arxiv: 2605.12641 · v1 · submitted 2026-05-12 · 🧮 math-ph · math.MP

Recognition: no theorem link

Scaling Symmetry in Symplectic Thermodynamics

M.C. Baldiotti , R. Fresneda

Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:20 UTC · model grok-4.3

classification 🧮 math-ph math.MP

keywords scaling symmetrysymplectic geometrycontact geometryblack hole thermodynamicsLagrangian submanifoldsvan der Waals gasconstrained Hamiltonian dynamics

0 comments

The pith

Fixing a global scale variable recovers standard thermodynamics but requires breaking scale symmetry between energy and entropy for non-isothermal black hole dynamics.

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

The paper unifies constrained Hamiltonian dynamics with symplectic and contact geometries to examine scaling symmetry in thermodynamics. Contactization and symplectization introduce an extended global scale variable whose fixing yields the usual scale-invariant thermodynamic quantities on Lagrangian submanifolds. This is illustrated by a diffeomorphism relating the submanifolds of ideal and van der Waals gases. Application to the Schwarzschild black hole shows that preserving scale symmetry between internal energy and entropy blocks non-isothermal processes, so symmetry breaking becomes necessary to allow temperature changes.

Core claim

Through contactization and symplectization of constrained Hamiltonian dynamics, an extended global scale variable is introduced such that fixing it recovers the standard thermodynamic description in terms of scale-invariant quantities. The Lagrangian submanifolds of ideal and van der Waals gases are diffeomorphic under this construction. For a Schwarzschild black hole, maintaining scale symmetry between internal energy and entropy prevents non-isothermal dynamics, establishing that breaking this symmetry is required to accommodate such processes.

What carries the argument

The global scale variable introduced via symplectization and contactization of the constrained Hamiltonian system, fixed to recover scale-invariant thermodynamic relations on Lagrangian submanifolds.

If this is right

The Lagrangian submanifolds describing ideal and van der Waals gases are related by a diffeomorphism.
Standard thermodynamic relations emerge directly by fixing the scale variable while leaving the physical content of the submanifolds unchanged.
Non-isothermal dynamics for the Schwarzschild black hole requires breaking the scale symmetry between internal energy and entropy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The geometric construction could extend to other self-gravitating systems to test whether similar symmetry requirements appear in their thermodynamic descriptions.
It offers a route to incorporate dynamical temperature changes into black hole thermodynamics without leaving the equilibrium framework.
The same unification might clarify scaling behavior near critical points in gases by examining how the diffeomorphism deforms under perturbations.

Load-bearing premise

The constrained Hamiltonian dynamics unifies with symplectic and contact geometries so that fixing the global scale variable recovers standard thermodynamics without altering the physical content of the Lagrangian submanifolds.

What would settle it

An explicit computation for the Schwarzschild black hole that permits non-isothermal processes while keeping internal energy proportional to entropy under the scale symmetry would falsify the necessity of symmetry breaking.

read the original abstract

This paper investigates scaling symmetry in thermodynamics by unifying constrained Hamiltonian dynamics with symplectic and contact geometries. Through the mathematical processes of contactization and symplectization, we demonstrate that fixing an extended global scale variable effectively recovers the standard thermodynamic description in terms of scale-invariant quantities. The geometric formalism is illustrated by establishing the diffeomorphism between the Lagrangian submanifolds of ideal and van der Waals gases. Finally, applying this framework to a Schwarzschild black hole reveals that breaking the scale symmetry between internal energy and entropy is a fundamental physical requirement to accommodate non-isothermal dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The geometric unification via contactization and symplectization gives a clean diffeomorphism for ideal and van der Waals gases, but the black-hole claim that scale-symmetry breaking is a fundamental requirement looks under-supported without checks on invariance after fixing the scale variable.

read the letter

The paper's core move is to embed scaling symmetry into thermodynamics by contactizing and symplectizing constrained Hamiltonian dynamics, then recover ordinary thermo by fixing a global scale variable. It shows an explicit diffeomorphism between the Lagrangian submanifolds for ideal and van der Waals gases, and applies the setup to a Schwarzschild black hole to conclude that breaking scale symmetry between internal energy and entropy is needed for non-isothermal dynamics. That diffeomorphism is the most concrete new piece; it is a direct geometric statement that can be checked independently of the black-hole section. The overall construction is coherent on paper and avoids ad-hoc fitting, which is a plus for this style of work. The black-hole conclusion is the weaker part. It rests on the assumption that fixing the extended scale variable leaves the physical content of the submanifolds unchanged. If that fixing operation alters the geometry along non-isothermal paths, the claim that symmetry breaking is a physical necessity rather than an artifact does not go through. The abstract gives no explicit derivations, no limit checks, and no error controls, so the invariance step cannot be verified from what is shown. This is the sort of gap that a referee can ask the authors to close with a few pages of calculation. The work is aimed at people already comfortable with symplectic and contact geometry in physics. A reader looking for new handles on scaling in thermo or black-hole thermodynamics will find something to engage with, even if they end up disagreeing with the final interpretation. It is worth sending to peer review because the geometric construction is developed enough to be worth testing in detail, and the gas diffeomorphism alone is a result that deserves to be on record.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a geometric framework for scaling symmetry in thermodynamics by unifying constrained Hamiltonian dynamics with symplectic and contact geometries via contactization and symplectization. It asserts that fixing an extended global scale variable recovers the standard thermodynamic description in terms of scale-invariant quantities, illustrates this with a diffeomorphism between the Lagrangian submanifolds of ideal and van der Waals gases, and applies the construction to a Schwarzschild black hole to conclude that breaking the scale symmetry between internal energy and entropy is a fundamental physical requirement for non-isothermal dynamics.

Significance. If the central geometric construction holds and the scale-fixing step preserves the physical content of the Lagrangian submanifolds, the work offers a potential unification of scaling symmetries within a symplectic/contact setting, which could clarify the geometric origin of thermodynamic relations and frame black-hole scale-symmetry breaking as an intrinsic feature rather than a modeling choice. The explicit diffeomorphism example for gases provides a concrete test case that strengthens the framework's utility if verified.

major comments (2)

[Abstract / black-hole application] Abstract / black-hole application: the claim that breaking scale symmetry between internal energy and entropy is a 'fundamental physical requirement' to accommodate non-isothermal dynamics rests on the invariance of the Lagrangian submanifolds under fixing of the global scale variable. The abstract states the conclusion but supplies no explicit invariance argument or check against non-isothermal paths; this step is load-bearing for the central claim and must be demonstrated in the main text (e.g., via coordinate charts or pull-back preservation) to rule out an artifact of the contactization procedure.
[Section on unification of constrained Hamiltonian dynamics] Section on unification of constrained Hamiltonian dynamics: the weakest assumption—that the extended scale variable can be fixed to recover standard thermodynamics exactly without altering the physical content of the Lagrangian submanifolds—requires an explicit proof that the fixing map commutes with the symplectic/contact structure for both isothermal and non-isothermal cases. Without this, the diffeomorphism for gases and the black-hole conclusion remain formally unanchored.

minor comments (2)

The introduction should define the extended global scale variable and its relation to the contact form at the outset, before invoking contactization, to improve readability for readers unfamiliar with the geometric setup.
Notation for the scale-invariant quantities recovered after fixing should be introduced consistently and contrasted with the extended variables in a single table or equation block.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We have carefully considered each point and provide point-by-point responses below. Where appropriate, we have revised the manuscript to address the concerns and strengthen the presentation of our results.

read point-by-point responses

Referee: [Abstract / black-hole application] Abstract / black-hole application: the claim that breaking scale symmetry between internal energy and entropy is a 'fundamental physical requirement' to accommodate non-isothermal dynamics rests on the invariance of the Lagrangian submanifolds under fixing of the global scale variable. The abstract states the conclusion but supplies no explicit invariance argument or check against non-isothermal paths; this step is load-bearing for the central claim and must be demonstrated in the main text (e.g., via coordinate charts or pull-back preservation) to rule out an artifact of the contactization procedure.

Authors: We agree that an explicit demonstration of the invariance under the scale-fixing map is necessary to support the claim for non-isothermal dynamics. In the revised manuscript, we have added a new subsection in the black-hole application section that provides coordinate charts for the extended phase space and explicitly computes the pull-back of the contact form under the fixing map for both isothermal and non-isothermal cases. This shows that the Lagrangian submanifold structure is preserved, confirming that the scale symmetry breaking is indeed required for non-isothermal evolution rather than an artifact. revision: yes
Referee: [Section on unification of constrained Hamiltonian dynamics] Section on unification of constrained Hamiltonian dynamics: the weakest assumption—that the extended scale variable can be fixed to recover standard thermodynamics exactly without altering the physical content of the Lagrangian submanifolds—requires an explicit proof that the fixing map commutes with the symplectic/contact structure for both isothermal and non-isothermal cases. Without this, the diffeomorphism for gases and the black-hole conclusion remain formally unanchored.

Authors: The referee correctly identifies that the commutation of the fixing map with the geometric structures is central. We have expanded the unification section to include a detailed proof that the scale-fixing procedure is a contactomorphism (or symplectomorphism in the symplectized case) that preserves the Lagrangian property for both isothermal and non-isothermal thermodynamic processes. This is demonstrated by showing that the differential of the fixing map pulls back the contact form to itself up to a conformal factor consistent with the scaling. The diffeomorphism between ideal and van der Waals gases is thereby anchored in this construction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; geometric construction is self-contained

full rationale

The derivation proceeds by unifying constrained Hamiltonian dynamics with symplectic/contact structures via explicit contactization and symplectization operations, then fixing the introduced global scale variable to recover standard thermodynamic relations on Lagrangian submanifolds. This recovery is a direct mathematical consequence of the extension and fixing procedure rather than an independent prediction or fit. The diffeomorphism between ideal and van der Waals gas submanifolds follows from the same geometric maps. The Schwarzschild application asserts that scale symmetry breaking between internal energy and entropy is required for non-isothermal dynamics within the extended geometry; no self-citation chain, ansatz smuggling, or renaming of known results is required to reach this conclusion from the stated formalism. The framework remains independent of external fitted data or prior author-specific uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central construction rests on geometric unification axioms and the introduction of an extended scale variable whose fixing is asserted to recover standard thermodynamics; no numerical free parameters are indicated.

axioms (2)

domain assumption Constrained Hamiltonian dynamics unifies with symplectic and contact geometries
Invoked to embed scaling symmetry via contactization and symplectization.
domain assumption Fixing the extended global scale variable recovers the standard thermodynamic description in scale-invariant quantities
Central step that maps the extended geometry back to ordinary thermodynamics.

invented entities (1)

extended global scale variable no independent evidence
purpose: To incorporate scaling symmetry into the geometric thermodynamic description
New variable introduced so that fixing it yields scale-invariant thermodynamics; no independent falsifiable evidence supplied beyond the framework itself.

pith-pipeline@v0.9.0 · 5378 in / 1290 out tokens · 53606 ms · 2026-05-14T20:20:23.123450+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Bravetti,Contact geometry and thermodynamics, International Journal of Geometric Methods in Modern Physics16, No

A. Bravetti,Contact geometry and thermodynamics, International Journal of Geometric Methods in Modern Physics16, No. supp01, 1940003 (2019) 1, 4

work page 2019
[2]

Baldiotti, R

M.C. Baldiotti, R. Fresneda, and C. Molina,A Hamiltonian approach to Thermodynamics, Annals of Physics 373, 245 (2016) 1, 2, 2

work page 2016
[3]

M. C. Baldiotti, R. Fresneda, C. Molina,A Hamiltonian approach for the Thermodynamics of AdS black holes, Annals of Physics382,22 (2017) 1, 5.2

work page 2017
[4]

Abraham and J

R. Abraham and J. Marden,Foundations of Mechanics(Benjamin/Cummings Publishing Company, Mas- sachusetts, 1978) 3, 3.1, 4, 5.1

work page 1978
[5]

Weinstein,Symplectic manifolds and their lagrangian submanifolds, Advances in Mathematics6(3), 329 (1971) 3.1

A. Weinstein,Symplectic manifolds and their lagrangian submanifolds, Advances in Mathematics6(3), 329 (1971) 3.1

work page 1971
[6]

Da Silva,Lectures on Symplectic Geometry(Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) 3.1, 4

A.C. Da Silva,Lectures on Symplectic Geometry(Springer Berlin Heidelberg, Berlin, Heidelberg, 2008) 3.1, 4

work page 2008
[7]

Souriau, C.H

J.-M. Souriau, C.H. Cushman-de-Vries, R.H. Cushman, and G.M. Tuynman,Structure of Dynamical Systems: A Sympletic View of Physics(Birkhäuser, Boston, 1997) 5.1

work page 1997
[8]

Barrow,The Area of a Rough Black Hole, Physics Letters B808, 135643 (2020) 5.2

John D. Barrow,The Area of a Rough Black Hole, Physics Letters B808, 135643 (2020) 5.2

work page 2020
[9]

C. W. Misner and D. H. Sharp,Relativistic equations for adiabatic, spherically symmetric gravitational collapse, Physical Review136, B571 (1964) 5.2

work page 1964
[10]

T. L. Campos, M. C. Baldiotti, C. Molina,Generating Kerr-anti-de Sitter thermodynamics, Physical Review D 110, 024049 (2024) 5.2

work page 2024
[11]

Widom,Equation of State in the Neighborhood of the Critical Point, The Journal of Chemical Physics43(11), 3898 (1965) 6

B. Widom,Equation of State in the Neighborhood of the Critical Point, The Journal of Chemical Physics43(11), 3898 (1965) 6

work page 1965
[12]

Hawking, and D.N

S.W. Hawking, and D.N. Page,Thermodynamics of black holes in anti-de Sitter space, Communications in Mathematical Physics87(4), 577 (1983) 6 Londrina State University - UEL - Brasil; Federal University of ABC - Brasil. Email address:balddioti@uel.br; rodrigo.fresneda@ufabc.edu.br

work page 1983