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arxiv: 2604.10225 · v1 · submitted 2026-04-11 · 🧮 math.DS · math-ph· math.MP· math.SG

Which Phases Are Thermodynamically Realizable? A Local Entropy Criterion

C. Evans Hedges This is my paper

Pith reviewed 2026-05-10 15:59 UTC · model grok-4.3

classification 🧮 math.DS math-phmath.MPmath.SG
keywords equilibrium statesentropy mapupper semicontinuityvariational principleamenable group actionsthermodynamic phasesfree energyergodic measures
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The pith

An ergodic measure is an equilibrium state for some continuous potential exactly when the entropy map is upper semicontinuous at that measure.

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

The paper proves that for continuous actions of locally compact amenable groups on compact metrizable spaces with finite topological entropy, an ergodic invariant measure arises as an equilibrium state for some continuous potential if and only if the entropy map is upper semicontinuous at the measure. Equivalently, the measures that cannot be realized as thermodynamic phases are precisely those concealed by the convex envelope of the free energy function. This local criterion answers which invariant measures can appear as physical phases in the variational approach to statistical mechanics. The result also supplies a corrected characterization of equilibrium faces and a realization theorem for C0 potentials on one-point compactifications, covering countable-state Markov shifts.

Core claim

For continuous actions of locally compact amenable groups on compact metrizable spaces with finite topological entropy, an ergodic measure μ is an equilibrium state for some continuous potential if and only if the entropy map h is upper semicontinuous at μ; equivalently, the unrealizable phases are exactly those hidden behind the convex envelope of the free energy. The same criterion applies whenever the system has bounded entropy and embeds as an invariant subsystem of a compact metrizable system. A weak-* closed set of ergodic measures determines an equilibrium face if and only if h restricted to the set is continuous and h is upper semicontinuous at each point of the set.

What carries the argument

The entropy map h that assigns to each invariant probability measure its measure-theoretic entropy; upper semicontinuity at μ ensures that μ can be a maximizer in the variational principle for pressure with a suitable continuous potential.

Load-bearing premise

The group action is continuous, the group is locally compact and amenable, the space is compact and metrizable, and the system has finite topological entropy or embeds into one with bounded entropy.

What would settle it

Exhibit an ergodic measure μ at which the entropy map fails to be upper semicontinuous, then demonstrate that no continuous potential exists for which μ maximizes the sum of entropy and the integral of the potential.

read the original abstract

In the variational approach to statistical mechanics, equilibrium states are the rigorous analogues of thermodynamic phases; the question of which invariant measures can arise as equilibrium states is therefore the question of which phases are thermodynamically realizable. We prove that for continuous actions of locally compact amenable groups on compact metrizable spaces with finite topological entropy, an ergodic measure $\mu$ is an equilibrium state for some continuous potential if and only if the entropy map $h$ is upper semicontinuous at $\mu$; equivalently, the unrealizable phases are exactly those hidden behind the convex envelope of the free energy. More generally, the same criterion applies whenever $(X, T)$ has bounded entropy and embeds as an invariant subsystem of a compact metrizable system. As a canonical case, one-point compactification yields a $C_0$-potential realization theorem for locally compact $\sigma$-compact systems, with applications to countable-state Markov shifts. We also show that the equilibrium-face realization stated by Jenkinson (2006) omits a necessary continuity hypothesis, exhibiting a counterexample on the full shift, and give the sharp corrected statement: a weak-$*$ closed set $\mathcal{E}$ of ergodic measures determines an equilibrium face if and only if $h|_{\mathcal{E}}$ is continuous and $h$ is upper semicontinuous at each point of $\mathcal{E}$.

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 manuscript establishes a local entropy criterion for thermodynamic realizability of ergodic measures in dynamical systems. Specifically, for continuous actions of locally compact amenable groups on compact metrizable spaces with finite topological entropy, or more generally for systems with bounded entropy that embed into such systems, an ergodic measure μ is an equilibrium state for some continuous potential if and only if the entropy map h is upper semicontinuous at μ. Equivalently, the unrealizable phases are those concealed by the convex envelope of the free energy. The paper also corrects a statement from Jenkinson (2006) by providing a counterexample on the full shift and giving the sharp condition for when a weak-* closed set of ergodic measures forms an equilibrium face, requiring continuity of h on the set and upper semicontinuity at each point. Applications to one-point compactifications and countable-state Markov shifts are discussed.

Significance. This result offers a precise characterization of which invariant measures can serve as equilibrium states, addressing a fundamental question in the variational principle for statistical mechanics. By linking realizability directly to the upper semicontinuity of the entropy map—a property already central in ergodic theory—it provides both theoretical insight and a practical test. The correction to prior work and the extension to C0-potentials via one-point compactification enhance the applicability to non-compact systems like countable Markov shifts. If verified, this advances the understanding of thermodynamic phases in amenable group actions.

minor comments (2)
  1. [Introduction] §1 (Introduction): the statement of the main theorem could explicitly reference the amenability hypothesis in the opening sentence to align with the abstract.
  2. [Correction to Jenkinson] The counterexample construction in the correction to Jenkinson (2006) is sketched but would benefit from an explicit verification that the entropy map fails to be continuous on the relevant closed set E while remaining usc at the points in question.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. The report provides a clear summary of our results but does not list any specific major comments or requested changes.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The central result—an ergodic measure is realizable as an equilibrium state for a continuous potential if and only if the entropy map is upper semicontinuous at that measure—rests on the standard variational principle for topological pressure together with weak* continuity of the integral functional. Necessity follows immediately from these classical facts without redefinition or fitting. Sufficiency is proved under explicitly stated hypotheses (locally compact amenable group action, compact metrizable space, finite or bounded topological entropy) that guarantee metrizability of the weak* topology and allow separation of measures; these hypotheses are independent of the target statement. The correction to Jenkinson (2006) is supplied by an explicit counter-example on the full shift, which is externally checkable and does not rely on self-citation. No step equates a derived quantity to its own inputs by construction, renames a fitted parameter as a prediction, or imports a uniqueness theorem from the authors' prior work. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts about topological entropy, equilibrium states, and amenable group actions; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Topological entropy is upper semicontinuous on the space of invariant measures for the systems under consideration
    Invoked to equate the realizability condition with the stated semicontinuity property.
  • standard math Existence of equilibrium states for continuous potentials on compact metrizable systems
    Background fact from thermodynamic formalism used to define the set of realizable measures.

pith-pipeline@v0.9.0 · 5545 in / 1348 out tokens · 85255 ms · 2026-05-10T15:59:59.104819+00:00 · methodology

discussion (0)

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Reference graph

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