REVIEW 4 major objections 5 minor 21 references
A geometric description of some thermodynamical systems
T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read This paper claims that the evolution equations of several classes of thermodynamic systems are exactly the integral curves of evolution vector fields defined by partially cosymplectic structures.
desk verdict A solid geometric reformulation of Gay-Balmaz–Yoshimura thermodynamics, but the entropy balance is put in by hand via the 1-form η and the abstract oversells future work. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is a partially cosymplectic structure: a manifold carrying a closed 2-form $\omega$ and one or more 1-forms $\eta_k$ whose top wedge product with $\omega$ is nonzero, with the $\eta_k$ not required to be closed. The machinery is the bundle isomorphism $\flat(X)=i_X\omega+\sum_k \eta_k(X)\eta_k$, which converts a chosen 1-form $dH+\eta-F^{\mathrm{ext}}$ into a unique evolution vector field $E_H$. The 1-form $\eta$ is built from the thermodynamic data, such as the entropy derivative $\partial H/\partial S$, the friction force, and in the extended cases the mass and heat fluxes, so that matching coefficients in the basis $\{dq^i,dp_i,\ldots,\eta\}$ forces the mechanical equations and the entropy balance simultaneously.
What would settle it
A concrete check: take a simple system with friction, choose a Hamiltonian and a friction force, and see whether every solution of the variational equations in [8] also satisfies the entropy-production identity $-\frac{\partial H}{\partial S}\frac{dS}{dt}=\frac{dq^j}{dt}F^{\mathrm{fr}}_j$; one trajectory that violates it would show that the paper's $\eta$ does not describe that system, and a solution of those equations that is not an integral curve of any evolution vector field $\flat(E_H)=dH+\eta-F^{\mathrm{ext}}$ with the paper's $\eta$ would separate the two formalisms.
Extended reading notes
Core claim
The central claim is that each of the four thermodynamic settings treated in the paper is governed by the same geometric prescription: choose a partially cosymplectic structure on the extended phase space, choose the 1-form $\eta$ that encodes entropy and dissipative effects, and define the evolution vector field $E_H$ by $\flat(E_H)=dH+\eta-F^{\mathrm{ext}}$. The integral curves of $E_H$ then satisfy exactly the equations of motion and entropy balance for that system, and in each case these equations coincide with those obtained in [8] from a variational principle. The paper establishes this in Propositions 4.1, 4.3, 4.5, and 4.7 and passes to the Lagrangian formulation through the Legendre transformation.
Load-bearing premise
The load-bearing premise is that the entropy-and-friction term can be chosen in advance as minus the entropy derivative times $dS$ minus the friction force, a formula the paper fixes so that the resulting equations match the variational ones rather than deriving it from a deeper principle.
Editorial extensions
If this is right
- The variational nonequilibrium thermodynamics of [8] and the partially cosymplectic description are equivalent for the four classes treated, so results transfer between the two languages.
- The Legendre transformation is a cosymplectomorphism, so the Lagrangian and Hamiltonian formulations of these thermodynamic systems are geometrically identical.
- Entropy production is not added by hand: in the simplest case it follows from the $\eta$-component of $\flat(E_H)=dH+\eta-F^{\mathrm{ext}}$, giving $-\frac{\partial H}{\partial S}\frac{dS}{dt}=\frac{dq^j}{dt}F^{\mathrm{fr}}_j$.
- The contact-geometry description of simple systems with friction is recovered as a special case, obtained when the friction force is chosen as $F^{\mathrm{fr}}_i=-R(H)p_i$.
- The geometric formulation opens the way to applying symmetry reduction, Hamilton-Jacobi theory, and structure-preserving discretization to thermodynamic systems.
Reading between the lines
- Editorial inference: the 1-form $\eta$ is chosen to match known equations; if no independent principle fixes $\eta$, the paper's contribution is a geometric reformulation of [8] rather than new dynamics.
- Editorial inference: the same construction should carry over to continuum thermodynamics by working on infinite-dimensional manifolds, but the paper does not prove that extension.
- Editorial inference: the division by $\partial H/\partial S$ restricts the description to states with nonzero temperature; a limiting or regularized version would be needed to cover zero-temperature processes.
- Editorial inference: the cosymplectomorphism property suggests that any symmetry preserving the pair $(\omega,\eta)$ yields a reduced thermodynamic system, which could be tested by applying symplectic reduction to a concrete heat-conducting system.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proposes a geometric framework for certain finite-dimensional nonequilibrium thermodynamical systems based on 'partially cosymplectic structures,' where the 2-form is closed but the 1-form(s) η need not be closed. The authors define an evolution vector field by ♭(E_f) = df + η − F and show, by direct coordinate computation, that for suitable choices of η the integral curves reproduce the evolution equations previously derived by Gay-Balmaz and Yoshimura using a variational approach. The cases treated are adiabatically closed simple systems, systems with internal mass transfer, adiabatically closed non-simple systems with several entropies, and open simple systems with ports. The paper also proves covariance of the construction under the Legendre transformation for hyperregular Lagrangians and sketches future applications to reduction, Hamilton-Jacobi theory, and discretization.
Significance. If the construction is made fully rigorous, the paper offers a compact geometric reorganization of a family of known thermodynamical evolution equations. Its strengths are the explicit and largely consistent coordinate computations, the systematic treatment of increasing physical complexity (friction, internal mass transfer, heat conduction, open ports), and the preservation of the structure under Legendre transforms. The main value is as a reformulation: it shows that the Gay-Balmaz-Yoshimura equations can be written as flows of a partially cosymplectic evolution vector field. However, the 1-form η is essentially chosen by hand to match the target equations, and the paper does not supply an independent principle that would make the framework 'natural' beyond this fitting. In addition, the general isomorphism theorem for higher-order partially cosymplectic structures has a proof gap when the manifold dimension exceeds 2n+p. These issues currently limit the significance of the contribution to a mostly notational reformulation.
major comments (4)
- [Section 3.2, Proposition 3.2] The proof that ♭ is an isomorphism assumes that ω^n ∧ η_1 ∧ ... ∧ η_p is a top-degree form on M, which requires dim M = 2n+p. In the thermodynamic applications of Sections 4.3 and 4.4, however, the manifold M has dimension 2n+5P (with p=P) and 2n+5 (with p=1), so the wedge is not a top-degree form and the argument does not apply. The nondegeneracy condition in Definition 3.2 is therefore not sufficient, as stated, to guarantee invertibility of ♭ in exactly the cases used later. Please either generalize Definition 3.2 to an appropriate top-degree condition (e.g., ω^N ∧ η_1 ∧ ... ∧ η_p ≠ 0 with N = (dim M − p)/2) and prove the isomorphism under that condition, or verify invertibility of ♭ directly for the explicit structures in Sections 4.3 and 4.4, which will require ∂H/∂S_A ≠ 0.
- [Section 4.1, before Eq. (9), and analogous definitions in 4.2–4.4] The 1-form η is chosen so that the entropy balance emerges as the η-component of Eq. (9). In Section 4.1, η = −(∂H/∂S)dS − F^fr is defined so that the dS term in dH cancels, and the entropy production law (15) is equivalent to the coefficient of η in Eq. (11) being zero. No independent geometric or physical principle selects this η; it is fixed by the requirement that the resulting equations coincide with those of Gay-Balmaz-Yoshimura [8]. If the contribution is intended as a reformulation, this should be stated explicitly and the status of η as a prescribed geometric datum should be clarified. If a derivation is intended, a principle determining η is missing. As written, the claim that almost cosymplectic structures are a 'natural framework' is not supported by the manuscript.
- [Section 4.1, Eqs. (14), (25), (40), (57), and the nondegeneracy condition] The evolution equations divide by ∂H/∂S (or ∂H/∂S_k in the multi-subsystem case), and the partially cosymplectic nondegeneracy condition also requires ∂H/∂S ≠ 0. This is equivalent to assuming nonzero temperature, but this hypothesis is nowhere stated. The propositions as written ('Every integral curve ... is a solution of ...') are false without this assumption. Please add the explicit hypothesis ∂H/∂S ≠ 0 (respectively ∂H/∂S_A ≠ 0) throughout and state its physical meaning as a nonzero-temperature condition.
- [Section 4.4, Eq. (48)] In the open-system case, the right-hand side of Eq. (48) contains an additional term proportional to η, namely (∑_a(J_a μ_a + J_a^S T^a) + ∑_b J_b^S T^b)η, which is not a semibasic form. This term is introduced solely to recover the open-system entropy balance (57). The geometric meaning of such a non-semibasic 'force' term is not discussed, and it reinforces the concern that η and the force decomposition are fitted to the target equations rather than derived from a structural principle. Please clarify the status of non-semibasic terms in Definition 3.3 and in the interpretation of forces.
minor comments (5)
- [Section 4.2 and Section 4.3, local expressions for ♭] In both sections the local expression for ♭(∂/∂p_i) is written as −dp_i, but the correct expression is −dq^i, as follows from ω = dq^i ∧ dp_i and as is used in the subsequent coefficient matching. Please correct these typos.
- [Section 4.3, Eq. (40) and Eq. (47)] In Eq. (40), the denominator should be ∂H/∂S_k, not ∂H/∂S, and in Eq. (47) the left-hand side should involve ∂L/∂S_k (or carry a subscript k). The coefficient matching in Eq. (33) shows that the entropy evolution for subsystem k divides by ∂H/∂S_k. As printed, the index structure is inconsistent.
- [Definition 3.2 and Section 4.3] The dimension statement '2n+p' in Definition 3.2 conflicts with the higher-dimensional applications in Sections 4.3 and 4.4. If the generalized definition is retained, the dimension condition should be updated and the notation adjusted consistently.
- [Section 4.2, definition of J] The definition J = ∑_l J_{l,k} dW^k := J_k dW^k is ambiguous because of the antisymmetry convention. Please clarify the meaning of J_k and the summation indices so that Eqs. (24)–(26) are unambiguous.
- [Throughout the paper] The paper would benefit from an explicit statement connecting ∂H/∂S (or ∂H/∂S_A) to the thermodynamic temperature, and from a brief physical interpretation of η in each of the examples. This would help the reader see why the chosen η is not completely arbitrary.
Circularity Check
Entropy balance is encoded in the hand-picked 1-form η, so the geometric formalism reproduces Gay-Balmaz–Yoshimura by construction rather than by independent derivation.
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self definitional
[Section 4.1, Eqs. (9)–(15)]
"We define the 1-form over M: η = −∂H/∂S dS − F^fr ... ♭(EH) = dH + η − F^ext (9) ... AiF^fr_i + C ∂H/∂S = 0 ... dS/dt = −1/(∂H/∂S) ∂H/∂p_j F^fr_j (14) ... −∂H/∂S dS/dt = dq^j/dt F^fr_j (15)"
The coefficient comparison in (11) is done in the basis {dqi, dpi, η}; because η was defined to contain −(∂H/∂S)dS, the dS term of dH cancels in (10) before the η-component is read off. Solving the resulting condition A^iF^fr_i + C∂H/∂S = 0 gives dS/dt = −(∂H/∂p_j F^fr_j)/(∂H/∂S), which is (14), and (15) is just this multiplied by ∂H/∂S. Thus the entropy balance is an algebraic restatement of the chosen η, which was chosen to match the Gay-Balmaz–Yoshimura entropy production; it is not a consequence of the partially cosymplectic geometry alone.
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self definitional
[Section 4.2, Eqs. (18)–(26)]
"Let J = Σ J_l,k dW^k := J_k dW^k ... Consider in M the 1-form given by: η = −∂H/∂S dS − F^fr − J ... dS/dt = −1/(∂H/∂S)(∂H/∂p_j F^fr_j + J_k ∂H/∂N_k) (25)"
The same mechanism is repeated for internal mass transfer: the mass-flux 1-form J is inserted into η, and then the η-component condition (20) forces the entropy equation (25) to contain J_k∂H/∂N_k. No independent variational or physical principle selects J; it is the Gay-Balmaz–Yoshimura input, so the resulting entropy balance is prescribed rather than predicted.
1 more flagged steps
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fitted input called prediction
[Section 4.4, Eqs. (48) and (57)]
"We define the evolution vector field with external forces as the vector field EH satisfying: ♭(EH) = dH + η − F^ext − (Σ_a(Jaμa + JaS T a) + Σ_b JbS T b)η (48) ... −∂H/∂S dΣ/dt = dq^i/dt F^fr_i + Σ_a Ja dW/dt + (Σ_a JaS + Σ_b JbS)dΓ/dt − (Σ_a(Jaμa + JaS T a) + Σ_b JbS T b) (57)"
The term added on the right of (48) is not a semibasic force from Section 2.4; it is an η-multiple inserted so that the coefficient of η in (50) yields the open-system entropy balance (57). The port terms appearing in (57) are exactly the terms that were put into (48). The 'prediction' of the open-system entropy balance is therefore a rearrangement of the fitted input, not an independent result of the cosymplectic framework.
full rationale
The paper is transparent that its aim is to reobtain the Gay-Balmaz–Yoshimura equations, and the coefficient matchings in Sections 4.1–4.4 are internally consistent; the comparison with [8] is an external check, so there is no self-citation or imported-uniqueness problem. The circularity is in the status of the entropy balance: it is not derived from the partially cosymplectic structure but is placed there by the hand-selected 1-form η. In Section 4.1, η = −(∂H/∂S)dS − F^fr cancels the dS part of dH, and the entropy balance (15) is exactly the η-component identity A^iF^fr_i + C∂H/∂S = 0. The same construction is repeated for mass transfer (Section 4.2) and is made overt in the open system, where Eq. (48) adds a non-semibasic η-term so that Eq. (57) contains the desired port terms. Thus the claimed equivalence to Gay-Balmaz–Yoshimura is a faithful reformulation, but any reading that the entropy-production law is a consequence of almost cosymplectic geometry would be circular: the law is in the input. A separate correctness gap, not counted as circularity, is that Eqs. (14), (25), (40), and (57) divide by ∂H/∂S, while the nonzero-temperature hypothesis ∂H/∂S ≠ 0 is never stated.
Assumptions & free parameters
assumptions (6)
- ad hoc to paper The 1-form η = −(∂H/∂S)dS − F^fr (and its order-p analogues with J_k dW^k, J_AB dΓ^B, and port terms) is the correct geometric encoding of the thermodynamic forces of the system.
- domain assumption The physical temperature is identified with ∂H/∂S (resp. ∂H/∂S_A) and is assumed nonzero.
- ad hoc to paper The auxiliary variables Σ_A track the physical entropy: dS_A/dt = dΣ_A/dt along trajectories (Eq. (39)).
- domain assumption The Lagrangian is regular/hyperregular, i.e., the Legendre transformation is a (local or global) diffeomorphism.
- domain assumption The thermodynamic phase space admits the global product structure T*Q × P1 × ... × P5 × R with global coordinates W, N, Γ, S, Σ supporting the forms dW∧dN and dΓ∧d(S−Σ).
- standard math Darboux-type theorems for presymplectic structures and the partition-of-unity argument establish the Reeb vector field and the splitting T*M = H ⊕ ⟨η⟩.
invented entities (2)
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Auxiliary entropy variables Σ_A
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Partially cosymplectic structures of order p
Cite this review
Pith. "Pith review of A geometric description of some thermodynamical systems." pith.science (2026). https://pith.science/paper/4FZYTIYA
@misc{pith2026241218478,
author = {Pith},
title = {Pith review of: A geometric description of some thermodynamical systems},
year = {2026},
howpublished = {\url{https://pith.science/paper/4FZYTIYA}},
note = {Machine review of arXiv:2412.18478}
}
read the original abstract
In this paper we show how almost cosymplectic structures are a natural framework to study thermodynamical systems. Indeed, we are able to obtain the same evolution equations obtained previously by Gay-Balmaz and Yoshimura (see Entropy, 21(8):39, 2019) using variational arguments. The proposed geometric description allows us to apply geometrical tools to discuss reduction by symmetries, the Hamilton-Jacobi equation or discretization of these systems.
Reference graph
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