Pith. sign in

REVIEW 3 major objections 6 minor 24 references

Entropy in Themodynamics: from Foliation to Categorization

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper reworks entropy as an ordering on state spaces and claims the minimal relation linking two entropy systems is a Galois connection—the Landauer connection—which localizes generated heat and resolves Maxwell's demon.

desk verdict A sound survey of geometric and axiomatic entropy whose final categorical section overreaches by presenting an order-theoretic Galois connection as if it derived Landauer heat. read the letter →

arxiv 1908.07583 v2 pith:SB56BVUL submitted 2019-08-20 math-ph cond-mat.stat-mechmath.MP

Rados{l}aw A. Kycia This is my paper · ORCID
classification math-phcond-mat.stat-mechmath.MP MSC 80A0580A1018B3518A4053D10
keywords entropythermodynamicscontactgeometryadiabaticaccessibilityposetGaloisconnectionLandauerprincipleMaxwell'sdemon
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 follows entropy through three mathematical formulations: as a foliation of a contact manifold, as an ordering induced by adiabatic processes, and as a category of ordered sets. Its central claim is that an entropy system is a (group-scaled) poset, and the minimal relation that preserves the entropy ordering between two systems is a Galois connection, written $S_2(Fc)\leq S_2(d)\iff S_1(c)\leq S_1(Gd)$ and named the Landauer connection. If this is right, the connection says exactly which part of a compound entropic system generates Landauer's heat, and it resolves Maxwell's demon paradox. The paper is presented as an overview that reorganizes known material rather than as a new proof.

What carries the argument

The central object is the entropy system: a set of equilibrium states carrying a scaling action by $(\mathbb{R}_+,\cdot,1)$ and a total order induced by an entropy function $S$, viewed as a poset-category. The decisive mechanism is the Galois connection between two such categories, a pair of order-preserving maps $F$ and $G$ satisfying the equivalence above; this is the minimal relation that preserves both orders in both directions and is a prototype of adjoint functors. Supporting machinery includes the contact form $\theta=dU-Q+W$ on the state space, whose integrability via Caratheodory and Frobenius produces the foliation $S=$ const, and the axiomatic framework in which adiabatic accessibility plus the Comparison Hypothesis force the existence of $S$.

What would settle it

Find or construct a simple system with two equilibrium states X and Y such that neither X≺Y nor Y≺X under adiabatic accessibility while the other axioms hold; then the Comparison Hypothesis fails, no total-order entropy function exists, and the Galois-connection treatment does not apply to that system.

Watch

Extended reading notes

Core claim

On the paper's own terms: entropy is not primarily a number but a total order on equilibrium states given by adiabatic accessibility, and the entropy function is a monotone numerical representation of that order. Treating an entropy system as an object of the category G-Pos—a poset with a scaling group action—the paper states that the minimal bidirectional order-preserving relationship between two entropy systems is a Galois connection, called the Landauer connection. Concretely, two entropy systems are connected when there are order-preserving functors $F$ and $G$ with $S_2(Fc)\leq S_2(d)\iff S_1(c)\leq S_1(Gd)$. The paper claims this connection distinguishes the physical part of a compound system where Landauer heat is generated and explains Maxwell's demon paradox. It also asserts the Comparison Hypothesis, total adiabatic comparability, as the condition under which the whole ordering picture holds.

Load-bearing premise

The construction stands on the Comparison Hypothesis: every two equilibrium states of a simple system are comparable by adiabatic accessibility, so the ordering is total; if some states are incomparable, the entropy function and the poset-category treatment need not exist.

Editorial extensions

If this is right

  • Comparing any two entropy systems becomes the question of whether a Galois connection exists between their state posets; when it does, entropy changes in one system are carried to the other by the two Landauer functors.
  • Landauer's principle appears as a special case: an irreversible logical operation in one entropy system forces a compensating entropy change in the physical system that implements it, so the generated heat is located on the physical side.
  • Maxwell's demon paradox is claimed to be resolved because the demon's measurement and erasure are described by the same Galois-connected transfer, so no entropy accounting gap remains.
  • The smooth and axiomatic routes to entropy are reconciled: both yield the same monotone entropy function, which is what makes the categorical layer a faithful summary rather than an additional assumption.

Reading between the lines

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

  • The minimality claim invites a test the paper does not perform: for a given pair of entropy systems, construct all order-preserving relations and check whether a non-Galois relation with fewer arrows exists; a counterexample would not disturb the thermodynamics but would change the categorical claim.
  • If the Comparison Hypothesis fails for some real system, for instance states on different sheets of a foliation that are adiabatically incomparable, the natural fallback is a partial-order variant in which the Landauer connection is replaced by residuated maps; the entropy picture would survive as a monotone but not total representation.
  • The template is portable: any process governed by a monotone quantity, such as information erasure or population dynamics, can be cast as the same adjunction, and one could test empirically whether measured entropy transfers obey the identity $S_2(Fc)\leq S_2(d)\iff S_1(c)\leq S_1(Gd)$ in those settings.
  • The paper leaves open whether the Landauer connection, when it exists, is unique; a concrete family of systems with multiple Galois connections would show whether the physical heat localization is independent of the choice of functors.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. This paper is an expository overview of entropy in thermodynamics, presented in three parts: the smooth/contact-geometric formulation (Section 2), the axiomatic ordering approach of Lieb and Yngvason (Section 3), and a categorical reformulation (Section 4). In Section 4, an entropy system is identified with a poset (or G-poset) and a 'Landauer connection' between two entropy systems is defined as a Galois connection, Eq. (23). The paper claims that this connection 'makes a sharp distinction, in which part of the compound entropic system such Landauer's heat is generated' and that it 'also explained Maxwell's demon paradox [13].' An appendix collects mathematical preliminaries on differential forms, Frobenius, Darboux, and jet spaces.

Significance. Sections 2 and 3 are a competent, if selective, review of standard results: the Maxwell relations are correctly derived from the contact form, and the Lieb-Yngvason axioms are accurately summarized. The categorical section, however, is largely a summary of the author's previous work [13] and contains unproved assertions that go beyond a simple review. The claimed explanatory power for Landauer's principle and Maxwell's demon is not supported by the formalism presented in the manuscript. The paper would be acceptable as an expository note if the categorical claims were explicitly marked as conjectural or if the missing derivations were supplied; in its current form, the central claim of the categorical section is not established.

major comments (3)
  1. [Section 4] The manuscript states 'We will consider only a simple (i.e., not compound) systems for simplicity' and then, two paragraphs later, asserts that the Landauer connection 'makes a sharp distinction, in which part of the compound entropic system such Landauer's heat is generated' and 'also explained Maxwell's demon paradox.' No definition of a compound entropy system is given in the categorical formalism, and Eq. (23) contains only order relations between entropies, with no energy scale, temperature, or heat variable. The explanatory claims about heat generation therefore do not follow from the presented framework; the authors should either remove these claims or provide a derivation from Eq. (23) and the definitions of Section 4.
  2. [Section 4, Definition 8] The statement that the minimal order-preserving relation between two entropy posets is a Galois connection is asserted without proof and without a precise definition of 'minimal.' Since this is the foundational claim on which the Landauer connection rests, the authors should provide a proof, or explicitly state the sense in which it is minimal, with a reference to a proof in [13].
  3. [Section 3, Definition 4; Section 4] The categorical treatment assumes a total order on the state space, which is only guaranteed if the Comparison Hypothesis holds. The proof of CH is not reproduced; the paper states that it can be proved from the definition of simple systems and the Zeroth Law. The paper should explicitly state that the results of Section 4 are conditional on CH; otherwise the poset-category description does not match the thermodynamic setup in cases where some states are incomparable.
minor comments (6)
  1. [Title] The title on the arXiv has a typo: 'Themodynamics' should be 'Thermodynamics'.
  2. [Section 2.2] In the last bullet, 'no ∆W and no ∆W can be calculated' should read 'no ∆Q and no ∆W can be calculated.'
  3. [Section 3, axiom list] The first axiom is labeled 'Monotonicity: X∼X,' but this is reflexivity, not monotonicity; this is a misnomer.
  4. [Section 4, Definition 7] 'with preserving ordering group (R+,·, 1) action' should be 'with order-preserving group action of (R+,·, 1).'
  5. [Section 4, Eq. (23)] The elements c and d are not introduced; specify c ∈ Γ1 and d ∈ Γ2.
  6. [Section 2.6] 'Lagrange manifolds' should be 'Legendre submanifolds.'

Circularity Check

1 steps flagged · score 4.0 of 10

Section 4's explanatory payoff (Landauer heat location, Maxwell's demon) is imported from the author's own [13]; the formal adjunction (23) is a definition, not a derivation.

  1. self citation load bearing [Section 4, Definition 8 and the concluding paragraph of the section]
    "In this section, we review some concepts from [13]. ... We will consider only a simple (i.e., not compound) systems for simplicity. ... Therefore, the categorical approach makes a sharp distinction, in which part of the compound entropic system such Landauer's heat is generated. This result also explained Maxwell's demon paradox [13]."

    The formal content of the section is Definition 8: an entropy system is 'implemented' in another exactly when a Galois connection exists, rewritten as S2(Fc)<=S2(d) iff S1(c)<=S1(Gd) in Eq. (23). That equation is the definition of a Galois connection between two entropy posets; it contains no heat, no energy scale, and no compound-system construction. The paper then asserts that this connection locates Landauer heat in a compound system and explains Maxwell's demon paradox. That explanatory claim is not derived from Eq. (23) or from any definition in the text, and the section explicitly restricts to simple systems. Its only cited support is the author's own previous paper [13], so the load-bearing conclusion reduces to a self-citation of an unproved framework.

full rationale

The paper is an explicit overview ('This presentation is by no means original research'), and its mathematical spine is external: the contact-geometric proof follows Frankel, the axiomatic entropy existence and uniqueness are quoted from Lieb-Yngvason ([16,17]), and the Galois-connection fact is attributed to standard references ([23,21]). No parameter is fitted and no quantity is 'predicted' from a subset of data. Section 4 is self-consciously a rephrasing ('Up to now, this is only rephrasing of the previous section in terms of "abstract nonsense"...'). The one genuinely load-bearing self-citation is the closing claim that the categorical framework 'makes a sharp distinction, in which part of the compound entropic system such Landauer's heat is generated' and 'explained Maxwell's demon paradox [13]'. That claim does not follow from Eq. (23), which merely restates the definition of a Galois connection between entropy orderings and contains no energy or compound-system structure; the section explicitly restricts to simple systems. Its support is solely the author's prior article [13]. This raises the score above the 0-2 baseline, but because the formal categorical content is independently standard and the paper makes no original predictive claim, a score of 4 rather than 6+ is appropriate.

Assumptions & free parameters 0 free parameters · 10 assumptions · 1 invented entities

The paper rests on the standard axioms of equilibrium thermodynamics (Zeroth, First, Second Laws), the smooth-manifold and Legendre-submanifold modeling assumptions, the Lieb-Yngvason axioms including the Comparison Hypothesis, and background theorems (Frobenius, Darboux, Caratheodory). The categorical section adds the assumption that order-preserving bidirectional relations are Galois connections, which is imported from [13,23,21]. No free parameters are fitted. The Landauer connection is a new conceptual term borrowed from [13].

assumptions (10)
  • domain assumption Assumption 1: An equilibrium state is fully described by a set of extensive variables.
    Section 2.1; foundational modeling assumption of equilibrium thermodynamics.
  • domain assumption Assumption 2: The equilibrium state is a point in a 2n+1 dimensional smooth manifold M of states.
    Section 2.1; enables the differential-form and contact treatment.
  • domain assumption Axiom 1 (Zeroth Law): thermal equilibrium is an equivalence relation.
    Section 2.1; used to define empirical temperature and thermometers.
  • domain assumption Axiom 2 (First Law): dU = Q - W.
    Section 2.3; conservation of energy, stated as an axiom for the mathematical formulation.
  • domain assumption Axiom 3 (Second Law, Caratheodory): near any state there are states not accessible by quasi-static adiabatic paths.
    Section 2.4; used via Caratheodory's theorem to get Q and dQ = 0 and Q = TdS.
  • domain assumption Axiom 4 (Second Law, Kelvin): in a quasi-static cyclic process heat cannot be entirely converted to work.
    Section 2.4; used to prove global behavior of the entropy foliation.
  • domain assumption Assumption 3: a non-degenerate thermodynamical system is a maximal (Legendre) submanifold of dimension n.
    Section 2.3; connects equations of state with contact geometry.
  • domain assumption Comparison Hypothesis: all pairs of states are comparable by adiabatic accessibility.
    Definition 4, Section 3; required for total ordering and existence of additive entropy in Theorems 2 and 3.
  • standard math Frobenius integrability theorem, Darboux theorem, and Caratheodory accessibility theorem.
    Appendix A; unproved background theorems used to derive Q = TdS from the Second Law.
  • standard math Category-theoretic fact that adjoint functors between posets are Galois connections.
    Section 4; imported from [23,21] and used to define the Landauer connection.
invented entities (1)
  • Landauer connection / Landauer functors (F,G)
    purpose: Formalizes implementation or simulation of one entropy system in another via a Galois connection, and is used to attribute Landauer heat to the physical part of a compound system.
    Defined in Section 4, Definition 8, and attributed to [13]. It is a mathematical or conceptual construct rather than a physical entity, with no falsifiable handle provided in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Entropy in Themodynamics: from Foliation to Categorization." pith.science (2026). https://pith.science/paper/SB56BVUL

@misc{pith2026190807583,
  author       = {Pith},
  title        = {Pith review of: Entropy in Themodynamics: from Foliation to Categorization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SB56BVUL}},
  note         = {Machine review of arXiv:1908.07583}
}
read the original abstract

We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on a set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

Figures

Figures reproduced from arXiv: 1908.07583 by the authors.

Figure 1
Figure 1. System of large degrees of freedom (atoms) in thermodynamics is [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The work along 2 is Z 2 W = Z 2 Q − dU = − Z 2 dU = − Z 1 dU = Z −1 dU = Z −1 Q, (12) 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 2
Figure 2. Path 1 is a cooling at constant volume along which no work is [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages

  1. [13]

    Kycia, Landauers Principle as a Special Case of Galois Connection , Entropy, 20(12), 971, (2018); DOI: https://doi.org/10.3390/e20120971

    R.A. Kycia, Landauers Principle as a Special Case of Galois Connection , Entropy, 20(12), 971, (2018); DOI: https://doi.org/10.3390/e20120971

    work page doi:10.3390/e20120971 2018

  2. [1]

    Babson, D.N

    E. Babson, D.N. Kozlov, Group Actions on Posets, J. Algebra, 285, 2, 439–450 (2005)

    work page 2005

  3. [2]

    Bamberg, S

    P. Bamberg, S. Sternberg, A Course in Mathematics for Students of Physics , Cambridge University Press, vol. 2, 1990

    work page 1990

  4. [3]

    Boyling, An Axiomatic Approach to Classical Thermodynamics, Proc

    J.B. Boyling, An Axiomatic Approach to Classical Thermodynamics, Proc. R. Soc. London, A 329, 35–70 (1972) 18

    work page 1972

  5. [4]

    Callen, Thermodynamics, John Wiley & Sons Inc., 1966

    H.B. Callen, Thermodynamics, John Wiley & Sons Inc., 1966

    work page 1966

  6. [5]

    tom Dieck, Transformation Groups and Representation Theory , Lecture Notes in Mathematics, 766, Springer, 1979

    T. tom Dieck, Transformation Groups and Representation Theory , Lecture Notes in Mathematics, 766, Springer, 1979

    work page 1979

  7. [6]

    Edelen, Applied Exterior Calculus , Dover, 2011

    D.G.B. Edelen, Applied Exterior Calculus , Dover, 2011

    work page 2011

  8. [7]

    Frankel, Geometry of Physics , Cambridge University Press, 2011

    T. Frankel, Geometry of Physics , Cambridge University Press, 2011

    work page 2011

Show all 24 references
  1. [8]

    Ingarden, A

    R. Ingarden, A. Jamio lkowski, R. Mruga la, Fizyka statystyczna, PWN, 1990 (in Polish)

    1990

  2. [9]

    Katok, B

    A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Revised edition, 1996

    1996

  3. [10]

    Kol´ aˇ r, P.W

    I. Kol´ aˇ r, P.W. Michor, J. Slov´ ak,Natural Operations in Differential Geometry, Springer-Verlag Berlin Heidelberg, 1993

    1993

  4. [11]

    Kushner, V

    A. Kushner, V. Lychagin, V. Rubtsov, Contact Geometry and Nonlinear Dif- ferential Equations, Cambridge University Press, 1 edition, 2007

    2007

  5. [12]

    Kushner, V

    A. Kushner, V. Lychagin, J. Slov´ ak,Lectures on Geometry of Monge–Amp` ere Equations with Maple in Nonlinear PDEs, Their Geometry, and Applications, Birkh¨ auser Basel, 2019

    2019

  6. [14]

    Ladyman, S

    J. Ladyman, S. Presnell, A.J. Short, B. Groisman, The Connection Between Logical and Thermodynamic Irreversibility, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 38, 1, 58–79 (2007); DOI: https://doi.org/10.1016/j.shp...

    2007 doi

  7. [15]

    Landauer, Irreversibility and Heat Generation in The Computing Process , IBM Journal of Research and Development, 5, 183–191 (1961)

    R. Landauer, Irreversibility and Heat Generation in The Computing Process , IBM Journal of Research and Development, 5, 183–191 (1961)

    1961

  8. [16]

    E.H. Lieb, J. Yngvason, A Guide to Entropy and the Second Law of Thermo- dynamics, Notices of The AMS, 1998

    1998

  9. [17]

    E.H. Lieb, J. Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics , Phys. Rept., 310, 1–96 (1999); DOI: 10.1016/S0370- 1573(98)00082-9

    1999 doi

  10. [18]

    Lychagin, Contact Geometry, Measurement, and Thermodynamics in Nonlinear PDEs, Their Geometry, and Applications, Birkh¨ auser Basel, 2019

    V.V. Lychagin, Contact Geometry, Measurement, and Thermodynamics in Nonlinear PDEs, Their Geometry, and Applications, Birkh¨ auser Basel, 2019

    2019

  11. [19]

    Lychagin, Contact Geometry and Nonlinear Second Order Differential Equations, Uspechi Mat

    V.V. Lychagin, Contact Geometry and Nonlinear Second Order Differential Equations, Uspechi Mat. Nauk, 34, 137–165 (1979) 19

    1979

  12. [20]

    Mac Lane, Categories for the Working Mathematician , Springer, 2nd edi- tion, 1978

    S. Mac Lane, Categories for the Working Mathematician , Springer, 2nd edi- tion, 1978

    1978

  13. [21]

    Ore, Galois Connexions, Transactions of the American Mathematical So- ciety, 55, 493–513 (1944)

    O. Ore, Galois Connexions, Transactions of the American Mathematical So- ciety, 55, 493–513 (1944)

    1944

  14. [22]

    Reza, An Introduction to Information Theory , Dover Publications, Re- vised edition, 1994

    F.M. Reza, An Introduction to Information Theory , Dover Publications, Re- vised edition, 1994

    1994

  15. [23]

    Smith, Category Theory: A Gentle Introduction , Script https://www

    P. Smith, Category Theory: A Gentle Introduction , Script https://www. logicmatters.net/categories/

  16. [24]

    W. Li, Y. Zhao, Q. Wang, J. Zhou, Twenty Years of Entropy Research: A Bibliometric Overview, Entropy, 21(7), 694 (2019); DOI: https://doi.org/ 10.3390/e21070694 20

    2019 doi

Pith tools

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