Mathematical methods of diagonalization of quadratic forms applied to the study of stability of thermodynamic systems

F. N. Lima; J. M. de Sousa

REVIEW 3 major objections 5 minor 25 references

Mathematical methods of diagonalization of quadratic forms applied to the study of stability of thermodynamic systems

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A consistent derivation ties the nonnegativity of $c_v$, $c_p$, $k_T$, and $k_s$ to the curvature classification of thermodynamic potentials by diagonalizing quadratic forms obtained from Taylor expansions.

desk verdict A well-intended teaching note whose central derivation rests on a false premise: setting the first derivatives of u(s,v) to zero at equilibrium, which is not true and not needed. read the letter →

arxiv 1908.03211 v1 pith:OVJWOZAH submitted 2019-08-08 cond-mat.stat-mech

F. N. Lima , J. M. De Sousa This is my paper · ORCID

classification cond-mat.stat-mech

keywords quadraticformsdiagonalizationthermodynamicstabilitypotentialsTaylorexpansioncurvatureminimumenergyprinciplecompressibility

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

Using second-order Taylor expansions and diagonalization of the resulting quadratic forms, the paper aims to derive the standard thermodynamic stability conditions from the curvature of the thermodynamic potentials. Starting from the minimum-energy principle, it obtains $u_{ss} \ge 0$, $u_{vv} \ge 0$, and $u_{ss}u_{vv} - u_{sv}^2 \ge 0$ for the internal energy, together with the corresponding concavity conditions for entropy and the mixed curvature signs for the Helmholtz, enthalpy, and Gibbs potentials. It then reads off the response coefficients from these potentials and concludes that the specific heats $c_v$ and $c_p$ and the compressibilities $k_T$ and $k_s$ are all nonnegative. The intended contribution is pedagogical: a self-contained, purely mathematical route from the postulational principles to thermal and mechanical stability that can supplement a standard thermodynamics course.

What carries the argument

The load-bearing object is the Hessian matrix of each thermodynamic potential, written as a symmetric quadratic form from the second-order Taylor expansion, with entries $a = \partial^2 u/\partial s^2$, $b = \partial^2 u/\partial v^2$, and $c = 2\,\partial^2 u/\partial s\partial v$. Diagonalizing this matrix reduces the quadratic form to $\lambda_1 \Delta s'^2 + \lambda_2 \Delta v'^2$, converting stability questions into eigenvalue-sign questions: nonnegative eigenvalues for a minimum, nonpositive for a maximum, and mixed signs for a saddle. The same procedure is repeated for the entropy, Helmholtz, enthalpy, and Gibbs potentials, and the relevant response coefficients are identified with the curvature signs.

What would settle it

Take any known stable simple fluid, such as an ideal gas with $u = c_v T$, and compute the first derivatives $\partial u/\partial s = T$ and $\partial u/\partial v = -p$ at an equilibrium state: neither is zero, so Eq. (2) is not satisfied while the stability inequalities still hold; this would show the paper's derivation route cannot be the actual source of the inequalities. Alternatively, if any empirical equation of state produced a stable state with negative $c_v$ or $k_T$, the claimed curvature-stability link would be refuted.

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Extended reading notes

Core claim

The paper's central claim is that the type of each thermodynamic potential surface—minimum, maximum, or saddle—is fixed by the signs of the eigenvalues of its second-derivative matrix, and that these eigenvalue signs are exactly what the stability principles select. For $u(s,v)$, the requirement that the canonical form $\lambda_1 \Delta s'^2 + \lambda_2 \Delta v'^2$ stay nonnegative forces both eigenvalues to be nonnegative, which is equivalent to the three inequalities in Eq. (13). The same diagonalization applied to $s(u,v)$ gives the opposite signs required by the maximum-entropy principle, while the Legendre-transformed potentials $f$, $h$, and $g$ inherit a sign flip in each transformed variable: $f$ and $h$ are saddle surfaces and $g$ is a maximum. In Section 4 the derivatives that define $c_v$, $c_p$, $k_T$, and $k_s$ are matched to these curvatures, yielding $c_v \ge 0$, $c_p \ge 0$, $k_T \ge 0$, and $k_s \ge 0$. Thus the paper presents a consistent demonstration that thermal and mechanical stability are immediate corollaries of the curvature classification of thermodynamic potentials.

Load-bearing premise

The whole eigenvalue analysis rests on Eq. (2), which sets $\partial u/\partial s = 0$ and $\partial u/\partial v = 0$ at the equilibrium state; for a real one-component system $T = \partial u/\partial s > 0$ and $p = -\partial u/\partial v > 0$, so those first derivatives do not vanish there, and if that premise is removed the derivation as written collapses.

Editorial extensions

If this is right

The nonnegativity of $c_v$ and $c_p$ follows directly from the concavity of the Helmholtz and Gibbs potentials in temperature, without invoking microscopic models.
The nonnegativity of $k_T$ and $k_s$ follows from the convexity of the Helmholtz potential in volume and the concavity of the enthalpy in pressure.
The determinant inequality $u_{ss}u_{vv} - u_{sv}^2 \ge 0$ is the same requirement that the energy surface be locally convex, so thermal and mechanical stability are two faces of one curvature condition.
The Legendre-transform sign-flip rule means every potential with a mix of extensive and intensive variables is a saddle, and only a fully transformed potential such as the Gibbs function can be a maximum.
Since $\alpha$ changes sign in real materials such as water below $4\,^\circ\mathrm{C}$, the curvature analysis does not constrain $\alpha$; stability only constrains combinations like $c_p - c_v = T v \alpha^2 / k_T \ge 0$.

Reading between the lines

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

The stationarity premise in Eq. (2) is not physically satisfied, so a charitable reconstruction of the argument would use convexity of $u$ and concavity of $s$ as the actual premises; the paper's inequalities would still follow without requiring the first derivatives to vanish.
The same Hessian-diagonalization logic generalizes to multicomponent or multi-variable systems, where stability would require the leading principal minors of the Hessian to satisfy alternating or nonnegative sign conditions.
The determinant inequality could be tested directly on empirical equations of state as a local stability diagnostic, independent of the pedagogical route used here.

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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

The stress-test note is right, and the reader's take is basically right. Section 2's proof starts from Eq. (2): ∂u/∂s = ∂u/∂v = 0 at equilibrium. For a one-component system those derivatives are T and -p, which are not zero in any ordinary state. The Taylor expansion cannot be reduced to the homogeneous quadratic form in Eq. (4) by dropping the linear terms on the grounds of stationarity. The minimum-energy principle does not require, and in fact forbids, zero first derivatives with respect to the unconstrained variables. The later inequalities for cv, cp, kT, and ks are correct, but they are quoted from a proof that does not hold as written. That is a load-bearing flaw. The paper is not trash. The pedagogical intent is clear: it walks through quadratic form diagonalization, Legendre transformations, and curvature signs, and connects them to positivity of specific heats and compressibilities. For a student who knows the answer, the exposition is readable, and the matrix diagonalization nicely illustrates the determinant conditions. The final inequalities are standard and correct. The paper honestly cites Callen and other textbooks and does not fabricate new results. Soft spots beyond Eq. (2): Eq. (25) writes ∂²h/∂u² but h is h(s,p), a typo; Eq. (29) gives the same expression for cv and cp, which should differ; there are assorted wording slips. The sign arguments for the Legendre-transformed potentials are asserted more than derived, acceptable in a pedagogical note but not as a 'consistent mathematical demonstration.' Who is this for? A well-prepared undergraduate or a graduate student reviewing thermodynamics. As a journal submission it does not clear the bar for a research paper, and the current derivation is formally wrong. With a revised Section 2 that derives stability from the standard entropy-maximum or subsystem-plus-reservoir argument, it could become a decent teaching supplement. As is, I would not cite it or bring it to reading group. If I were the editor, I would not send it to referees in present form; the foundational error is obvious and novelty is nil. A teaching venue might consider it after major revision, but I would not encourage submission of this version anywhere serious.

Referee Report

3 major / 5 minor

Summary. The manuscript claims to provide a consistent mathematical demonstration of the minimum, maximum, and saddle conditions of thermodynamic potentials by expanding the energy, entropy, Helmholtz, enthalpy, and Gibbs functions in Taylor series, expressing the second-order terms as quadratic forms, and diagonalizing the associated matrices. From this procedure the authors derive the standard stability inequalities u_ss≥0, u_vv≥0, u_ss u_vv - u_sv²≥0 and, via the curvatures of the Legendre-transformed potentials, the positivity of c_v, c_p, k_T, and k_s. Section 4 connects these curvature conditions to the signs of physical quantities and concludes that thermal and mechanical stability follow from the minimum-energy principle. The paper is pedagogical in intent and treats a one-component system in the entropy and energy representations.

Significance. If the derivation were correct, the paper would offer an accessible, self-contained derivation of standard thermodynamic stability conditions. Its strength is that Section 4 correctly identifies the derivative identities (e.g., ∂²f/∂T² = -∂s/∂T and ∂v/∂p = ∂²h/∂p²) that link potential curvatures to measurable coefficients, and the final inequalities c_v≥0, c_p≥0, k_T≥0, k_s≥0 are the correct textbook results. However, the central derivation in Section 2 rests on an incorrect physical premise and an inconsistent matrix formulation, so the claimed 'consistent mathematical demonstration' is not achieved as written. The paper also provides no machine-checkable proofs, and several sign assignments for Legendre-transformed potentials are asserted rather than derived. The correct results are standard and are not placed in a new or more rigorous framework, so the contribution is at best a draft of a pedagogical note.

major comments (3)

[Section 2, Eq. (2)] The derivation of the quadratic form in Eq. (4) depends on setting ∂u/∂s=0 and ∂u/∂v=0 at the expansion point (s0,v0). For a one-component thermodynamic system these derivatives are the intensive fields: ∂u/∂s=T>0 and ∂u/∂v=-p<0 (with p>0) in any ordinary equilibrium state. Thus Eq. (2) is false unless one is considering an artificial stationary point of the energy surface, which is not the physical equilibrium of a simple system. The minimum-energy principle does not imply stationarity of u(s,v) with respect to independent variations of s and v; it implies convexity of u(s,v). Without Eq. (2), the Taylor expansion retains the linear terms TΔs - pΔv, and the truncation cannot be written as the homogeneous quadratic form of Eq. (4). The subsequent eigenvalue analysis, and hence Eq. (13), therefore does not apply to the physical energy difference. Since the inequalities for c_v, c_p, k_T, and k_s in Section 4 are presented as consequences of these curvature conditions, the central claim of the abstract is unsupported.
[Section 2, Eqs. (4)-(11)] There is an inconsistency between the quadratic form and its matrix representation. With a≡u_ss, b≡u_vv, and c≡2u_sv, the quadratic form in Eq. (4) is a(Δs)² + b(Δv)² + cΔsΔv, which corresponds to the symmetric matrix [[a, c/2], [c/2, b]], not [[a, c], [c, b]] as written in Eq. (5). The eigenvalue equation should involve (a-λ)(b-λ) - c²/4 = 0, not (a-λ)(b-λ) - c² = 0. Consequently the determinant condition for positive eigenvalues is ab - c²/4 > 0, which, with c=2u_sv, is exactly u_ss u_vv - u_sv² > 0. The paper's calculation ab - c² > 0 would give u_ss u_vv - 4u_sv² > 0. Thus the third inequality in Eq. (13) does not follow from the stated matrix; the correct factor arises only if the off-diagonal matrix element is c/2. This is a load-bearing mathematical error in the derivation of the stability determinant.
[Section 2, Eqs. (21), (25), and (26)] The sign assignments for the Legendre-transformed potentials f, h, and g are asserted rather than derived from the stated quadratic-form methodology. For f the paper states that one 'needs to set eigenvalues with opposite signs', and for h and g that the signs follow 'by analogy with the previous cases', but no Taylor expansion or diagonalization is actually performed for these functions. In particular, Eq. (21) is not derived from the minimum-energy principle or from the Legendre transformation; the correct result ∂²f/∂T² = -1/(∂²u/∂s²) is stated in the text but is not used to derive the determinantal inequality. Moreover, Eq. (25) contains an undefined derivative: h(s,p) is a function of s and p, yet the first inequality is written as ∂²h/∂u²≥0. This appears to be a typo for ∂²h/∂s², but as printed the expression is meaningless and the claimed derivation is incomplete. The same applies to the eigenvalue assignments for g in Eq. (26).

minor comments (5)

[Section 3, Eq. (29)] The definitions of c_v and c_p are both written as T ∂s/∂T without indicating the variable held constant; c_v should have v constant and c_p should have p constant. The subscript is present in the names but absent in the formulas.
[Section 4, around Eq. (51)] The text says 'Helmholtz's potential h' when it is describing the enthalpy h(s,p); the Helmholtz potential is f(T,v). This naming error appears just before Eq. (51).
[Throughout] The English is often ungrammatical or awkward (e.g., 'the almost infinite fredom degrees envolved', 'the solid princicle of minimum energy'), and there are numerous typos. A careful language revision is needed for a journal submission.
[References] Reference [2] and [3] are in Portuguese, which is acceptable, but the capitalization and formatting are inconsistent (e.g., 'V ol.', 'Wreszinski, Termodinâmica V ol. 50'). Reference [18] is in Portuguese; this is fine, but all entries should be formatted uniformly.
[Section 2, Eqs. (12) and (13)] The definitions a, b, c in Eq. (12) omit the evaluation at (s0,v0), despite the earlier definitions in the text including it. Adding the evaluation point would make the notation consistent.

Circularity Check

2 steps flagged · score 3.0 of 10

Mild circularity: Eq. (13) restates the d2u≥0 premise, and the f/h/g sign patterns are imposed before diagonalization.

self definitional [Sec. 2, Eqs. (1)–(13)]
"Using the physical principle of minimum energy at stationary point (s0,v 0) (d2u≥ 0, see [1]) ... The physical principle of minimum energy imposes the function ˜u(s,v ) ... ≥ 0 that occurs only when λ1≥ 0 and λ2≥ 0 ... ∂2u ∂s2 ≥ 0 , ∂2u ∂v 2 ≥ 0 and ∂2u ∂s2 ∂2u ∂v 2 − ( ∂2u ∂s∂v )2 ≥ 0. (13)"

The derivation starts from the premise d2u≥0, which already states that the Hessian quadratic form is positive semidefinite. For a symmetric 2×2 matrix, positive semidefiniteness is equivalent, by the principal-minor criterion, to exactly the three inequalities in Eq. (13). The diagonalization step (λ1≥0 and λ2≥0) is the same statement in the eigenbasis. Thus Eq. (13) is not a new consequence derived from the minimum-energy principle; it is an entrywise restatement of the input 'd2u≥0' that was assumed at the outset.
self definitional [Sec. 2, Eqs. (21), (25), (26)]
"If we employ the same method used foru in the Helmholtz potential, in this case we need to set eigenvalues with opposite signs in canonical form off (similarly to the Eq. 6). Remembering that we achievedf by Legendre transformation ons parameter ofu function whereby we introduceT as variable off. This surface of two variablesf has a maximum in relation to the temperature but a minimum in relation to the volume."

To obtain Eq. (21), the paper first asserts that f is a saddle surface and then imposes opposite-sign eigenvalues in the diagonalization; the resulting inequalities ∂2f/∂T2≤0, ∂2f/∂v2≥0, and det≤0 are exactly that prior assertion. The same pattern is used for h and g: h is declared to be a maximum in p and a minimum in s, and for g the signs are announced before Eq. (26), with the diagonalization 'obtained by analogy' or by a Taylor expansion after the signs have already been put in. The sign patterns therefore are not consequences of the diagonalization; they are inputs smuggled in as requirements.

full rationale

The paper contains no data fitting, no benchmark prediction, and no load-bearing self-citation chain; its references to Callen and standard mathematics are external and non-circular. The core energy-function analysis is a pedagogical unpacking: once 'minimum energy' is written as d2u≥0, the eigenvalue argument leading to Eq. (13) restates positive semidefiniteness of the Hessian in terms of principal minors. That is mathematically correct but does not add physical content beyond the postulate. The more genuinely circular element is in the treatment of the Legendre-transformed potentials: the saddle/maximum character of f, h, and g is asserted or imposed before the diagonalization is performed, so Eqs. (21), (25), and (26) are not independently derived from the minimum-energy principle. Separately, Eq. (2) sets ∂u/∂s=0 and ∂u/∂v=0 at equilibrium, which is physically incorrect for a one-component system because those derivatives are T and -p; this is a correctness problem for the motivating expansion, not itself a circularity, but it reinforces that the quadratic-form argument is not a faithful representation of the physical energy difference. Overall the stability conclusions are standard and the proof is mostly self-contained, but the imposed Legendre-potential signs and the restatement of d2u≥0 as Eq. (13) warrant a mild circularity score of 3.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. It depends on the postulational foundations of thermodynamics and on a stationary-point assumption (Eq. 2) that is physically unjustified. The Legendre transformation curvature rules are also asserted rather than fully derived.

assumptions (5)

domain assumption A well-defined fundamental equation u(s,v) exists and is twice differentiable.
The paper assumes the existence of u(s,v) and expands it in a Taylor series without proving differentiability. This is standard in thermodynamics.
domain assumption The minimum energy principle and maximum entropy principle are valid.
The paper explicitly invokes the empirical nature of these postulates and uses them to set the sign of the quadratic forms.
ad hoc to paper At the equilibrium state, the first derivatives ∂u/∂s and ∂u/∂v vanish.
This is Eq. (2). It is not a standard thermodynamic fact because T and -p are nonzero at equilibrium. The derivation depends on this assumption.
domain assumption Legendre transformation changes the curvature of the transformed function with respect to the introduced intensive variable and leaves other curvatures unchanged.
The paper uses this property to assign signs to f_TT, h_pp, g_TT, and g_pp without a full proof. It is correct in a qualified sense but not rigorously established here.
standard math Taylor's theorem and the spectral theorem for symmetric matrices are applicable.
Used to expand the thermodynamic functions and to put the quadratic forms into eigenvalue form.

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Pith. "Pith review of Mathematical methods of diagonalization of quadratic forms applied to the study of stability of thermodynamic systems." pith.science (2026). https://pith.science/paper/OVJWOZAH

@misc{pith2026190803211,
  author       = {Pith},
  title        = {Pith review of: Mathematical methods of diagonalization of quadratic forms applied to the study of stability of thermodynamic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OVJWOZAH}},
  note         = {Machine review of arXiv:1908.03211}
}

read the original abstract

In this paper, we use quadratic forms diagonalization methods applied to the function thermodynamic energy to analyze the stability of physical systems. Taylor's expansion was useful to write a quadratic expression for the energy function. We consider the same methodology to expanding the thermodynamic entropy and investigate the signs of the second-order derivatives of the entropy as well as previously to the thermodynamic energy function. The signs of the second-order derivatives to the Helmholtz, enthalpy and Gibbs functions are also analysed. We show the immediate consequences on the stability of physical systems due to the signs or curvatures of the second-order derivatives of these thermodynamic functions. The thermodynamic potentials are presented and constructed pedagogically as well as demonstrated the main mathematical aspects these surfaces. We demonstrate the power of superposition of mathematical and physical aspects to understand the stability of thermodynamic systems. Besides, we provide a consistent mathematical demonstration of the minimum, maximum, and saddle conditions of the potentials. We present here a detailed approach on aspects related to the curvature of the thermodynamic functions of physical interest with consequences on stability. This work can be useful as a part or supplement material of the traditional physics curriculum that requires a solid formation in thermodynamics, particularly about formal aspects on the stability.

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Pith tools

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

[1] [1]

H. B. Callen, Thermodynamics and an introduction to thermostatistics (1998)

work page 1998

[2] [2]

M. J. de Oliveira, Termodinâmica, Editora Livraria da Física, 2005

work page 2005

[3] [3]

W. F. Wreszinski, Termodinâmica V ol. 50, Edusp, 2003

work page 2003

[4] [4]

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman lectures on physics, V ol. I: The new millennium edition: mainly mechanics, radiation, and heat, V ol. 1, Basic books, 2011

work page 2011

[5] [5]

P. M. Harman, P. M. Harman, Energy, force and matter: the conceptual development of nineteenth-century physics, Cambridge University Press, 1982

work page 1982

[6] [6]

D. H. Perkins, D. H. Perkins, Introduction to high energy physics, CAMBRIDGE university press, 2000

work page 2000

[7] [7]

Shrimpton, Charge injection systems: physical principles, experimental and theoretical work, Springer Science & Business Media, 2009

J. Shrimpton, Charge injection systems: physical principles, experimental and theoretical work, Springer Science & Business Media, 2009

work page 2009

[8] [8]

D. Zhou, G. G. Zhang, D. Law, D. J. Grant, E. A. Schmitt, Physical stability of amorphous pharmaceuticals: importance of conﬁgurational thermodynamic quantities and molecular mobility, Journal of pharmaceutical sciences 91 (8) (2002) 1863–1872

work page 2002