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arxiv: 2604.15773 · v1 · submitted 2026-04-17 · ❄️ cond-mat.stat-mech · cs.AI· stat.ME

Phase Transitions as the Breakdown of Statistical Indistinguishability

Taiyo Narita , Hideyuki Miyahara This is my paper

Pith reviewed 2026-05-10 08:05 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cs.AIstat.ME
keywords phase transitionsstatistical indistinguishabilityhypothesis testingtwo-dimensional Ising modelcritical pointBinder parameterrun testthermodynamic limit
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The pith

A phase transition is the breakdown of statistical indistinguishability between system states under vanishingly small parameter changes in the thermodynamic limit.

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

The paper redefines phase transitions using hypothesis testing on data samples. It claims a transition occurs precisely when ensembles generated at nearby parameter values become distinguishable as system size approaches infinity. This view treats the Binder parameter and similar tools as specific ways to perform that distinguishability check. The authors apply a simple distribution-free run test to the two-dimensional Ising model and recover the known critical point without any prior information on the order parameter. The result supplies an order-parameter-free route to locating transitions that depends only on comparing raw configurations from perturbed conditions.

Core claim

We introduce a novel characterization of phase transitions based on hypothesis testing. In our formulation, a phase transition is defined as the breakdown of statistical indistinguishability under vanishing parameter perturbations in the thermodynamic limit. This perspective provides a general, order-parameter-free framework that does not rely on model-specific insights or learning procedures. We show that conventional approaches, such as those based on the Binder parameter, can be reinterpreted as special cases within this framework. As a concrete realization, we employ a distribution-free two-sample run test and demonstrate that the critical point of the two-dimensional Ising model isaccur

What carries the argument

Hypothesis testing that checks whether distributions of system configurations remain statistically indistinguishable when a control parameter is infinitesimally perturbed, taken in the infinite-size limit.

If this is right

  • Methods such as the Binder cumulant become particular instances of checking for indistinguishability breakdown.
  • Critical points can be located in systems where no order parameter is known in advance.
  • The framework requires no model-specific knowledge or training procedures.
  • Distribution-free tests suffice to recover the transition location in the Ising model.

Where Pith is reading between the lines

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

  • The same testing idea could be applied directly to experimental time series or imaging data where theoretical order parameters are unavailable.
  • Substituting other hypothesis tests for the run test might increase sensitivity for systems with different noise characteristics.
  • Phase transitions could be reinterpreted in terms of information distinguishability across a wider range of physical and non-physical systems.

Load-bearing premise

The chosen test must remain powerful enough to detect true distinguishability only at the actual transition and must not produce false signals from finite-size effects or correlations when the system is very large.

What would settle it

Applying the two-sample run test to Monte Carlo samples of the two-dimensional Ising model at many temperatures near the known critical point and observing that the detected transition temperature deviates significantly from the accepted value on sufficiently large lattices would falsify the central claim.

Figures

Figures reproduced from arXiv: 2604.15773 by Hideyuki Miyahara, Taiyo Narita.

Figure 1
Figure 1. Figure 1: FIG. 1: Schematics for (a) the proposed approach and (b) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
read the original abstract

We introduce a novel characterization of phase transitions based on hypothesis testing. In our formulation, a phase transition is defined as the breakdown of statistical indistinguishability under vanishing parameter perturbations in the thermodynamic limit. This perspective provides a general, order-parameter-free framework that does not rely on model-specific insights or learning procedures. We show that conventional approaches, such as those based on the Binder parameter, can be reinterpreted as special cases within this framework. As a concrete realization, we employ a distribution-free two-sample run test and demonstrate that the critical point of the two-dimensional Ising model is accurately identified without prior knowledge of the order parameter.

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

2 major / 2 minor

Summary. The manuscript proposes a definition of phase transitions as the breakdown of statistical indistinguishability between ensembles at infinitesimally nearby parameter values in the thermodynamic limit. It shows that standard diagnostics such as the Binder cumulant arise as special cases of this hypothesis-testing perspective and demonstrates the framework by applying a distribution-free two-sample run test to Monte Carlo samples of the 2D Ising model, claiming that the critical coupling is recovered without prior knowledge of the order parameter.

Significance. If the central claim is substantiated, the work supplies a genuinely order-parameter-free and model-independent route to locating phase transitions that relies only on the ability to sample configurations at nearby parameter values. The distribution-free character of the run test is a clear methodological strength, and the reinterpretation of existing methods as special cases within the same framework is conceptually useful. These features could extend the approach to systems where order parameters are unknown or difficult to define.

major comments (2)
  1. [Abstract and numerical demonstration] Abstract and the numerical demonstration section: the claim that the critical point of the 2D Ising model 'is accurately identified' is asserted without any reported error bars, finite-size scaling collapse, or direct numerical comparison to the known analytic value K_c ≈ 0.4406868. In the absence of such quantitative controls, it is impossible to judge whether the detected transition coincides with the true critical point within statistical uncertainty.
  2. [Run-test implementation] Section describing the run-test implementation: the two-sample run test relies on the assumption that the two sets of samples are exchangeable under the null hypothesis of indistinguishability. At criticality, however, the correlation length diverges with system size L, so that Monte Carlo samples separated by finite autocorrelation times become strongly dependent; this violates the exchangeability assumption and can systematically alter the run-length distribution, potentially shifting the parameter value at which the test rejects the null away from the true critical coupling.
minor comments (2)
  1. [Abstract] The abstract refers to 'vanishing parameter perturbations' but does not specify the scaling of the perturbation amplitude with system size L; a concrete protocol (e.g., ΔK ∝ 1/L^2 or fixed ΔK) should be stated explicitly.
  2. [Methods] Notation for the run-test statistic and the precise definition of the two samples (energy histograms, spin configurations, or magnetization) should be introduced with an equation or pseudocode to allow reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help strengthen the manuscript. We address each major point below, indicating the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and numerical demonstration] Abstract and the numerical demonstration section: the claim that the critical point of the 2D Ising model 'is accurately identified' is asserted without any reported error bars, finite-size scaling collapse, or direct numerical comparison to the known analytic value K_c ≈ 0.4406868. In the absence of such quantitative controls, it is impossible to judge whether the detected transition coincides with the true critical point within statistical uncertainty.

    Authors: We agree that the current presentation lacks the quantitative controls needed to substantiate the accuracy claim. In the revised manuscript we will add error bars on the detected transition point obtained from multiple independent Monte Carlo runs, perform a finite-size scaling analysis across several system sizes to extrapolate to the thermodynamic limit, and directly compare the extrapolated value to the known analytic K_c ≈ 0.4406868. These additions will allow readers to assess agreement within statistical uncertainty. revision: yes

  2. Referee: [Run-test implementation] Section describing the run-test implementation: the two-sample run test relies on the assumption that the two sets of samples are exchangeable under the null hypothesis of indistinguishability. At criticality, however, the correlation length diverges with system size L, so that Monte Carlo samples separated by finite autocorrelation times become strongly dependent; this violates the exchangeability assumption and can systematically alter the run-length distribution, potentially shifting the parameter value at which the test rejects the null away from the true critical coupling.

    Authors: This is a substantive concern about the validity of the exchangeability assumption under critical correlations. We will revise the manuscript to (i) specify the decorrelation protocol (number of sweeps between retained samples chosen to exceed the measured autocorrelation time at each K), (ii) report numerical checks that the identified transition location remains stable when the sampling interval is varied by factors of two or more, and (iii) add a brief discussion of the limitations of the run test for dependent samples. While these steps mitigate the practical impact, a fully rigorous treatment of the test under long-range dependence lies outside the present scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines phase transitions as the breakdown of statistical indistinguishability under vanishing perturbations in the thermodynamic limit and implements this via a standard off-the-shelf two-sample run test. This definition is introduced independently rather than derived from any fitted parameter or prior result within the paper. Conventional methods such as the Binder parameter are reinterpreted as special cases but are not used as inputs that force the new result. No equations reduce by construction to the target claim, no self-citations are load-bearing for the central premise, and the run test is not tuned using knowledge of the critical point. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard thermodynamic-limit assumption of statistical mechanics and the validity of the run test as a distribution-free procedure; no new entities or fitted parameters are introduced in the abstract.

axioms (2)
  • domain assumption The thermodynamic limit exists and statistical properties become sharply defined as system size tends to infinity
    Invoked directly in the definition of phase transition as breakdown under vanishing perturbations.
  • standard math A distribution-free two-sample run test can detect statistical indistinguishability between nearby parameter values
    Used as the concrete realization for identifying the critical point.

pith-pipeline@v0.9.0 · 5404 in / 1332 out tokens · 67824 ms · 2026-05-10T08:05:58.147241+00:00 · methodology

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Reference graph

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