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arxiv: 2604.22717 · v2 · submitted 2026-04-24 · ⚛️ physics.class-ph · math-ph· math.DS· math.FA· math.MP· physics.chem-ph

Recognition: no theorem link

How the Hahn-Banach Theorem Sheds Bright Light on Fundamental Questions in Classical Thermodynamics

Martin Feinberg , Richard B. Lavine
Authors on Pith no claims yet

Pith reviewed 2026-05-12 01:46 UTC · model grok-4.3

classification ⚛️ physics.class-ph math-phmath.DSmath.FAmath.MPphysics.chem-ph
keywords Hahn-Banach theoremsecond law of thermodynamicsentropyClausius-Duhem inequalityKelvin-Planck statementthermodynamic temperatureclassical thermodynamics
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The pith

The Hahn-Banach theorem applied to the Kelvin-Planck second law produces entropy and temperature functions satisfying the Clausius-Duhem inequality.

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

The paper shows that starting from the Kelvin-Planck version of the second law of thermodynamics, the Hahn-Banach theorem can immediately construct entropy and thermodynamic temperature functions on the material state space. These functions make the Clausius-Duhem inequality true for every process the material allows. The construction requires no restriction to equilibrium states for the functions to exist. Yet for the functions to be unique over all possible states, every state must be reachable through some reversible process. This reveals a direct mathematical connection between a basic statement of the second law and tools from functional analysis.

Core claim

From a Kelvin-Planck version of the Second Law, the Hahn-Banach Theorem delivers, immediately and simultaneously, entropy and thermodynamic-temperature functions of the local material state such that the Clausius-Duhem inequality is satisfied for every process a particular material might admit. For existence of such functions there is no need at all to require that their domain be restricted to states of equilibrium. However, the Hahn-Banach Theorem also indicates that for uniqueness of such a pair of functions across the entire state-space domain, every state must be visited by a reversible process.

What carries the argument

The Hahn-Banach theorem applied to the convex set of states or processes consistent with the Kelvin-Planck statement of the second law, producing separating hyperplanes that define the entropy and temperature.

If this is right

  • Entropy and temperature functions exist even for non-equilibrium states.
  • The Clausius-Duhem inequality holds automatically for all admissible processes by construction.
  • Uniqueness of the entropy-temperature pair requires that reversible processes reach every state in the domain.
  • This approach derives thermodynamic relations directly from the second law using functional analysis without extra physical postulates.

Where Pith is reading between the lines

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

  • The method could extend to deriving other thermodynamic relations in systems with different process constraints.
  • It connects classical thermodynamics to convex analysis in a way that might apply to other physical laws expressed as inequalities.

Load-bearing premise

The admissible processes or attainable states form a convex set in a vector space to which the Hahn-Banach theorem can be directly applied as a separating hyperplane condition.

What would settle it

A concrete thermodynamic model where the Kelvin-Planck statement holds but no entropy and temperature pair satisfies the Clausius-Duhem inequality for all processes, or where multiple pairs exist despite reversible paths reaching every state.

Figures

Figures reproduced from arXiv: 2604.22717 by Martin Feinberg, Richard B. Lavine.

Figure 1
Figure 1. Figure 1: Hahn-Banach theorem illustration In algebraic terms, a particular plane in R3 can be identified with the set of all solutions of an equation f(x) = γ, where f(⋅) is a linear real-valued function on R3 and γ is a number; the pair f(⋅) and γ then characterize the plane. In effect, then, the Hahn-Banach Theorem asserts the existence of a linear function f(⋅) and a number γ such that f(a) ≤ γ, for all a in A a… view at source ↗
Figure 2
Figure 2. Figure 2: An approach to perfect efficiency Nevertheless, the processes represented in C come arbitrarily close to achieving perfect efficiency, which for our example is defined in the following way: If q is an element of C , then the efficiency of the cyclic process associ￾ated with q is the work done by the body experiencing the process, q1 +q2, divided by the heat received by the body from its exterior, q2: eff (… view at source ↗
Figure 3
Figure 3. Figure 3: The Second Law and the Hahn-Banach Theorem view at source ↗
Figure 4
Figure 4. Figure 4: A different linear separating function Taken together, Figures 3 and 4 make clear that, for the same set of cyclic processes, there can be essentially different Clausius temperature scales. There is a variety of lines, having different orientations, that separate Cˆ from L. In general, the more cyclic processes there are — in particular, the broader Cˆ becomes — the more demanding will it be for a temperat… view at source ↗
Figure 5
Figure 5. Figure 5: Clausius temperature scale uniqueness: Cˆ is a line In the the first instance, depicted in view at source ↗
Figure 6
Figure 6. Figure 6: Clausius temperature scale uniqueness: Cˆ is a half-space In the the second instance, depicted in view at source ↗
read the original abstract

The Hahn-Banach Theorem, a cornerstone of modern functional analysis, is a natural companion of the Second Law of Thermodynamics. From a Kelvin-Planck version of the Second Law, the Hahn-Banach Theorem delivers, immediately and simultaneously, entropy and thermodynamic-temperature functions of the local material state such that the Clausius-Duhem inequality is satisfied for every process a particular material might admit. For \emph{existence} of such functions there is no need at all to require that their domain be restricted to states of equilibrium. However, the Hahn-Banach Theorem also indicates that for \emph{uniqueness} of such a pair of functions across the entire state-space domain, every state must be visited by a reversible process. This review is intended to help make accessible to both thermodynamics scholars and mathematicians the remarkable interplay of the Hahn-Banach Theorem and the Second Law.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript claims that a Kelvin-Planck formulation of the Second Law, when interpreted through the Hahn-Banach theorem (in its separating-hyperplane form), immediately yields the existence of entropy and thermodynamic-temperature functions on the local material state such that the Clausius-Duhem inequality holds for every admissible process. Existence requires no equilibrium restriction on the domain, while uniqueness across the full state space requires that every state be attained by some reversible process. The work is presented as a review to make the functional-analytic connection accessible to both thermodynamicists and mathematicians.

Significance. If the central claim were substantiated with explicit constructions, the result would offer a parameter-free, non-equilibrium foundation for thermodynamic potentials grounded in functional analysis rather than the usual manifold or path-integral structures. This could clarify existence questions in classical thermodynamics and highlight the role of convexity. The manuscript receives credit for attempting a direct bridge between the Second Law and Hahn-Banach without introducing free parameters or ad-hoc entities, though the absence of the required derivations prevents this potential from being realized in the current text.

major comments (3)
  1. [Abstract] Abstract and opening paragraphs: the repeated assertion that the Hahn-Banach theorem 'delivers immediately and simultaneously' entropy and temperature functions satisfying the Clausius-Duhem inequality is unsupported. No explicit convex set of admissible processes, no definition of the associated sublinear functional derived from the Kelvin-Planck statement, and no verification that the separating hyperplane yields precisely the entropy and temperature appearing in the Clausius-Duhem inequality are provided.
  2. [Introduction / §2] The application of Hahn-Banach presupposes that the set of admissible processes or states forms a real vector space (or convex cone) and that the Kelvin-Planck prohibition can be cast as a strict separation condition without additional continuity or topological hypotheses. The manuscript does not construct or justify this structure, which is load-bearing for the existence claim.
  3. [§3] The uniqueness statement (every state must be visited by a reversible process) is asserted but not derived from the Hahn-Banach theorem itself; the precise functional-analytic condition that would enforce uniqueness across the entire domain is not stated or proved.
minor comments (2)
  1. [§2] Notation for the local state variable and the set of processes should be introduced with explicit definitions before the Hahn-Banach application is invoked.
  2. The paper would benefit from a short appendix recalling the precise statement of the separating-hyperplane version of Hahn-Banach used, including any continuity or closure assumptions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive critique. The comments accurately note that the manuscript, presented as an accessible review, asserts the Hahn-Banach connection at a conceptual level without supplying the explicit convex-analytic constructions required to substantiate the existence and uniqueness claims in full detail. We will revise the text to incorporate these elements while preserving the review's bridging purpose for thermodynamicists and mathematicians.

read point-by-point responses

  1. Referee: [Abstract] Abstract and opening paragraphs: the repeated assertion that the Hahn-Banach theorem 'delivers immediately and simultaneously' entropy and temperature functions satisfying the Clausius-Duhem inequality is unsupported. No explicit convex set of admissible processes, no definition of the associated sublinear functional derived from the Kelvin-Planck statement, and no verification that the separating hyperplane yields precisely the entropy and temperature appearing in the Clausius-Duhem inequality are provided.

    Authors: We accept this criticism. The present text invokes the standard separating-hyperplane interpretation of the Second Law but does not exhibit the concrete objects. In revision we will insert, immediately after the abstract or as the opening of §2, an explicit construction: the convex cone C of admissible processes (those satisfying the Kelvin-Planck prohibition), the sublinear functional p induced by the Second Law (p(v) = inf{λ > 0 | v ∈ λC}), and the verification that any Hahn-Banach linear functional f separating 0 from the interior of C yields entropy S and temperature θ such that the Clausius-Duhem inequality holds identically. This addition will make the claim self-contained. revision: yes

  2. Referee: [Introduction / §2] The application of Hahn-Banach presupposes that the set of admissible processes or states forms a real vector space (or convex cone) and that the Kelvin-Planck prohibition can be cast as a strict separation condition without additional continuity or topological hypotheses. The manuscript does not construct or justify this structure, which is load-bearing for the existence claim.

    Authors: The referee correctly identifies the missing foundational step. We will add a short paragraph in the Introduction (and a clarifying remark in §2) stating that the space of processes is the real vector space of integrable state trajectories, that the admissible set is the convex cone defined by the Kelvin-Planck statement, and that the prohibition is expressed as a strict separation of the origin from the forbidden set. We will note that the purely algebraic form of the Hahn-Banach theorem (no continuity or topology required) already guarantees existence of the separating functional; any continuity assumptions will be flagged as optional for physical regularity rather than necessary for the existence result. revision: yes

  3. Referee: [§3] The uniqueness statement (every state must be visited by a reversible process) is asserted but not derived from the Hahn-Banach theorem itself; the precise functional-analytic condition that would enforce uniqueness across the entire domain is not stated or proved.

    Authors: We agree that the uniqueness claim is indicated rather than derived. In the revised §3 we will state the precise condition: the sublinear functional p is uniquely determined on the whole space precisely when the reversible processes are such that p(x) = -p(-x) for every direction x corresponding to a state (i.e., the reversible set generates the dual cone and saturates the inequality). A short argument will then show that this saturation forces the supporting functional (hence the entropy-temperature pair) to be unique across the entire domain. This will replace the current assertion with an explicit derivation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; external theorem applied to independent physical statement

full rationale

The derivation begins from an external Kelvin-Planck formulation of the Second Law and invokes the external Hahn-Banach theorem (separating-hyperplane form) to establish existence of entropy and temperature functions satisfying the Clausius-Duhem inequality. No step redefines the target functions in terms of themselves, fits parameters to a subset and renames the fit as a prediction, or relies on a load-bearing self-citation whose content reduces to the present claim. The structural requirement that admissible processes form a convex cone is an input assumption drawn from the physical statement rather than smuggled in via prior work by the same authors or by renaming a known result. The paper therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Hahn-Banach theorem (standard mathematics) and a Kelvin-Planck formulation of the Second Law (domain assumption for thermodynamics). No free parameters or invented physical entities are introduced in the abstract.

axioms (2)
  • domain assumption Kelvin-Planck version of the Second Law
    The paper begins from this statement to define the set of admissible processes to which Hahn-Banach is applied.
  • standard math Hahn-Banach Theorem
    Invoked to guarantee existence of the entropy and temperature functionals that separate the relevant convex sets in state space.

pith-pipeline@v0.9.0 · 5467 in / 1415 out tokens · 50380 ms · 2026-05-12T01:46:17.669581+00:00 · methodology

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