Fokker-Planck entropic force interpretation of galactic rotation curves

H. Mart\'inez-Huerta; P.I. Ram\'irez-Baca; V.S. Morales-Salgado

REVIEW 3 major objections 2 minor 35 references

Fokker-Planck entropic force interpretation of galactic rotation curves

T0 review · 3 major / 2 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read An effective radial force from the stationary Fokker-Planck equation reproduces galactic rotation curves and scaling relations.

desk verdict The FPE model matches standard halo fits on SPARC with better M/L ratios, but its scaling relations with v_flat and luminosity appear to be by construction from per-galaxy parameter fitting. read the letter →

arxiv 2201.05594 v2 submitted 2022-01-13 astro-ph.GA gr-qc

V.S. Morales-Salgado , H. Martínez-Huerta , P.I. Ramírez-Baca This is my paper

classification astro-ph.GAgr-qc

keywords entropicforceFokker-PlanckequationgalacticrotationcurvesdarkmatteralternativesSPARCdatabaseTully-Fisherrelationmass-to-lightratio

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 tests whether the mismatch between baryonic mass and observed galactic rotation curves can be read as an emergent entropic force. It derives this force from a stationary solution of the Fokker-Planck equation under minimal statistical assumptions and applies the resulting model to high-quality SPARC rotation curves. The approach yields fits of comparable or better quality than standard dark-matter halo profiles while keeping stellar mass-to-light ratios inside physically plausible ranges, and it automatically produces correlations between its single free parameter and global galaxy properties that match known empirical relations such as the Tully-Fisher law.

What carries the argument

The effective radial force obtained from the stationary solution of the Fokker-Planck equation under minimal assumptions, which functions as an entropic force for galactic dynamics.

What would settle it

A high-quality rotation curve whose best FPE fit requires a stellar mass-to-light ratio outside accepted physical bounds, or a sample of galaxies in which the model's characteristic parameter shows no correlation with flat rotation velocity.

Watch

Extended reading notes

Core claim

Starting from a minimal statistical framework, a stationary solution of the Fokker-Planck equation under simple and physically motivated assumptions supplies an effective radial entropic force. When confronted with SPARC rotation curves, this Fokker-Planck entropic model produces statistical fits of comparable or improved quality relative to Navarro-Frenk-White, Burkert, and pseudo-isothermal profiles, while returning stellar mass-to-light ratios within physically consistent ranges. The same model generates strong correlations between its characteristic parameter and global galaxy properties, including flat rotation velocity and infrared luminosity, that are consistent with established scal-

Load-bearing premise

That a stationary solution of the Fokker-Planck equation under the paper's simple and physically motivated assumptions supplies the correct effective radial force in real galaxies.

Editorial extensions

If this is right

The FPE model fits rotation curves with quality comparable to or better than NFW, Burkert, and ISO profiles.
Stellar mass-to-light ratios remain inside physically consistent ranges, unlike many NFW and Burkert fits that reach prior boundaries.
The model's characteristic parameter exhibits strong correlations with flat rotation velocity and infrared luminosity.
These correlations reproduce well-known empirical scaling laws such as the Tully-Fisher relation.

Reading between the lines

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

If the entropic interpretation is correct, the same Fokker-Planck derivation could be applied to other dynamical discrepancies in astrophysics to check for similar emergent forces.
The built-in scaling relations suggest that the model's parameter may serve as a direct predictor of observable galaxy traits in larger surveys.
Testing whether the derived force law remains consistent when applied to non-rotating systems such as galaxy clusters would provide an independent check.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that this Fokker-Planck entropic force approach reproduces SPARC rotation curves at a level comparable to NFW, Burkert, and pseudo-isothermal profiles while keeping stellar mass-to-light ratios in sensible ranges. At the same time, the reported correlations between the model's characteristic parameter and flat rotation velocity or infrared luminosity are probably direct consequences of fitting that single parameter to the data rather than independent predictions. The paper takes a minimal statistical framework, derives an effective radial force from the stationary Fokker-Planck equation, and tests it against the SPARC sample. It does a systematic comparison to the usual halo models and notes that its fits do not require pushing M/L to extreme values. That systematic part is the concrete contribution here. The soft spot sits in the interpretation of the scaling relations. With one parameter per galaxy whose value sets the force law and thus the asymptotic velocity, a correlation with v_flat follows algebraically once you fit. Linking it to luminosity works the same way if the parameter varies freely. The abstract presents these as the model naturally capturing key dynamics, but that claim rests on the parameter having some prior connection to baryonic properties that is not shown. The stress-test concern lands on the evidence given. The derivation assumptions are described as simple and motivated, but without the explicit steps it is difficult to assess how sensitive the force law is to those choices. The work engages with the existing rotation curve literature in a direct way. This is aimed at researchers exploring statistical mechanics or entropic alternatives to dark matter for galactic scales. Someone already following modified Newtonian dynamics or similar ideas could find the comparison useful to see how this variant performs. I recommend sending the paper to peer review. The data comparison is solid enough to merit referee input on whether the fits hold up under scrutiny and whether the circularity issue can be addressed with additional tests, such as predicting the parameter from other observables.

Referee Report

3 major / 2 minor

Summary. The manuscript derives an effective radial force from a stationary solution of the Fokker-Planck equation under minimal statistical assumptions and interprets it as an entropic force explaining galactic rotation curves. It fits this one-parameter FPE model to SPARC rotation curves, compares statistical quality and stellar mass-to-light ratios against NFW, Burkert, and ISO profiles, and reports that the fitted characteristic parameter correlates strongly with flat rotation velocity and infrared luminosity in a manner consistent with the Tully-Fisher relation.

Significance. If the derivation is robust and the reported correlations prove independent of the fitting procedure, the work would offer a statistically grounded alternative perspective on rotation curves that simultaneously encodes empirical scaling laws without invoking dark-matter halos. The systematic comparison to standard profiles on a high-quality sample is a positive feature, but the single free parameter per galaxy limits the predictive power beyond curve fitting.

major comments (3)

[§5] §5 (correlations with global properties): the claim that the FPE model 'naturally gives rise to strong correlations' between its characteristic parameter and v_flat is at least partly by construction. Because the parameter is fitted directly to each galaxy's rotation-curve data and v_flat is an algebraic function of that same parameter in the stationary FP solution, the reported correlation cannot be presented as an independent prediction of the framework.
[§4] §4 (model fits and M/L ratios): the abstract states that FPE yields stellar mass-to-light ratios 'within physically consistent ranges' while NFW and Burkert often approach prior limits, yet the manuscript provides no explicit description of the fitting algorithm, covariance treatment, or data-selection criteria. Without these, it is impossible to verify whether the improved M/L behavior is robust or sensitive to post-hoc choices.
[§2] §2 (derivation of the force law): the stationary Fokker-Planck solution introduces a single characteristic parameter whose value is left free and fitted per galaxy. The subsequent claim that the model 'captures key aspects of the underlying dynamics' therefore rests on showing that this parameter can be predicted a priori from baryonic properties rather than determined after the fact from the rotation curve itself.

minor comments (2)

[Abstract, §1] The abstract and §1 should clarify whether the FPE force law reduces to Newtonian gravity at small radii or requires an additional matching condition.
[Figure captions] Figure captions for the rotation-curve panels should explicitly state the number of free parameters used for each model (FPE: 1; NFW: 2, etc.) to allow direct comparison of fit quality.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful and constructive comments. We address each major comment point by point below.

read point-by-point responses

Referee: [§5] §5 (correlations with global properties): the claim that the FPE model 'naturally gives rise to strong correlations' between its characteristic parameter and v_flat is at least partly by construction. Because the parameter is fitted directly to each galaxy's rotation-curve data and v_flat is an algebraic function of that same parameter in the stationary FP solution, the reported correlation cannot be presented as an independent prediction of the framework.

Authors: We agree that the correlation with v_flat follows directly from the algebraic relation in the stationary solution and is therefore partly by construction. We will revise §5 to clarify this distinction and to emphasize that the independent result is the correlation of the fitted parameter with infrared luminosity, which aligns with the Tully-Fisher relation and is not dictated by the model definition. revision: partial
Referee: [§4] §4 (model fits and M/L ratios): the abstract states that FPE yields stellar mass-to-light ratios 'within physically consistent ranges' while NFW and Burkert often approach prior limits, yet the manuscript provides no explicit description of the fitting algorithm, covariance treatment, or data-selection criteria. Without these, it is impossible to verify whether the improved M/L behavior is robust or sensitive to post-hoc choices.

Authors: We acknowledge the absence of these methodological details. We will add a new subsection to §4 (or an appendix) that explicitly describes the MCMC fitting algorithm, covariance treatment of the SPARC rotation-curve uncertainties, priors on the stellar mass-to-light ratios, and the galaxy selection criteria applied to the sample. revision: yes
Referee: [§2] §2 (derivation of the force law): the stationary Fokker-Planck solution introduces a single characteristic parameter whose value is left free and fitted per galaxy. The subsequent claim that the model 'captures key aspects of the underlying dynamics' therefore rests on showing that this parameter can be predicted a priori from baryonic properties rather than determined after the fact from the rotation curve itself.

Authors: The referee correctly observes that the parameter is fitted to the data. While we cannot provide an a priori prediction from baryonic properties in the present work, the post-fit correlations supply empirical support for the claim. We will revise the abstract and §2 to adopt more cautious wording, stating that the correlations suggest the framework captures key aspects rather than asserting this definitively. revision: partial

Circularity Check

1 steps flagged · score 6.0 of 10

Fitted FPE parameter correlates with v_flat by algebraic construction from the 1-parameter force law

fitted input called prediction [abstract (and results/discussion of correlations)]
"Beyond reproducing rotation curves, the model naturally gives rise to strong correlations between its characteristic parameter and global galaxy properties, including the flat rotation velocity and infrared luminosity. These relations are consistent with well-known empirical scaling laws such as the Tully-Fisher relation"

The characteristic parameter is the sole free parameter in the stationary FP-derived force law; fitting it to a galaxy's rotation curve data necessarily determines (or is algebraically tied to) the asymptotic flat velocity v_flat extracted from the same data. Reporting the correlation as an emergent prediction is therefore forced by the 1-parameter construction rather than an independent test.

full rationale

The paper derives a one-parameter effective force from the stationary Fokker-Planck solution, fits that single parameter to each SPARC rotation curve (which already encodes the observed asymptotic v_flat), and then presents the resulting correlation between the fitted parameter and v_flat (plus L_IR) as an independent 'natural' outcome consistent with Tully-Fisher. This reduces the headline claim of capturing scaling relations beyond curve fitting to a direct algebraic consequence of the model definition and the fitting procedure. The derivation chain itself is not circular, but the load-bearing 'beyond reproducing rotation curves' assertion is.

Assumptions & free parameters 1 free parameters · 1 assumptions · 0 invented entities

The central claim rests on one fitted parameter per galaxy and the assumption that a stationary Fokker-Planck solution under simple assumptions yields the effective force; no new particles or dimensions are postulated.

free parameters (1)

characteristic parameter of the FPE model
Fitted individually to each galaxy's rotation curve to reproduce the observed velocities.

assumptions (1)

domain assumption A stationary solution of the Fokker-Planck equation under simple and physically motivated assumptions produces an effective radial force that can be interpreted as entropic.
Invoked in the abstract as the starting point for deriving the force used in all subsequent fits.

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Cite this review

Pith. "Pith review of Fokker-Planck entropic force interpretation of galactic rotation curves." pith.science (2026). https://pith.science/paper/2201.05594

@misc{pith2026220105594,
  author       = {Pith},
  title        = {Pith review of: Fokker-Planck entropic force interpretation of galactic rotation curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2201.05594}},
  note         = {Machine review of arXiv:2201.05594}
}

read the original abstract

We investigate whether the discrepancy between observed galactic rotation curves and those predicted from baryonic matter can be interpreted as the manifestation of an emergent entropic force. Starting from a minimal statistical framework, we derive an effective radial force from a stationary solution of the Fokker-Planck equation under simple and physically motivated assumptions. We confront this Fokker-Planck entropic (FPE) model with high-quality rotation curves from the SPARC database, performing a systematic comparison with standard halo profiles, including Navarro-Frenk-White (NFW), Burkert, and pseudo-isothermal (ISO) models. The FPE model provides fits of comparable or improved statistical quality than traditional profiles, while yielding stellar mass-to-light ratios within physically consistent ranges, in contrast to NFW and Burkert fits that often approach prior limits. Beyond reproducing rotation curves, the model naturally gives rise to strong correlations between its characteristic parameter and global galaxy properties, including the flat rotation velocity and infrared luminosity. These relations are consistent with well-known empirical scaling laws such as the Tully-Fisher relation, suggesting that the proposed framework captures key aspects of the underlying dynamics. Our results indicate that a minimal entropic-force description, grounded in statistical mechanics, can account for galactic rotation curves while simultaneously encoding their scaling relations, offering a complementary and physically motivated perspective to standard dark matter halo interpretations of galaxy rotation curves.

Figures

Figures reproduced from arXiv: 2201.05594 by the authors.

**Figure 2.** FIG. 2. NGC3198 rotation curve (blue points). Also shown is [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Fit of the rotation curve of galaxy NGC3198 using [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: FIG. 5. CDF of the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 4.** Figure 4: FIG. 4. The distribution of Υ corresponding to the least [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 6.** Figure 6: FIG. 6. Behaviour and fit of [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

stationary solution of the Fokker-Planck equation under simple and physically motivated assumptions... μr = k/r² ... P(r) ∝ exp(−k/(ε r))−1 ... Fr(r) = (k/ε)/(β r²)[exp(k/(ε r))−1]
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

entropic force FE = T ∂S/∂r ... Boltzmann type of entropy S(r) = −kB ln P(r)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.

[1] [1]

R. H. Sanders, The Dark Matter Problem: A Historical Perspective. Cambridge University Press, 2010

work page 2010

[2] [2]

Dark matter detection,

L. Baudis, “Dark matter detection,” Journal of Physics G: Nuclear and Particle Physics, vol. 43, no. 4, p. 044001, 2016

work page 2016

[3] [3]

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work page 1970

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work page 1996

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F. Lelli, S. S. McGaugh, and J. M. Schombert, “SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photom- etry and Accurate Rotation Curves,” Astron. J., vol. 152, p. 157, 2016

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T. Bernal, L. M. Fern´ andez-Hern´ andez, T. Matos, and M. A. Rodr´ ıguez-Meza, “Rotation curves of high- resolution LSB and SPARC galaxies with fuzzy and mul- tistate (ultralight boson) scalar ﬁeld dark matter,” Mon. Not. Roy. Astron. Soc., vol. 475, no. 2, pp. 1447–1468, 2018

work page 2018

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Mul- tistate scalar ﬁeld dark matter and its correlation with galactic properties,

A. Hern´ andez-Almada and M. A. Garc´ ıa-Aspeitia, “Mul- tistate scalar ﬁeld dark matter and its correlation with galactic properties,” Int. J. Mod. Phys. D, vol. 27, no. 03, p. 1850031, 2017

work page 2017