558 journal journal The Journal of Chemical Physics \ volume 40 ,\ pages 2527 ( year 1964 ) NoStop

    1964 doi

  37. [45]

    Frenkel \ and\ author B

    author author D. Frenkel \ and\ author B. Smit ,\ @noop title Understanding Molecular Simulation: From Algorithms to Applications ,\ edition 2nd \ ed.,\ series Computational Science Series , Vol. volume 1 \ ( publisher Academic Press ,\ address San Diego ,\ year 2002 ) NoStop

    2002

  38. [46]

    author author S. H. \ Glenzer \ and\ author R. Redmer ,\ 10.1103/RevModPhys.81.1625 journal journal Rev. Mod. Phys. \ volume 81 ,\ pages 1625 ( year 2009 ) NoStop

    2009 doi

  39. [47]

    author author S. G. \ Brush , author H. L. \ Sahlin , \ and\ author E. Teller ,\ 10.1063/1.1727895 journal journal The Journal of Chemical Physics \ volume 45 ,\ pages 2102 ( year 1966 ) ,\ http://arxiv.org/abs/https://pubs.aip.org/aip/jcp/article-pdf/45/6/2102/18845826/2102\_...

    1966 doi

  40. [48]

    Baus \ and\ author J.-P

    author author M. Baus \ and\ author J.-P. \ Hansen ,\ https://doi.org/10.1016/0370-1573(80)90022-8 journal journal Physics Reports \ volume 59 ,\ pages 1 ( year 1980 ) NoStop

    1980 doi

  41. [49]

    Tsednee \ and\ author T

    author author T. Tsednee \ and\ author T. Luchko ,\ 10.1103/PhysRevE.99.032130 journal journal Phys. Rev. E \ volume 99 ,\ pages 032130 ( year 2019 ) NoStop

    2019 doi

  42. [50]

    author author A. V. \ Filinov , author V. O. \ Golubnychiy , author M. Bonitz , author W. Ebeling , \ and\ author J. W. \ Dufty ,\ 10.1103/PhysRevE.70.046411 journal journal Phys. Rev. E \ volume 70 ,\ pages 046411 ( year 2004 ) NoStop

    doi

  43. [51]

    Dornheim , author M

    author author T. Dornheim , author M. Bonitz , author Z. A. \ Moldabekov , author S. Schwalbe , author P. Tolias , \ and\ author J. Vorberger ,\ 10.1103/PhysRevB.111.115149 journal journal Phys. Rev. B \ volume 111 ,\ pages 115149 ( year 2025 ) NoStop

    doi

  44. [52]

    Bonitz , author J

    author author M. Bonitz , author J. Vorberger , author M. Bethkenhagen , author M. P. \ Böhme , author D. M. \ Ceperley , author A. Filinov , author T. Gawne , author F. Graziani , author G. Gregori , author P. Hamann , author S. B. \ Hansen , author M. Holzmann , author S. X....

  45. [53]

    \ van Leeuwen ,\ 10.1051/jphysrad:01921002012036100 journal journal Journal de Physique et le Radium \ volume 2 ,\ pages 361 ( year 1921 ) NoStop

    author author H.-J. \ van Leeuwen ,\ 10.1051/jphysrad:01921002012036100 journal journal Journal de Physique et le Radium \ volume 2 ,\ pages 361 ( year 1921 ) NoStop

    1921 doi

  46. [54]

    Balescu ,\ in\ https://doi.org/10.1016/B978-0-444-87091-9.50009-9 booktitle Classical Transport ,\ series and number Transport Processes in Plasmas ,\ editor edited by\ editor R

    author author R. Balescu ,\ in\ https://doi.org/10.1016/B978-0-444-87091-9.50009-9 booktitle Classical Transport ,\ series and number Transport Processes in Plasmas ,\ editor edited by\ editor R. Balescu \ ( publisher North-Holland ,\ address Amsterdam ,\ year 1988 )\ pp.\ pag...

    doi

  47. [55]

    author author D. A. \ Gurnett \ and\ author A. Bhattacharjee ,\ @noop title Introduction to plasma physics: with space and laboratory applications \ ( publisher Cambridge university press ,\ year 2005 ) NoStop

    2005

  48. [56]

    author author T. J. M. \ Boyd \ and\ author J. J. \ Sanderson ,\ @noop title The Physics of Plasmas \ ( publisher Cambridge University Press ,\ year 2003 ) NoStop

    2003

  49. [57]

    author author J. P. \ Freidberg ,\ @noop title Ideal MHD \ ( publisher Cambridge University Press ,\ year 2014 ) NoStop

    2014

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