Statistics of stochastic entropy for recorded transitions between ENSO states

S\'ilvio M. Duarte Queir\'os

REVIEW 4 major objections 5 minor 68 references

Statistics of stochastic entropy for recorded transitions between ENSO states

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper finds that none of the 11 recorded ENSO phase transitions since 1991 produced an extreme entropy variation; only the 1999–2000 La Niña to 2002–2003 El Niño change approaches the 4σ threshold.

desk verdict New stochastic-thermodynamics application to ENSO transitions, but an internal sign inconsistency and a tautological IFT check leave the specific extreme-event claim unverified. read the letter →

arxiv 2507.10516 v1 pith:2CRBAJJR submitted 2025-07-14 cond-mat.stat-mech physics.ao-phphysics.data-an

Sílvio M. Duarte Queirós This is my paper · ORCID

classification cond-mat.stat-mechphysics.ao-phphysics.data-an MSC 82C3160J6086A10 PACS 05.40.-a05.70.Ln

keywords stochasticthermodynamicsentropyproductionElNiño–SouthernOscillationSouthernIndexintegralfluctuationtheoremKramers–Moyalcoefficientsclimatedynamicspathintegrals

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 asks whether the dramatic weather difference between El Niño and La Niña shows up in the entropy produced by the Southern Oscillation Index as the climate moves from one phase to the other. Using stochastic thermodynamics, it computes the entropy variation for each of the 11 recorded ENSO phase transitions in the daily SOI record, both along the full index path and from boundary values alone. It finds that none of these transitions qualifies as an extreme entropic event: most are below one-tenth of a standard deviation, and only the transition from the strong 1999–2000 La Niña to the moderate 2002–2003 El Niño sits near the extreme-event threshold. All transitions also satisfy the integral fluctuation relation, the probabilistic form of the second law. If the calculation is right, this informational entropy is decoupled from the thermal/heat picture of these climate events.

What carries the argument

The load-bearing object is the stochastic entropy production defined through the log-ratio of the probability of the recorded SOI trajectory under the forward protocol to its probability under the reversed protocol. To get those probabilities, the paper separates the daily SOI series $s_t$ into a deterministic climate protocol $\Phi(t)$ (reconstructed by the Multi-Taper Method at seven significant periods) and a residual $\xi_t = s_t - \Phi(t)$ modelled as a stationary stochastic process with empirically fitted Kramers–Moyal coefficients. A change of variables converts the multiplicative noise into additive noise, and a path-integral (Onsager–Machlup) saddle-point approximation yields the forward and reverse transition probabilities used in the entropy ratio.

What would settle it

Refit the residual dynamics with a model that includes memory or non-Gaussian noise (for example, a nonzero third-order Kramers–Moyal coefficient or a non-Markovian term) and recompute the 11 entropy variations; if the 1999–2000 La Niña to 2002–2003 El Niño event drops below 3σ or another transition rises above 4σ, the extreme-event classification is an artifact of the stochastic model rather than a property of the data.

Watch

Extended reading notes

Core claim

The central claim is that the entropy variation $\Delta S(\vec s) = -\ln[p_F(\vec s)/p_R(\vec s)]$ computed from forward and reverse path probabilities of the daily SOI is statistically unremarkable for all 11 ENSO transitions. Only transition 4, the strong 1999–2000 La Niña to the moderate 2002–2003 El Niño, reaches about 4σ above the mean in the full-trajectory calculation (3σ when only the endpoints are used), placing it on the brink of being an extreme event but not clearly beyond it. The paper also verifies the integral fluctuation theorem, $\langle e^{-\Delta S}\rangle = 1$, in both calculation schemes, and finds no relation between the entropy variation rate and the intensity classification of the phases.

Load-bearing premise

The calculation assumes that after subtracting the climate protocol the daily SOI fluctuations are a stationary Markov process driven by Gaussian white noise with no third- or higher-order Kramers–Moyal coefficients, so the fitted model, not the raw data alone, fixes the path probabilities and hence the entropy values.

Editorial extensions

If this is right

All 11 recorded ENSO phase transitions satisfy the integral fluctuation relation in both the full-trajectory and boundary-only calculations, so the computed entropy changes are consistent with the probabilistic second law.
Most transitions have entropy variations below one-tenth of the trajectory-to-trajectory spread, so El Niño and La Niña phase shifts have so far been mild in this informational-entropy sense.
The 1999–2000 La Niña to 2002–2003 El Niño transition is the only case near the extreme threshold, making it the natural candidate for targeted study of large entropy excursions in the SOI.
The absence of a relation between entropy variation rate and the intensity classification of the phases suggests the index-based informational entropy and the thermal character of ENSO are not simply connected.

Reading between the lines

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

If the result survives, it implies that extreme weather associated with ENSO is not mirrored by extreme stochastic-entropy production of the SOI itself, so the index may not be the right observable for detecting thermodynamically exceptional transitions.
The same machinery could be applied to other ENSO indicators such as sea-surface-temperature indices or multivariate ENSO measures; a 4σ entropy event appearing in one of those would test whether the mildness found here is a property of the phenomenon or of the chosen index.
The boundary-only entropy variation $\Delta\tilde{S}(t_f,t_i)$ discards most of the signal (all but two transitions fall below $\sigma/10$), so comparing the two tables offers a concrete way to quantify how much of the entropy information lives in the path rather than in the endpoints.
A future strong La Niña to El Niño transition similar to 1999–2000 could be monitored in real time; if its entropy variation exceeds the 4σ threshold, the paper's conclusion would shift from 'on the brink' to observed extreme behaviour.

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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

Dear [Colleague],

This paper applies stochastic thermodynamics to daily Southern Oscillation Index data, computing path entropies for the 11 recorded ENSO phase transitions and checking both integral fluctuation relations. That application is new; I do not know of prior work doing entropy fluctuations for ENSO indices, though the claim to 'inaugurate' the mindset is overblown given the existing stochastic SOI literature. The supplemental material is thorough in documenting the Kramers-Moyal fits, the protocol reconstruction, and the long propagator formulas, and the author is transparent about the modeling assumptions and the modest scope. The broad qualitative conclusion that most transitions are not extreme entropic events is plausible and survives most concerns.

The quantitative claims, however, are not yet robust. The most concrete problem is the drift coefficient b: main-text Eq. (1) reports b = +0.08 ± 0.01, while the SM fit reports b = −0.08 ± 0.04 from the same data. The sign enters the transformed drift, the stationary density, and every propagator term, so all of Tables II and III are non-reproducible as printed. This is not a modeling disagreement; it should be settled by recomputation. A second problem is that the 'verification' of the fluctuation theorems is logically empty: forward and reverse path probabilities are generated from the same fitted model, so ⟨exp(−ΔS)⟩ = 1 is an identity, not an independent check. Third, the paper calls transition 4 a 4σ event, but its own CDF in Fig. 4d shows a probability of about 0.98 at the realized ΔS, which is roughly 2σ, not 4σ. The conclusion correctly downgrades this to 'on the brink,' but the main-text claim is not supported by the figure. Fourth, parameter uncertainties are not propagated into the reported ΔS or the extremeness classification; the σ in the tables is the model's trajectory spread, not the fit uncertainty.

The transitions also differ widely in duration (from one day to several years), which makes raw path-entropy comparisons tricky; the author does compute entropy rates, which helps. None of these issues invalidates the broad finding, since the entropy values are small relative to the noise scale, but the specific extreme-event classification needs to be redone with the sign fixed and the CDF discrepancy explained.

This paper is for readers interested in stochastic thermodynamics applied to climate indicators. It deserves a serious referee, not a desk reject, but the referee should require the internal inconsistency to be fixed before publication.

Best, [Name]

Referee Report

4 major / 5 minor

Summary. The paper analyzes daily Southern Oscillation Index (SOI) data from 1991–2023 to study entropy production during 11 catalogued El Niño/La Niña transitions. The SOI series is decomposed into a deterministic 'protocol' Φ(t) obtained by multi-taper spectral analysis and a residual ξ(t), which is modeled as a stationary Markov process with a linear drift and a quadratic diffusion coefficient (Eq. 1). After a Lamperti transformation to additive noise and a saddle-point evaluation of the Onsager–Machlup path integral, the paper derives forward and reverse transition probabilities and computes the stochastic entropy variation ΔS for each transition, both along the full observed trajectory (Table II) and using only the endpoints (Table III). The paper reports that all transitions satisfy the integral fluctuation theorem and finds that only transition 4 (1999–2000 La Niña to 2002–2003 El Niño) is near the threshold of an extreme entropic event (4σ in the full-trajectory measure, 3σ in the endpoint measure).

Significance. If the underlying stochastic model and the approximations were reliable, this would be a novel application of stochastic thermodynamics to a geophysical index, with a concrete, falsifiable ranking of ENSO transitions by entropy production. The paper is self-contained in its derivations, uses publicly available data, and provides considerable technical detail in the Supplemental Material, including explicit propagator formulas and numerical simulations of the fitted model. However, the significance is limited by the fact that the path probabilities and stationary distributions used to compute entropy are all generated from the same fitted model, so the fluctuation-theorem 'verification' is an internal consistency check rather than an empirical test. The usefulness of the extreme-event classification also depends on the correctness of a heavily parameterized model with several uncontrolled approximations.

major comments (4)

[Main text Eq. (1) vs SM 'Computation of the Kramers-Moyal coefficients'] The fitted parameter b is reported with opposite signs in the two parts of the manuscript: the main text (Eq. (1) and the following line) gives b = 0.08 ± 0.01, while the Supplemental Material (section 'Computation of the Kramers-Moyal coefficients', after Eq. (19)) reports b = −0.08 ± 0.04 from the same data. The sign of b enters the transformed dynamics (Eq. (7)), the stationary density (Eq. (6)), and every forward/reverse propagator term Ω_i^F and Ω_i^R in the SM (Eqs. (41)–(54)), and therefore it changes all values of ΔS, ⟨ΔS⟩, and σ in Tables II and III. The paper does not state which value was actually used in the reported computations. This internal inconsistency makes the headline numbers irreproducible and must be resolved by recomputation or an explicit statement before the results can be assessed.
[Fig. 4d and the '4σ' claim after Table II] The main text states that transition 4 'is a 4σ event and could statistically be considered an extreme event.' However, the paper's own cumulative distribution function for transition 4, shown in Fig. 4d, has the realized value ΔS = 0.330 marked at a CDF value of approximately 0.97–0.98, which corresponds to roughly 2σ under a normal distribution, not 4σ. The σ reported in Table II (8.13 × 10^-2) gives (ΔS − ⟨ΔS⟩)/σ ≈ 4.05, but the empirical CDF built from the same model indicates that the distribution is strongly non-Gaussian and that the 4σ classification is not supported by the paper's own figure. The claim should be revised, or the discrepancy between the Gaussian-sigma calculation and the CDF must be explained.
[Main text, after Table II; SM Eqs. (41)–(42)] The paper presents the verification of the integral fluctuation theorem as a positive check on the analysis. Because the forward and reverse transition probabilities pF and pR are both derived from the same fitted stochastic differential equation (SM Eqs. (41)–(42)), the relation ⟨exp(−ΔS)⟩ = 1 is an identity that holds by construction when the forward and reverse path probabilities are normalized and the entropy is defined as their log ratio. This check verifies internal consistency of the numerical implementation, but it does not provide independent evidence that the model or the computed entropy values describe the actual SOI data. The text should explicitly label this as a self-consistency check rather than an empirical validation.
[Main text Eq. (1); SM 'Computation of the Kramers-Moyal coefficients' and Fig. 2] The entire entropy calculation rests on the assumption that the residual ξ_t = s_t − Φ(t) is stationary and Markovian with Kramers–Moyal coefficients exactly zero for n ≥ 3 and Gaussian white noise. The paper's validation (SM Fig. 2c,d) compares third and fourth moments of the data with simulations generated from the same fitted model; this is not an independent test of those assumptions. If real SOI fluctuations have memory (for example, due to the multi-taper reconstruction or unresolved low-frequency variability) or non-Gaussian noise, the computed path probabilities and all entropy values in Tables II and III are model artifacts rather than measured quantities. This limitation should be stated explicitly, and the Markov property and noise statistics should be checked with a data-driven, non-parametric method before the central claim is made.

minor comments (5)

[Throughout] The presentation contains several typos and inconsistent notation; examples include 'Jarzinsky' in Ref. [30], the missing journal information in Ref. [61], and the use of both 'integral fluctuation relations' (abstract) and 'the Integral Fluctuation Theorem' (main text) for the same quantity.
[SM Eqs. (15)–(18)] The variables z and w used in the effective Kramers–Moyal coefficients are defined only after Eq. (15); moving the definitions before the equations would improve readability.
[Tables II and III] The numeric entries use inconsistent formatting, such as '1 .61 × 10−1' and '3 .30 × 10−1' with a space after the integer part; this should be cleaned up for consistency.
[Conclusion and Table II] The conclusion states that transition 4 is 'the only case on the brink of being an extreme event,' but Table II also shows transitions 1 and 2 with |ΔS| above 2σ. The text should clarify that 'on the brink' refers specifically to the 4σ threshold used for extreme events, not to all deviations above 2σ.
[SM Fig. 2] The axis labels and tick labels in the Supplemental Material figures appear garbled (e.g., '/Minus30' instead of −30); the figures should be regenerated with proper typography.

Circularity Check

1 steps flagged · score 4.0 of 10

The reported IFT verification is a tautology because the forward and reverse probabilities come from the same fitted model; the extreme-event classification is model-relative but not definitionally forced by the fit.

self definitional [Main text, paragraph after Table II; Eq. (4)]
"Even so, all the transitions abide by the Integral Fluctuation Theorem, ⟨exp [−∆S]⟩ = 1, a probabilistic version of the 2nd Law of Thermodynamics ∆ S ≥ 0 [26]."

Eq. (4) defines ∆S(s⃗) = −ln[p*_ti(s_ti) pF(s_ti+1|s_ti,Φ)... / (p*_tf(s_tf) pR(s_tf−1|s_tf,Φ)...)], with the stationary densities p* and the forward/reverse propagators pF/pR all constructed from the same fitted model (main-text Eqs. 1–3 and SM Eqs. 41–42). For any pair of normalized forward and reverse path probabilities, the average of exp(−∆S) over forward paths equals ∫ p*_tf(s_tf) pR(reverse path) d(path) = 1 identically. Thus the reported 'verification' is a mathematical identity inherited from the definition of ∆S, not an empirical check of the stochastic model. The fitted parameters cancel out of this relation, so the agreement carries no independent evidence for the entropy analysis.

full rationale

The paper contains no load-bearing self-citation chain: the use of Anteneodo & Duarte Queirós [48] in the SM is a published methodological derivation, not an unverified premise or a uniqueness theorem, and it is not what forces the central conclusion. The extreme-event classification is also not a simple fitted-input-called-prediction: the model is fitted to Kramers–Moyal coefficients of the residual ξ, while the realised ΔS values are computed from the recorded SOI paths, so the verdict is not a numerically forced re-statement of the fit. The single clear circular element is the IFT check, which is tautological because forward and reverse probabilities are generated from the same model; this inflates the appearance of validation but does not by itself determine the paper's substantive claim that only transition 4 approaches extreme-event status. That claim is model-relative and in-sample, which is a limitation rather than a definitional circularity. The sign discrepancy for b (+0.08 in main-text Eq. (1) vs −0.08 in the SM fit) is a reproducibility/correctness issue, not a circularity, and is noted here only to keep it distinct from the circularity verdict.

Assumptions & free parameters 7 free parameters · 4 assumptions · 0 invented entities

The entire entropy computation is conditioned on a model whose parameters are fitted to the same SOI data. The protocol (23 coefficients plus 7 periods) and the five stochastic coefficients are all data-derived, so the computed entropy tables and extreme-event classification are in-sample properties of the model, not independent measurements.

free parameters (7)

a = 0.15 ± 0.01
Drift coefficient in D1(ξ) = -a ξ + b, fitted to the empirical first Kramers-Moyal moment of the detrended SOI residual.
b = 0.08 ± 0.01 (main text), -0.08 ± 0.04 (SM)
Constant term in the drift; the sign inconsistency between main text and SM is not resolved in the paper.
α2 = 0.010 ± 0.002
Quadratic coefficient in the noise D2(ξ) = α2 ξ^2 + β ξ + γ2.
β = -0.14 ± 0.03
Linear coefficient in the noise term.
γ2 = 31.65 ± 0.8
Constant coefficient in the noise term.
Protocol coefficients A0, B0, Ai, Bi, θi (i=1..7) = 23 values, listed in Table IV of the SM
Fitted by mean-square minimization over 10000 initial conditions to reconstruct the deterministic protocol Φ(t).
Periods T_i = 24, 28, 36, 74, 102, 365, 2168 days
Selected from significant Multi-Taper spectral peaks of the same SOI dataset.

assumptions (4)

standard math Standard stochastic calculus and path integral formalism (Itô/Stratonovich, Fokker-Planck, Onsager-Machlup).
Used throughout to transform the SDE, derive the Fokker-Planck equation, and obtain saddle-point transition probabilities.
domain assumption The detrended residual ξ_t is stationary and Markovian with Gaussian white noise.
Required to define empirical Kramers-Moyal coefficients and to justify the form of Eq. (1). No test for Markovianity or non-stationarity is provided.
domain assumption The Multi-Taper reconstruction Φ(t) with seven significant periods captures all deterministic forcing.
The residual ξ_t = s_t - Φ(t) is treated as purely stochastic; if Φ misses real deterministic structure, the fitted noise is contaminated.
ad hoc to paper Small-parameter expansions and the saddle-point approximation are valid for the SOI fitted parameters.
The Euler-Lagrange equation is not exactly solvable; the paper neglects the c2 term and uses expansions based on specific fitted values (e.g., c2 << c1).

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Pith. "Pith review of Statistics of stochastic entropy for recorded transitions between ENSO states." pith.science (2026). https://pith.science/paper/2CRBAJJR

@misc{pith2026250710516,
  author       = {Pith},
  title        = {Pith review of: Statistics of stochastic entropy for recorded transitions between ENSO states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2CRBAJJR}},
  note         = {Machine review of arXiv:2507.10516}
}

abstract

We analyse the transitions between established phases of the El Ni\~no Southern Oscillation (ENSO) by surveying the daily data of the Southern Oscillation Index from an entropic viewpoint using the framework of stochastic Statistical Physics. We evaluate the variation of entropy produced due to each recorded path of that index during each transition as well as taking only into consideration the beginning and the end of the change between phases and verified both integral fluctuation relations. The statistical results show that these entropy variations have not been extreme entropic events; only the transition between the strong $1999-2000$ La Ni\~na to the moderate $2002-2003$ El Ni\~no is at the edge of being so. With that, the present work opens a long and winding avenue of research over the application of stochastic Statistical Physics to Climate Dynamics.

Figures

Figures reproduced from arXiv: 2507.10516 by the authors.

**Figure 2.** FIG. 2. First, second, third and fourth empirical Kramers-Moyal moments, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Joint (forward) probability density function [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Cumulative distribution functions, [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

Discussion (0). Continue with ORCID to comment.

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