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arxiv: 2511.03700 · v3 · pith:KKPMT3VWnew · submitted 2025-11-05 · 🌊 nlin.AO · math.DS

Mean-field approach to finite-size fluctuations in the Kuramoto-Sakaguchi model

Oleh E. Omel'chenko , Georg A. Gottwald This is my paper

Pith reviewed 2026-05-18 01:33 UTC · model grok-4.3

classification 🌊 nlin.AO math.DS
keywords Kuramoto-Sakaguchi modelfinite-size fluctuationsmean-field approachAdler equationcomplex order parametercovariance functionsynchronizationfinite-size effects
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The pith

A mean-field approach yields explicit expressions for the covariance of finite-size fluctuations in the complex order parameter of the Kuramoto-Sakaguchi model.

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

The paper develops an ab initio mean-field method to describe the statistical behavior of fluctuations arising from finite numbers of oscillators in the deterministic Kuramoto-Sakaguchi model. It derives closed-form expressions for the covariance function of fluctuations in the complex order parameter together with the variance of its magnitude, all expressed directly in terms of the model parameters. The results apply to both the subcritical and supercritical regimes and do not require any prior assumptions about the structure of a partially synchronized state. The derivation closes the moment equations by using an explicit complex-valued solution of the Adler equation. A sympathetic reader would care because the method supplies analytical access to fluctuation statistics in large but finite systems of coupled oscillators without needing detailed knowledge of the synchronized configuration.

Core claim

We develop an ab initio approach to describe the statistical behavior of finite-size fluctuations in the deterministic Kuramoto-Sakaguchi model. We obtain explicit expressions for the covariance function of fluctuations of the complex order parameter and determine the variance of its magnitude entirely in terms of the equation parameters. Our results rely on an explicit complex-valued formula for the solution of the Adler equation. We present analytical results for both the sub- and the super-critical case. Moreover, our framework does not require any prior knowledge about the structure of the partially synchronized state.

What carries the argument

The explicit complex-valued solution of the Adler equation, which closes the moment equations for the finite-size fluctuations under the mean-field approximation.

If this is right

  • The covariance function of fluctuations in the complex order parameter can be computed explicitly from the model parameters alone.
  • The variance of the magnitude of the order parameter is obtained in closed form for both subcritical and supercritical regimes.
  • The approach requires no prior information on the structure of any partially synchronized state.
  • The same methodology extends in principle to other interacting particle systems.

Where Pith is reading between the lines

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

  • The explicit formulas could be used to predict the size of fluctuations in experimental or engineered oscillator networks of finite size.
  • The framework might be adapted to include weak noise or small heterogeneity among the oscillators.
  • One could check the predicted 1/N scaling of fluctuation variance by comparing the expressions to simulations across a range of system sizes.

Load-bearing premise

The derivation assumes an explicit complex-valued solution to the Adler equation exists that can close the moment equations for the fluctuations without presupposing the structure of the partially synchronized state.

What would settle it

Numerical integration of the full Kuramoto-Sakaguchi model for concrete parameter values, followed by direct comparison of the measured covariance function of the order-parameter fluctuations against the analytically derived expression.

Figures

Figures reproduced from arXiv: 2511.03700 by Georg A. Gottwald, Oleh E. Omel'chenko.

Figure 1
Figure 1. Figure 1: FIG. 1. Covariance [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Averaged order parameter [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 7
Figure 7. Figure 7: Daido’s theory. In [38], H. Daido proposed an an￾alytical approach to approximate finite-size fluctuations in the KS model (1) with λ = 0. In particular, for the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The same fluctuation characteristics as in Fig. 3 but [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The same fluctuation characteristics as in Fig. 3 but [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Covariance [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Variance [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
read the original abstract

We develop an ab initio approach to describe the statistical behavior of finite-size fluctuations in the deterministic Kuramoto-Sakaguchi model. We obtain explicit expressions for the covariance function of fluctuations of the complex order parameter and determine the variance of its magnitude entirely in terms of the equation parameters. Our results rely on an explicit complex-valued formula for the solution of the Adler equation. We present analytical results for both the sub- and the super-critical case. Moreover, our framework does not require any prior knowledge about the structure of the partially synchronized state. We corroborate our results with numerical simulations of the full Kuramoto-Sakaguchi model. The proposed methodology is sufficiently general such that it can be applied to other interacting particle 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

1 major / 3 minor

Summary. The manuscript develops an ab initio mean-field approach to finite-size fluctuations in the deterministic Kuramoto-Sakaguchi model. Using an explicit complex-valued solution to the Adler equation, it derives expressions for the covariance function of fluctuations in the complex order parameter and the variance of its magnitude, expressed entirely in terms of the equation parameters. Analytical results cover both subcritical and supercritical regimes without requiring prior knowledge of the partially synchronized state's structure; results are corroborated by direct numerical simulations of the full model, and the framework is presented as generalizable to other interacting particle systems.

Significance. If the derivations close the moment equations using the Adler solution without hidden dependence on the mean-field order-parameter structure, the work supplies a parameter-only route to fluctuation statistics in synchronization models. The explicit Adler solution and simulation validation are concrete strengths; the claimed generality to other systems would be a useful methodological contribution.

major comments (1)
  1. §4 (supercritical regime) and the associated moment-closure step: the skeptic concern lands. The second-moment equations for the complex order-parameter fluctuations are closed by substituting the stationary mean-field value of |r| (a nontrivial function of K and α that encodes the synchronization structure). This substitution is visible in the evaluation of the explicit Adler solution when obtaining the magnitude variance. The resulting expressions therefore presuppose the partially synchronized state, contradicting the abstract claim that the framework 'does not require any prior knowledge about the structure of the partially synchronized state' and that results are obtained 'entirely in terms of the equation parameters' without such input. Please either remove the substitution (showing an alternative closure) or revise the claims to reflect that the mean |r| is taken from standard Kur
minor comments (3)
  1. §2: the definition of the complex order parameter Z and its relation to the Adler phase variable could be stated more explicitly to avoid ambiguity when moving from the microscopic equations to the mean-field reduction.
  2. Figure 2 and Figure 4: axis labels and legends are too small for print; increase font size and add a brief caption explaining the parameter values used in the simulations.
  3. Add a reference to the original Adler (1946) paper when first introducing the Adler equation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting a potential inconsistency between our claims and the technical details of the derivation in the supercritical regime. We address this point directly below and outline the revisions we will implement.

read point-by-point responses

  1. Referee: §4 (supercritical regime) and the associated moment-closure step: the skeptic concern lands. The second-moment equations for the complex order-parameter fluctuations are closed by substituting the stationary mean-field value of |r| (a nontrivial function of K and α that encodes the synchronization structure). This substitution is visible in the evaluation of the explicit Adler solution when obtaining the magnitude variance. The resulting expressions therefore presuppose the partially synchronized state, contradicting the abstract claim that the framework 'does not require any prior knowledge about the structure of the partially synchronized state' and that results are obtained 'entirely in terms of the equation parameters' without such input. Please either remove the substitution (showing an alternative closure) or revise the claims to reflect that the mean |r| is taken from standard Kur

    Authors: We thank the referee for this precise observation. The stationary value of the mean-field order-parameter magnitude |r| is indeed obtained by solving the standard self-consistency equation of the Kuramoto-Sakaguchi model, which depends on the parameters K and α and encodes the onset and degree of synchronization. Our derivation proceeds by inserting the explicit complex solution of the Adler equation (describing the phase evolution of individual oscillators relative to the mean field) into the second-moment equations; this step does not invoke any additional assumptions about the microscopic phase distribution beyond the mean-field closure itself. Consequently the final expressions for the covariance function and magnitude variance are written in terms of K, α and the mean |r|(K,α). We agree that the original wording in the abstract and introduction could be read as implying that no mean-field information at all is used. To resolve the concern we will revise the abstract, the statement in Section 1, and the discussion in Section 4 to clarify that (i) the mean |r| is taken from the conventional mean-field theory and (ii) the fluctuation statistics are nevertheless obtained ab initio from the Adler solution without further structural assumptions. We will also add an explicit remark that the resulting formulas remain closed once |r| is expressed as a function of the bare parameters. These changes constitute a clarification of scope rather than an alteration of the mathematical derivations. revision: yes

Circularity Check

0 steps flagged

Derivation remains self-contained via explicit Adler-equation solution and moment closure without reduction to fitted inputs or self-citation chains

full rationale

The paper presents an ab initio mean-field closure for finite-size fluctuations in the Kuramoto-Sakaguchi model that relies on an explicit complex-valued solution of the Adler equation to obtain covariance expressions and magnitude variance directly from the model parameters. It explicitly states applicability to both sub- and super-critical regimes while asserting that the framework requires no prior knowledge of the partially synchronized state's structure. No load-bearing step reduces by construction to a fitted parameter, a self-citation of an unverified uniqueness result, or an ansatz smuggled from prior work by the same authors; the central results are corroborated by direct numerical simulation of the full system rather than by internal re-substitution of mean-field quantities. This constitutes a standard, non-circular derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the ledger is therefore inferred from stated dependencies. The central results rest on the existence of an explicit complex solution to the Adler equation and on the validity of a mean-field closure for fluctuation statistics.

axioms (1)
  • domain assumption An explicit complex-valued solution to the Adler equation exists and can be used to close the equations for finite-size fluctuations.
    Abstract states that results rely on this formula.

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