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arxiv: 2604.17531 · v2 · submitted 2026-04-19 · 🧮 math.DS · math-ph· math.FA· math.MP

The Convex-Analytic Structure of Thermodynamic Equilibrium: Pressure, Subdifferentials, and Phase Transitions

Abdoulaye Thiam This is my paper

Pith reviewed 2026-05-10 05:15 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.FAmath.MP
keywords thermodynamic formalismpressure functionalconvex analysisequilibrium statesphase transitionsvariational principlesdynamical systems
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The pith

The pressure functional is the Legendre-Fenchel transform of the negative entropy, with equilibrium states as its subdifferentials.

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

This paper establishes a convex-analytic structure for the thermodynamic formalism of continuous maps on compact metric spaces. It proves that the pressure is the Legendre-Fenchel transform of the negative entropy, and that the biconjugate recovers the entropy, yielding a complete duality between these functionals. Equilibrium states are identified as the subdifferentials of the pressure functional, so that uniqueness of equilibrium states is equivalent to Gâteaux differentiability of the pressure and first-order phase transitions correspond to non-differentiability points. Under additional assumptions of specification and Hölder potentials, the pressure becomes Fréchet differentiable with its second derivative equal to the asymptotic variance of Birkhoff sums. The work also provides a universal variational principle that encompasses the classical additive, subadditive, and relative variational principles through convexity, lower semi-continuity, coercivity, and cocycle invariance.

Core claim

The pressure functional is the Legendre-Fenchel transform of the negative entropy, and the biconjugate recovery of the entropy from the pressure establishes a complete duality. Equilibrium states are elements of the subdifferential of the pressure, uniqueness of equilibrium states corresponds to Gâteaux differentiability, and first-order phase transitions correspond to non-differentiability. For systems with specification and Hölder potentials, the pressure is Fréchet differentiable in the Hölder norm, and the second derivative of the pressure equals the asymptotic variance of the Birkhoff sums. A universal variational principle unifies the classical additive, subadditive, and relative vari

What carries the argument

The Legendre-Fenchel transform relating the pressure functional to the negative entropy functional, which carries the duality and allows characterization of equilibria via subdifferentials.

Load-bearing premise

The functionals satisfy convexity, lower semi-continuity, coercivity, and cocycle invariance, with systems having the specification property and Hölder potentials for the differentiability results.

What would settle it

A continuous map on a compact metric space where the pressure computed directly fails to equal the Legendre-Fenchel transform of the negative entropy, or an equilibrium state that lies outside the subdifferential of the pressure.

read the original abstract

We develop the convex-analytic structure of the thermodynamic formalism for continuous maps on compact metric spaces. The pressure functional is the Legendre-Fenchel transform of the negative entropy, and the biconjugate recovery of the entropy from the pressure establishes a complete duality. Equilibrium states are elements of the subdifferential of the pressure, uniqueness of equilibrium states corresponds to G\^{a}teaux differentiability, and first-order phase transitions correspond to non-differentiability. For systems with specification and H\"{o}lder potentials, the pressure is Fr\'{e}chet differentiable in the H\"{o}lder norm, and the second derivative of the pressure equals the asymptotic variance of the Birkhoff sums. We prove a universal variational principle that unifies the classical additive, the subadditive, and the relative variational principles through a single theorem on convex functionals satisfying convexity, lower semi-continuity, coercivity, and cocycle invariance. Extensions to systems with the specification property and to non-compact spaces under coercivity conditions are included, with applications to countable Markov shifts via Sarig's recurrence classification. This Part constitutes Part II of a six-part series on the thermodynamic formalism for hyperbolic dynamical systems.

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

0 major / 2 minor

Summary. The paper develops the convex-analytic structure of the thermodynamic formalism for continuous maps on compact metric spaces. The pressure functional is the Legendre-Fenchel transform of the negative entropy, and the biconjugate recovery of the entropy from the pressure establishes a complete duality. Equilibrium states are elements of the subdifferential of the pressure, uniqueness of equilibrium states corresponds to Gâteaux differentiability, and first-order phase transitions correspond to non-differentiability. A universal variational principle unifies the classical additive, subadditive, and relative variational principles under the conditions of convexity, lower semi-continuity, coercivity, and cocycle invariance. For systems with specification and Hölder potentials, the pressure is Fréchet differentiable in the Hölder norm with second derivative equal to the asymptotic variance of Birkhoff sums. Extensions to systems with the specification property, non-compact spaces under coercivity, and applications to countable Markov shifts via Sarig's recurrence classification are included.

Significance. If the results hold, the manuscript offers a significant unification of thermodynamic formalism via convex analysis, providing structural insights into equilibrium states, phase transitions, and differentiability. The universal variational principle is a clear strength, consolidating multiple classical principles under four explicit conditions without free parameters. The explicit correspondence between subdifferentials/differentiability and physical phenomena (uniqueness, phase transitions) and the variance formula for the second derivative add value. The extensions and applications enhance applicability, and the work is grounded in standard convex analysis tools applied to dynamical systems.

minor comments (2)
  1. The abstract refers to this as 'Part II of a six-part series'; a brief sentence in the introduction on how the convex-analytic results here support the broader program on hyperbolic systems would improve context.
  2. Notation for the pressure functional P and the entropy functional could be introduced with a short table or explicit list of properties (convexity, lsc, etc.) in §2 to aid readability before the main theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the detailed and positive summary of our manuscript, the recognition of its significance in unifying thermodynamic formalism via convex analysis, and the recommendation for minor revision. No specific major comments were provided in the report, so we have no points to address point-by-point at this stage. We will incorporate any minor suggestions during revision and ensure the manuscript remains consistent with the six-part series structure.

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard convex analysis to independently defined pressure functional

full rationale

The paper defines the pressure functional via dynamical systems (topological pressure for continuous maps on compact metric spaces) and then applies the Legendre-Fenchel transform and subdifferential calculus from convex analysis to obtain the duality, equilibrium states, and differentiability characterizations. The universal variational principle is proven as a theorem under four explicit, verifiable conditions (convexity, lower semi-continuity, coercivity, cocycle invariance) that unify additive/subadditive/relative cases without presupposing the target conclusions. Fréchet differentiability and variance formulas for Hölder potentials follow from specification property assumptions external to the convex structure. No step reduces a claimed prediction or first-principles result to a fitted parameter, self-definition, or unverified self-citation chain; the central claims remain independently verifiable against the dynamical assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The claims rest on standard domain assumptions from dynamical systems and convex analysis; no free parameters or invented entities are introduced.

axioms (3)
  • domain assumption Continuous maps on compact metric spaces
    Stated as the setting for the thermodynamic formalism.
  • domain assumption Functionals satisfy convexity, lower semi-continuity, coercivity, and cocycle invariance
    Required for the universal variational principle theorem.
  • domain assumption Systems have specification property and Hölder potentials
    Used for Fréchet differentiability and second-derivative formula.

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Forward citations

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