On the Equivalence of Equilibrium and Freezing States in Dynamical Systems

C. Evans Hedges

REVIEW 3 major objections 6 minor 1 cited by

On the Equivalence of Equilibrium and Freezing States in Dynamical Systems

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Freezing a system adds no new restriction beyond the equilibrium condition.

desk verdict The central equivalence is a good idea, but the proof of Theorem 1 has a real gap in the Edwards separation step, and the proposed repair doesn't fix it as written. read the letter →

arxiv 2412.05639 v2 pith:SVFKZFRE submitted 2024-12-07 math.DS math-phmath.MP

C. Evans Hedges This is my paper · ORCID

classification math.DSmath-phmath.MP MSC 37D3537A35

keywords freezingphasetransitionsequilibriumstateszero-temperaturelimitsentropymapthermodynamicformalisminvariantmeasuresspecificationpropertypressurefunction

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 aims to prove that freezing a system—arranging that below some temperature the equilibrium states of $\beta\varphi$ no longer change—adds no new constraint beyond the ordinary equilibrium condition. Its main theorem says that a non-empty collection $F$ of invariant measures can be frozen by some continuous potential, in the sense that $F$ is the equilibrium set of $\beta\psi$ for every $\beta \ge 1$, exactly when $F$ is already the equilibrium set of some potential and entropy is constant on $F$. If correct, this makes freezing states and equilibrium states the same class of measures, so any ergodic measure that can appear as an equilibrium state can also be made the unique frozen state at low temperature. It also yields a concrete picture of when freezing potentials are common or rare, and shows that a true freeze is always accompanied by a non-analytic pressure function at the transition temperature.

What carries the argument

The key identity is Lemma 3.1: a potential $\varphi$ freezes on $F$ exactly when $F$ is the equilibrium set for $\varphi$ and all measures in $F$ attain the same maximal integral $\int \varphi\,d\mu$. The sufficiency part of Theorem 1 then runs through a separation theorem for the compact convex simplex of invariant measures (Theorem 4 from [6]): after adding a constant so that $\int \varphi\,d\mu = 0$ on $F$, the proof interpolates a weak-$*$ continuous affine function $A$ between the convex function $f(\nu) = |\int \varphi\,d\nu|$ and the concave lower semicontinuous function $g$ that is zero on $F$ and $\|\varphi\|$ off $F$, and sets $\psi = \varphi - A$. This makes $\psi$ maximized precisely on $F$ while preserving the equilibrium condition, so Lemma 3.1 applies at every $\beta \ge 1$.

What would settle it

Take the non-closed equilibrium set $F$ from Example 2.2 (the convex hull of the two maximal-entropy measures on disjoint full shifts, with the fixed point $\delta_\infty$ in its closure) and determine whether any continuous $\psi$ has $F$ as the equilibrium set of $\beta\psi$ for every $\beta \ge 1$; the absence of such a $\psi$ would refute Theorem 1 as stated, while its existence would show the closedness gap is repairable.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1: for any dynamical system $(X,T)$ with finite topological entropy and any non-empty $F \subset M_T(X)$, there exists $\psi \in C(X)$ with $F$ equal to the equilibrium set of $\beta\psi$ for all $\beta \geq 1$ if and only if $h$ is constant on $F$ and $F$ is the equilibrium set of some continuous potential. The mechanism is Lemma 3.1, which shows that freezing on $F$ is equivalent to $F$ being an equilibrium set for the same potential while every measure in $F$ maximizes the integral of that potential. In the upper semicontinuous-entropy case the theorem yields freezing on the closed convex hull of any closed, constant-entropy collection of ergodic measures; in particular every ergodic equilibrium state can be frozen. The paper further proves that freezing potentials are dense in $C(X)$ when entropy is upper semicontinuous, that under specification for $\mathbb{Z}$-actions the non-freezing potentials form a dense $G_\delta$, and that freezing at a minimal inverse temperature forces $P_\varphi$ to be non-analytic there.

Load-bearing premise

The proof of the 'if' direction in Theorem 1 assumes that a convex set of equilibrium states is automatically closed, because the separation theorem it uses requires a lower semicontinuous bounding function; the paper's own Example 2.2 gives a convex equilibrium set that is not closed when the entropy map is not upper semicontinuous, so the general theorem rests on this unproved closedness.

Editorial extensions

If this is right

For a single ergodic measure, being the freezing state of some potential is equivalent to being an equilibrium state of some potential; the low-temperature regime imposes no extra restriction.
When the entropy map is upper semicontinuous, any closed collection of ergodic measures with constant entropy can be frozen on its closed convex hull, covering lattice models from statistical physics.
Freezing potentials are dense in the uniform topology whenever entropy is upper semicontinuous, so every potential can be perturbed slightly to one that freezes.
For $\mathbb{Z}$-actions with specification, the typical potential does not freeze: the non-freezing potentials contain a dense $G_\delta$ even though freezing potentials are dense.
A potential that first freezes at inverse temperature $\beta_0$ exhibits a phase transition: the pressure function is not analytic at $\beta_0$.

Reading between the lines

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

Because the theorem equates freezing states with constant-entropy equilibrium states, a complete classification of which measures are equilibrium states would automatically settle which measures can be frozen; the two questions are one.
A natural test of the theorem's boundary is the non-closed equilibrium set from Example 2.2: determining whether a freezing potential exists for it would decide whether the equivalence needs an explicit closedness hypothesis.
The contrast between density of freezing potentials and genericity of non-freezing potentials suggests that 'most potentials freeze' and 'most potentials do not freeze' are both true in different topological senses; the physically relevant notion of typicality would need a measure on $C(X)$, not just a topology.
The rapid-switching construction (maximal-entropy equilibrium for $\beta<1$, zero-entropy equilibrium for $\beta>1$) is a toy model for first-order-like transitions; one could try to build analogous potentials with nonzero entropy on both sides of the switch.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The central claim is attractive and probably true: a collection of measures can freeze exactly when it is an equilibrium set with constant entropy. Corollary 2 and the density/genericity theorems (Theorems 4 and 5) are genuinely new and useful. But the proof of Theorem 1 is not gap-free, and I don't think the repair is as easy as the stress-test note suggests.

The proof asserts 'since F is convex, F is a closed and convex set.' That's false, and the paper's own Example 2.2 is a counterexample. The closedness is needed for the lower semicontinuity of the barrier function g in Edwards's theorem. The stress-test says one can work with cl(F) instead, but the proposed g' (0 on cl(F), ||phi|| elsewhere) is not concave in general: a convex combination of two points outside cl(F) can land in cl(F) if cl(F) is not a face, and no argument is given that cl(F) is a face in the non-u.s.c. case. So the separation step, as written and as repaired in the note, doesn't go through.

That said, the theorem may well be salvageable with a different separation argument, because F is a face for the affine function h+∫phi and its complement is convex. A referee with expertise in Choquet theory could sort this out. The rest of the paper is in better shape. Lemma 4.4 has an inequality direction error, but the result is true with a one-line fix. Theorem 4's proof is fine. Theorem 5's argument is fairly standard, though the claim that Hölder potentials lie in A needs to be checked carefully.

So my overall take: novel results, serious but repairable-seeming gaps in the main proof, and some minor errors elsewhere. I wouldn't take it as it stands, but I would send it to a good referee, especially one who can evaluate the Edwards step. The author clearly knows the literature and the question is legitimate.

Referee Report

3 major / 6 minor

Summary. The paper studies freezing phase transitions for continuous potentials on general dynamical systems. Its main theorem (Theorem 1) claims that a nonempty set F of invariant measures can be realized as the set of equilibrium states for βψ for every β ≥ 1 (a freezing set) if and only if entropy is constant on F and F is already an equilibrium set for some potential. From this it derives corollaries for single ergodic measures and for upper semicontinuous entropy maps, plus density of freezing potentials (Theorem 4) and genericity of non-freezing potentials under specification (Theorem 5). Additional observations concern analyticity of the pressure function at freezing temperatures and a rapid switching example. The paper is clearly written and the main idea is appealing, but the proof of the main theorem contains a substantive separation-theoretic gap, and a later proof contains an inequality-direction error.

Significance. If the main theorem is established, it gives a complete characterization of freezing sets in arbitrary dynamical systems and directly answers, in broad generality, the question of whether equilibrium and freezing states impose the same restriction on a measure. The Edwards-separation strategy is natural and the corollaries, especially the dense freezing potentials and the residual non-freezing potentials under specification, would be valuable contributions. The manuscript also correctly draws on external theorems (Ruelle, Jenkinson, Bowen, Morris) without circularity. However, the current proof of Theorem 1 is not valid as written, so the advertised generality is not yet established; the gaps appear repairable, but they are load-bearing.

major comments (3)

[§3.1, proof of Theorem 1] The proof asserts: 'since F is convex, F is a closed and convex set.' This is false in general, and the paper's own Example 2.2 gives a convex equilibrium set that is not closed. Therefore the function g defined by g|F=0 and g|MT(X)\F=||φ|| need not be weak-* lower semicontinuous, and Edwards' separation theorem cannot be applied as stated. Moreover, concavity of g requires the face property of equilibrium sets, not mere convexity: if F were only convex, two points outside F could have their midpoint in F and concavity would fail. The proof can likely be repaired by replacing F with cl(F), proving that cl(F) is a face of the Choquet simplex, and checking that any ν∈cl(F)\F has h(ν)<h0 and hence is not an equilibrium state for the constructed potential. This repair is substantive and must be written out before Theorem 1 can be considered proved.
[§3.3, proof of Theorem 5] The proof states: 'by continuity of Ptop and Max(·), we know A = ⋂β>0 Aβ = ⋂β∈Q+ Aβ.' Continuity alone does not justify this equality: a continuous function can satisfy a strict inequality on a dense set yet attain equality at a limit point. Because the intersection over all β>0 is uncountable, A is not automatically a Gδ set. To make the argument work one must use convexity of β↦Pφ(β) and the finite asymptote βMax(φ)+h∞(φ) to show that if Pφ(q)>qMax(φ) for all rational q, then Pφ(β)>βMax(φ) for every real β>0. This argument is absent. In addition, the assertion that Bowen's theorem implies every Hölder potential lies in A is asserted without proof; the needed implication and the relevant reference should be stated explicitly.
[§4.2, Lemma 4.4] The displayed inequality in the proof has the wrong direction. Since β≥1, the factor (1−β) is nonpositive; because ∫φdµ is the maximum of ∫φdν, it follows that (1−β)∫φdµ ≤ (1−β)∫φdν, not ≥. The statement of the lemma is true and can be proved by writing h(ν)+β∫φdν = h(ν)+∫φdν+(β−1)∫φdν ≤ ∫φdµ+(β−1)∫φdµ = β∫φdµ, but the proof as printed is incorrect. Since Lemma 4.4 is used in Observation 4.5, this error must be corrected.

minor comments (6)

[§4.1, Lemma 4.1] The proof only computes the right derivative and concludes equality for all equilibrium states. One should also compute the left derivative (for β<β0) and use the existence of the derivative to obtain the equality ∫φdµ = ∂Pφ/∂β(β0).
[§3.1, proof of Theorem 1] There is a typo in the displayed inequality: 'Fdνu' should be 'Fdν'.
[Theorem 1 statement] The phrase 'there there exists' should be 'there exists'.
[§4.2, Observation 4.5] The text refers to 'Observation 4.4' in the proof, but the relevant statement is Lemma 4.4.
[§2.2, Examples 2.1 and 2.2] The two examples are asserted without proof. Example 2.2 is used later to illustrate the non-closedness issue, so providing at least a brief justification or a reference for the assertions would improve the paper.
[§3.3, proof of Theorem 5] When defining B = {φ : h∞(φ)=0}, the text says h∞ is obtained by any cluster point of equilibrium states (μβ). This presumes existence of equilibrium states for βφ; under specification existence is standard, but it would be helpful to state this explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found; the main theorem is proved from independent external results (Edwards separation theorem, Ruelle, Jenkinson, Bowen, Morris), with no self-citations or fitted inputs.

full rationale

The paper's derivation chain is self-contained in the relevant sense: Theorem 1 takes as input an equilibrium-state set F for some φ with constant entropy, and constructs a ψ via Edwards' separation theorem on the Choquet simplex M_T(X). The functions f(ν)=|∫φ dν| and g=0 on F, g=||φ|| on M_T(X)\F are constructed from the given φ; Edwards' theorem (an external, published result of D.A. Edwards) supplies an affine continuous F with f≤F≤g, and ψ=φ−F is then shown to satisfy Lemma 3.1. No parameter is fitted, no 'prediction' is a rename of an input, and no load-bearing claim depends on a self-citation. Corollaries 2 and 3 apply Ruelle's and Jenkinson's external theorems rather than prior work of the present author. The one notable defect in the written proof — the claim 'since F is convex, F is a closed and convex set' (Section 3.1) — is a soundness gap, not a circularity: the false closedness assertion is not used to smuggle in the conclusion, and the skeptic's repair via cl(F) shows the equivalence is likely true. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central theorem rests on standard results in thermodynamic formalism (Choquet simplex structure of invariant measures, affine entropy, Edwards separation, Ruelle and Jenkinson realization theorems) plus an implicit assumption that equilibrium sets are closed, which is not generally true.

assumptions (6)

standard math MT(X) is a Choquet simplex and ergodic measures are its extreme points.
Used throughout, stated in Section 2.1; standard for compact metric spaces with continuous actions.
standard math The entropy map h is affine on MT(X) and topological entropy is finite.
Used in the variational principle and in Lemma 3.1; standard in amenable group settings.
standard math Edward's theorem (Theorem 4 of [6]) applies to the functions f and g defined in Theorem 1.
The theorem requires appropriate semicontinuity and concavity; the author's application requires F to be closed, which is not established.
ad hoc to paper The equilibrium set F is closed in MT(X).
Asserted in the proof of Theorem 1 without justification; contradicted by Example 2.2 in the same paper.
standard math The realization theorems of Ruelle (Corollary 3.17 [15]) and Jenkinson (Theorem 5 [9]) allow arbitrary closed collections of ergodic measures to be equilibrium sets when h is upper semicontinuous.
Used in Corollary 3 and Observation 4.5.
standard math Under specification, Holder potentials have unique equilibrium states (Bowen [1]) and generic continuous potentials have zero residual entropy (Morris [14]).
Used in the proof of Theorem 5.

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Cite this review

Pith. "Pith review of On the Equivalence of Equilibrium and Freezing States in Dynamical Systems." pith.science (2026). https://pith.science/paper/SVFKZFRE

@misc{pith2026241205639,
  author       = {Pith},
  title        = {Pith review of: On the Equivalence of Equilibrium and Freezing States in Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SVFKZFRE}},
  note         = {Machine review of arXiv:2412.05639}
}

abstract

This paper is concerned with freezing phase transitions in general dynamical systems. A freezing phase transition is one in which, for a given potential $\phi$, there exists some inverse temperature $\beta_0 > 0$ such that for all $\alpha, \beta > \beta_0$, the collection of equilibrium states for $\alpha \phi$ and $\beta \phi$ coincide. In this sense, below the temperature $1 / \beta_0$, the system "freezes" on a fixed collection of equilibrium states. We show that for a given invariant measure $\mu$, it is no more restrictive that $\mu$ is the freezing state for some potential than it is for $\mu$ to be the equilibrium state for some potential. In fact, our main result applies to any collection of equilibrium states with the same entropy. In the case where the entropy map $h$ is upper semi-continuous, we show any ergodic measure $\mu$ can be obtained as a freezing state for some potential. In this upper semi-continuous setting, we additionally show that the collection of potentials that freeze at a single state is dense in the space of all potentials. However, in the $\Z$ action setting where the dynamical system satisfies specification, the collection of potentials that do not freeze contains a dense $G_\delta$.

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Forward citations

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Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Some systems with unique equilibrium state s

Rufus Bowen. Some systems with unique equilibrium state s. Math. Systems Theory, 8(3):193–202, 1974/75

work page 1974

[2] [2]

Renormalization, the rmodynamic formalism and quasi-crystals in subshifts

Henk Bruin and Renaud Leplaideur. Renormalization, the rmodynamic formalism and quasi-crystals in subshifts. Comm. Math. Phys., 321(1):209–247, 2013

work page 2013

[3] [3]

Renormalization, fre ezing phase transitions and Fibonacci qua- sicrystals

Henk Bruin and Renaud Leplaideur. Renormalization, fre ezing phase transitions and Fibonacci qua- sicrystals. Ann. Sci. ´Ec. Norm. Sup´ er.(4) , 48(3):739–763, 2015

work page 2015

[4] [4]

Nonlinear thermodynamical formalism

J´ erˆ ome Buzzi, Beno ˆ ıt Kloeckner, and Renaud Leplaideur. Nonlinear thermodynamical formalism. arXiv preprint arXiv:2002.00576, 2020

work page Pith review arXiv 2002

[5] [5]

On the zero-t emperature limit of Gibbs states

Jean-Ren´ e Chazottes and Michael Hochman. On the zero-t emperature limit of Gibbs states. Communications in Mathematical Physics, 297(1):265–281, 2010

work page 2010

[6] [6]

On separation and approximation of real func tions deﬁned on a Choquet simplex

DA Edwards. On separation and approximation of real func tions deﬁned on a Choquet simplex. General Topology and its Relations to Modern Analysis and Algebra, pages 122–128, 1967

work page 1967

[7] [7]

Nicolai T. A. Haydn. Phase transitions in one-dimension al subshifts. Discrete Contin. Dyn. Syst., 33(5):1965–1973, 2013

work page 1965

[8] [8]

Examples for the nonuniqueness of the eq uilibrium state

Franz Hofbauer. Examples for the nonuniqueness of the eq uilibrium state. Transactions of the American Mathematical Society, 228:223–241, 1977

work page 1977