Variational principles and thermodynamics

P\'eter V\'an; R\'obert Kov\'acs

REVIEW 3 major objections 5 minor 68 references

Variational principles and thermodynamics

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Pith's one-line read The paper argues that the second law alone can substitute variational principles and construct both dissipative and nondissipative evolution equations.

desk verdict Programmatic survey with a clean Cahn-Hilliard example, but the one novel derivation (inertial gravitation) fails at the key elimination step, so the universal claim is unsupported as written. read the letter →

arxiv 1908.02679 v1 pith:72KCCYYT submitted 2019-08-07 cond-mat.stat-mech

Péter Ván , Róbert Kovács This is my paper · ORCID

classification cond-mat.stat-mech

keywords secondlawofthermodynamicsnonequilibriumvariationalprincipleinternalvariablesentropyproductionCahn-HilliardequationNewtoniangravitationHamiltonianmechanics

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 tries to establish that the second law of thermodynamics, joined with internal variables and gradient-dependent entropies, can serve as a complete generator of evolution equations for both ideal and dissipative systems. It shows that Hamiltonian point-mass mechanics emerges from dual thermodynamic variables when entropy production vanishes, that phase-field and coupled fluid equations follow from the entropy inequality, and that a dissipative theory of Newtonian gravitation with inertia can be derived the same way, including a wave equation for the gravitational potential in the nondissipative limit. If this is right, the search for variational principles in dissipative physics can be redirected toward thermodynamic construction, with the second law acting as the selection rule for the laws of nature.

What carries the argument

The central object is entropy treated as a generating potential, a field-theoretic density rather than a statistical quantity. The working machinery is the entropy balance: a Gibbs relation extended to internal variables and their gradients, followed by separation of divergences to identify the entropy flux and the entropy production, and then linear constitutive relations between thermodynamic fluxes and forces with coefficients constrained by the inequality. Dual internal variables provide the antisymmetric part of the transport matrix, which becomes the symplectic, Hamiltonian structure when the dissipative coefficients vanish, while weak nonlocality, meaning dependence on gradients, supplies the higher-derivative terms that produce the phase-field and gravitational field equations.

What would settle it

Measure the propagation speed of gravitational-potential disturbances in a self-gravitating fluid at known density $\rho$: the paper's nondissipative limit predicts a scalar wave with speed $c^2 = a^2/(4\pi G K \rho)$, which depends on the ambient density. Observing a density-independent speed, or no such scalar mode at all, would falsify the inertial-gravitation derivation; more broadly, finding a dissipative system whose empirically correct evolution equation cannot be produced from any concave entropy with internal variables and gradients would falsify the general program.

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Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the entropy inequality is constructive rather than merely restrictive. Writing entropy as a concave potential that increases in time, and choosing the entropy flux by separating divergences, yields linear flux-force relations whose antisymmetric part becomes ideal, symplectic dynamics and whose symmetric part becomes dissipation. Three demonstrations carry the claim: a single point mass whose position and momentum are dual internal variables; a two-component heat-conducting diffusive fluid whose gradient-dependent entropy produces Fourier-Navier-Stokes-Cahn-Hilliard-Korteweg equations; and a self-gravitating fluid with two scalar internal fields whose additive energy decomposition produces a dissipative massive Newtonian gravitation, reducing to a wave equation for the gravitational potential in the nondissipative limit.

Load-bearing premise

For the inertial-gravitation example, everything rests on assuming the internal energy splits additively as $u = e - \varphi - \frac{\partial_i\varphi\partial^i\varphi}{8\pi G\rho} - \frac{\psi^2}{2K}$, with the second scalar $\psi$ introduced solely to produce inertial effects; no independent physical evidence is offered for that decomposition, and without it the derived gravitational wave equation has no basis.

Editorial extensions

If this is right

Hamiltonian mechanics, usually derived from a variational principle, can be obtained as the zero-dissipation limit of dual-variable thermodynamics, with dissipative corrections carrying fixed signs required by the second law.
The Cahn-Hilliard equation and the coupled Fourier-Navier-Stokes-Cahn-Hilliard-Korteweg fluid equations can be derived from the entropy inequality, fixing cross-coupling coefficients and correcting variational constructions.
Newtonian gravitation can be given a dissipative, inertial extension in which the gravitational potential obeys a wave equation in the ideal limit and couples to the spherical part of the stress tensor.
The same constructive procedure works for pure memory, pure nonlocality, and mixed cases, so the second law can serve as a unified selection rule for evolution equations.
The need for dissipation potentials and other ad hoc additions to variational principles disappears, because dissipative and nondissipative parts arise from one thermodynamic calculation.

Reading between the lines

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

Inference: if the program is right, variational principles become a shortcut valid mainly for the nondissipative sector, while dissipative physics would be derived from entropy rather than patched onto Lagrangians.
Inference: the density-dependent wave speed predicted for gravitational-potential disturbances is a testable signature that could constrain the inertial coefficient $K$ from observations or experiments.
Inference: because the method requires choosing which fields count as internal variables and how entropy depends on them, its predictive power ultimately depends on a yet-unstated selection principle for the thermodynamic state space; without one, many entropy ansatze may yield the same equations.
Inference: the equivalence shown in nondissipative limits suggests a duality between Hamilton's principle and the second law, with conservation laws from symmetry corresponding to thermodynamically imposed balance constraints.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a programmatic paper arguing that the second law can replace variational principles for constructing both dissipative and nondissipative equations. The first two examples are clean restatements of known results; the third, the inertial-gravitation derivation, has a real algebraic error. I checked the stress-test note and it is correct: eq. (45) does not follow from (42)-(43). With l1=l2=0 and l12=-l21=a, those equations give φ¨ = (a²ρ/K)F plus a ρ˙/ρ term, so for constant ρ you get (K/a²)φ¨ = ρF, not eq. (45). Equation (44) is also garbled, with an 'L' that is never defined. That collapses the paper's only novel support.

What the paper does well: it is honest about its lineage. It explicitly credits GENERIC, metriplectic dynamics, and extended thermodynamics, and presents itself as a programmatic unification rather than a new formalism. The point-mass section is a nice pedagogical illustration of dual internal variables and conserved entropy giving Hamiltonian dynamics. The Cahn-Hilliard section is a competent use of the divergence-separation method; the comparison between entropic and energy representations, including the coefficient relations in eq. (32), is correct and useful. None of this is new, but it is sound.

The weak point is exactly where the paper makes its strongest claim. The second-law-as-generator thesis is already present in the frameworks the authors cite; their only new support is Example III, and that example has a wrong elimination step. To make matters worse, the internal-energy decomposition (34) is ad hoc: ψ is introduced purely to produce inertial terms, with no physical or observational justification. That would be acceptable in a programmatic paper if the algebra were right, but together with the error it leaves the universal claim resting on restatements.

This paper is for readers interested in the conceptual relationship between thermodynamics and variational principles. The survey and the first two examples are worth a skim, but as written the paper does not establish its advertised conclusion. I would still send it to peer review rather than desk-reject: the program is worth discussing, and the flaw is fixable. The authors can either correct the elimination or explicitly limit the claim to the two known examples. A competent referee will catch the gravity issue; the editor should not have to.

Referee Report

3 major / 5 minor

Summary. The paper argues that modern nonequilibrium thermodynamics—using the entropy inequality, internal variables, and weak nonlocality—can construct both dissipative and nondissipative evolution equations, thereby offering an alternative to variational principles. Three examples are presented: (i) Hamiltonian mechanics of a point mass from dual internal variables and a conserved entropy; (ii) a two-component compressible fluid leading to Fourier-Navier-Stokes-Cahn-Hilliard-Korteweg equations; and (iii) Newtonian gravitation with inertia, derived from an extended entropy balance with two scalar internal fields. The central claim, stated in the abstract and conclusions, is that the second law alone can be an effective, universal tool for generating evolution equations, potentially replacing variational principles.

Significance. If fully supported, the paper would provide a conceptually valuable unification: it would show that the entropy principle can generate both the symplectic (nondissipative) and dissipative structures of evolution equations, matching and in some cases correcting variational constructions. The first two examples are useful didactic demonstrations that are largely consistent with existing metriplectic/GENERIC and rational thermodynamics results. The third example, however, is the main novelty—it claims to derive inertial gravitational wave equations from thermodynamics. The validity of that derivation is crucial to the paper's universal claim. Because the elimination step leading to the wave equation is algebraically incorrect, the strongest novel support for the paper's thesis fails as written. The paper remains a competent survey and a plausible research program, but it does not currently substantiate its most ambitious claim.

major comments (3)

[Section 5, Eqs. (42)-(45)] The claimed reduction to the wave equation (45) is algebraically incorrect. With l1=l2=0 and l12=-l21=a, Eqs. (42)-(43) become φ̇ = -aρψ/K and ψ̇ = -a(∂i∂iφ/(4πG)-ρ). Differentiating the first equation and substituting the second gives, for constant ρ, (4πGK/a²)φ̈ - ρ∂i∂iφ + 4πGρ² = 0, which is not Eq. (45), namely (4πGK/a²)ρφ̈ - ∂i∂iφ - 4πGρ = 0. The two equations differ by a factor ρ in the leading term and by the sign and form of the source term. Therefore the advertised nondissipative wave equation does not follow from the stated transport equations, and Example III cannot support the paper's central claim as written.
[Section 5, Eq. (44)] Equation (44) is not derived from Eqs. (42)-(43) and contains an undefined coefficient 'L'. The two displayed expressions on the right-hand side are not shown to be equal, and the term '-LK(∂i∂iφ/(4πG) - l2/L φ̇ - ρ)' is dimensionally unclear. A step-by-step elimination of ψ from (42)-(43) is needed; as it stands, Eq. (44) cannot be used to justify the subsequent ideal limit (45).
[Section 5, Eq. (34)] The additive decomposition u = e - φ - (∂iφ∂iφ)/(8πGρ) - ψ²/(2K) introduces a second scalar internal variable ψ and an inertial coefficient K without independent physical or observational justification. Since the derivation of the inertial gravitational equations depends entirely on this decomposition, Example III is a construction that assumes the desired inertial behavior rather than a derivation from the second law alone. The authors should either provide a physical motivation for (34) or explicitly frame the example as an illustrative constitutive assumption, not as a predictive derivation.

minor comments (5)

[Section 6] The section references in the Summary and conclusions are off by one: Section 2 (point mass) should be Section 3, Section 3 (phase fields) should be Section 4, and Section 4 (gravity) should be Section 5.
[Section 4, Eq. (33)] Equation (33) contains a garbled expression involving the functional derivative: '∂cs − 1/ρ ∂k∂∂ kc(ρs)' should presumably read '∂c s − (1/ρ)∂k(∂_{∂k c}(ρs))' or a similar clear notation.
[Section 3, Eq. (14)] The inequality 'l11l22 − (l11+l11)²/2 > 0' appears to be a typo; based on the subsequent text it should be 'l11l22 − l² > 0' (where l=(l12+l21)/2).
[Throughout] The manuscript contains numerous typographical errors, including 'no ndissipative' in the abstract, 'V ARIATIONAL' in the title header, 'devide' in Section 5, and 'Tams Flp' in the acknowledgment. These should be corrected.
[References] References [11] and [36] are listed as 'this volume' with no other bibliographic information; for an arXiv preprint these should be completed or removed.

Circularity Check

2 steps flagged · score 5.0 of 10

Examples I and III construct the target equations by inserting them as potentials, so the claimed universality of the second-law program is partially circular, although the thermodynamic formalism and Example II retain independent content.

self definitional [Section 3.3, Eqs. (10)-(15)]
"However, in case of pure mechanics without temperature, H may play the role of free energy and S := −H. ... If the dissipative (symmetric) part is zero, the conservation of S follows. The antisymmetric part LI generates ideal evolution, a symplectic dynamics. It is identical with (1)-(2) if k = 1."

With S = −H and k = 1, equations (11)-(12) become xdot = ∂pH and pdot = −∂xH, which are exactly the input Hamilton equations (1)-(2). The Hamiltonian is chosen as the thermodynamic potential, and the antisymmetric matrix is chosen with unit coefficient. The entropy inequality only constrains the symmetric dissipative part; it does not select H or the symplectic structure. Thus the 'thermodynamic derivation' of Hamiltonian mechanics is a restatement of the chosen input in thermodynamic notation, not an independent output of the second law.
fitted input called prediction [Section 5, Eqs. (34), (42)-(43), (45)]
"The last quadratic term is responsible for inertial effects, K is an inertial coefficient, as usual in variational principles. ... In this case (44) reduces to the wave equation, as it is expected: K4πG a2 ρ ¨ϕ −∂i∂iϕ − 4πGρ = 0."

The inertial variable ψ enters only through the ad hoc term −ψ^2/(2K) in the internal energy ansatz (34), and the text says this term is 'responsible for inertial effects.' The wave equation (45) is then announced 'as it is expected.' The inertial structure of the output is therefore built into the chosen potential, not derived from the second law. Moreover, the reduction from (42)-(43) to (45) is asserted rather than shown; even accepting the ansatz, the stated derivation does not exhibit the promised wave equation as an independent consequence.

full rationale

The paper is largely a self-contained thermodynamic construction: given an entropy/energy potential and a state space, the entropy inequality is exploited to obtain Onsager-type constitutive relations. Example II (Fourier–Navier–Stokes–Cahn–Hilliard–Korteweg) is a standard and largely independent demonstration of that logic. The circularity burden lies in the framing of Examples I and III. Example I inserts S = −H and a unit antisymmetric matrix, making Hamilton's equations an exact restatement of the input. Example III inserts a quadratic internal variable specifically chosen to produce inertia and then presents the wave equation 'as it is expected,' so the output is constructed from the ansatz in Eq. (34). The paper's own concluding caveat that entropy, 'like a Lagrangian, is the generating potential of the evolution of the fields,' concedes that the second law alone does not fix the dynamics. No load-bearing self-citation was found; reference [64] is motivational only. The apparent algebraic mismatch between Eqs. (42)-(43) and (45) is a correctness concern rather than a circularity step, but it reinforces that the inertial-gravitation example does not provide the independent support claimed for the universal program.

Assumptions & free parameters 4 free parameters · 5 assumptions · 1 invented entities

The central claim rests on the assumption that a concave increasing entropy can generate all relevant evolution equations, supplemented by the quasilinear Onsager form and particular choices of state-dependent potentials. The gravity example further assumes a specific additive decomposition of the internal energy, introducing a new internal variable ψ and free coefficients K and a with no external constraints.

free parameters (4)

Inertial coefficient K
Introduced in eq. (34) as the coefficient of the ψ^2 term; governs the inertial response of the gravitational potential. No independent determination; chosen to make the nondissipative limit a wave equation.
Symplectic coupling a
In the nondissipative limit l12 = -l21 = a; sets the speed of the gravitational wave in eq. (45). Not fixed by data.
Onsager coefficients (l1, l2, l12, l21, l13, l23, l31, l32, l3)
Constitutive coefficients in eqs. (37)-(41), constrained only by positive semidefiniteness of the dissipative part. Not determined by the theory.
Onsager coefficients in point-mass example (l11, l22, l, k)
Coefficients in eqs. (11)-(12); k=1 recovers Hamilton equations; others are free and constrained by the second law.

assumptions (5)

domain assumption There exists a concave thermodynamic potential (entropy or negative free energy) that is nondecreasing along the evolution.
The entire construction in Sections 3.2-3.3 rests on this postulate.
domain assumption Evolution equations can be written in the quasilinear Onsager form (11)-(12) with a positive semidefinite symmetric part and an antisymmetric part.
Used in Sections 3, 4, and 5 to derive all examples; standard in nonequilibrium thermodynamics.
domain assumption The entropy depends on the gradients of state variables, with the extended Gibbs relation (25).
Needed in Section 4 to obtain Cahn-Hilliard type couplings.
domain assumption Linear flux-force relations and Onsager reciprocity hold.
Used to obtain constitutive equations (28) and (31).
ad hoc to paper The internal energy of a self-gravitating fluid has the additive decomposition (34).
This specific form is introduced in Section 5 to generate inertial effects; no independent justification is given.

invented entities (1)

Second scalar internal variable ψ
purpose: Generates inertial effects in the evolution of the gravitational potential, yielding a wave-like equation.
Introduced in eq. (34); no observable consequence is derived or compared with data.

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Cite this review

Pith. "Pith review of Variational principles and thermodynamics." pith.science (2026). https://pith.science/paper/72KCCYYT

@misc{pith2026190802679,
  author       = {Pith},
  title        = {Pith review of: Variational principles and thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/72KCCYYT}},
  note         = {Machine review of arXiv:1908.02679}
}

read the original abstract

Variational principles play a fundamental role in deriving evolution equations of physics. They are working well in case of nondissipative evolution but for dissipative systems they are not unique, not predictive and not constructive. With methods of modern nonequilibrium thermodynamics, one can derive evolution equations for dissipative phenomena and, surprisingly, can also reproduce the Euler-Lagrange form of the evolution equations for ideal processes. In this work, we examine some demonstrative examples and compare thermodynamic and variational techniques. Then, we argue that instead of searching for variational principles for dissipative systems, a different program can be more fruitful: the second law alone can be an effective tool to construct both dissipative and nondissipative evolution equations.

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Pith tools

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

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