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arxiv: 2606.17179 · v2 · pith:NGYLIHQInew · submitted 2026-06-15 · ❄️ cond-mat.stat-mech

Why dimensional analysis works: general classification of self-similarity based on scale-invariance

Hirokazu Maruoka This is my paper

Pith reviewed 2026-06-27 02:26 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords self-similarityscale invariancedimensional analysissimilarity of the first kindsimilarity of the second kindself-similar solutionsphysical parameters
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The pith

Dimensional analysis works because scale invariance is partially shared between units and physical parameters.

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

The paper formulates self-similarity as a function transformed into a form invariant under scale transformations. When this formulation is applied to physical parameters composed of numerical values and units, it shows that dimensional analysis succeeds because the scale invariance is only partially shared between the units and the parameters themselves. This partial sharing creates a clear distinction between similarity of the first kind, where the scale functions match exactly, and similarity of the second kind, where they do not. The second kind is further divided according to whether the power exponents depend on functions of dimensionless numbers, producing an overall classification into three kinds of self-similar solutions.

Core claim

A self-similar form is understood as the transformation of a function into a form invariant under scale transformations. By applying this formulation to physical parameters, which consist of numerical values and units, it is demonstrated that dimensional analysis works for physical problems because scale invariance is partially shared between units and physical parameters. This naturally leads to the distinction between similarity of the first kind and similarity of the second kind according to whether the scale functions induced by units and those associated with physical parameters are equivalent or not. Self-similar solutions of the second kind can be further classified according to wheth

What carries the argument

Decomposition of physical parameters into numerical values and units, with separate application of scale-invariance transformations to each component for comparison of equivalence.

If this is right

  • Self-similar solutions fall into exactly three kinds based on the equivalence or nonequivalence of scale functions and the dependence of exponents on dimensionless numbers.
  • Similarity of the first kind occurs when units and parameters induce fully equivalent scale functions.
  • Similarity of the second kind occurs when the scale functions differ, and it splits further when exponents become functions of dimensionless numbers.
  • Dimensional analysis applies precisely in the cases where partial scale invariance is shared between units and parameters.

Where Pith is reading between the lines

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

  • The three-kind classification could be used to decide in advance whether a given nonlinear equation will admit a simple power-law similarity solution or require a more involved form.
  • The framework suggests that problems lacking dimensional homogeneity might still possess self-similarity of the second kind if the mismatch in scale functions can be tracked explicitly.
  • One could test the classification by re-deriving known exact solutions from fluid mechanics or heat transfer and checking which of the three kinds each belongs to.

Load-bearing premise

Physical parameters can be decomposed into numerical values and units such that scale-invariance transformations can be applied separately to each component and then compared for equivalence.

What would settle it

A physical system in which dimensional analysis produces correct scaling relations but the scale functions induced by units and parameters are neither equivalent nor non-equivalent in the stated manner, or a documented self-similar solution that falls outside all three classified kinds.

read the original abstract

In this work, we formulate self-similarity from the perspective of scale invariance, where a self-similar form is understood as the transformation of a function into a form invariant under scale transformations. By applying this formulation to physical parameters, which consist of numerical values and units, it is demonstrated that dimensional analysis works for physical problems because scale invariance is partially shared between units and physical parameters. This naturally leads to the distinction between similarity of the first kind and similarity of the second kind according to whether the scale functions induced by units and those associated with physical parameters are equivalent or not. Self-similar solutions of the second kind can be further classified according to whether the power exponents of the similarity parameters include functions of dimensionless numbers. This leads to the conclusion that there are three kinds of self-similar solutions. The present work provides a unified framework for understanding dimensional analysis and a universal classification of self-similarity in physical problems.

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 / 0 minor

Summary. The paper formulates self-similarity via scale invariance, where a self-similar form is a function invariant under scale transformations. By decomposing physical parameters into numerical values and units and applying scale-invariance transformations separately, it argues that dimensional analysis succeeds due to partial sharing of scale invariance between units and parameters. This yields a distinction between similarity of the first kind (equivalent scale functions) and second kind (non-equivalent), with the second kind further split according to whether similarity exponents depend on dimensionless numbers, producing a three-kind classification of self-similar solutions.

Significance. If the classification is shown to be non-tautological and independent of the introduced definitions, the framework could supply a unified conceptual basis for why dimensional analysis works and for organizing self-similar solutions across physical problems. The approach builds on existing distinctions (first vs. second kind) but attempts to ground them directly in separate scale transformations on units versus parameters.

major comments (1)
  1. [Abstract] Abstract, paragraph beginning 'By applying this formulation to physical parameters': the three-kind taxonomy is generated directly by the definitions of 'scale functions induced by units' and 'scale functions associated with physical parameters.' Without an independent benchmark, external validation against known solutions, or a demonstration that the classification can fail or be falsified, the scheme risks being tautological by construction rather than a derived result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comment point by point below, providing what we believe is a substantive defense of the framework while agreeing to strengthen the presentation with additional validation.

read point-by-point responses

  1. Referee: [Abstract] Abstract, paragraph beginning 'By applying this formulation to physical parameters': the three-kind taxonomy is generated directly by the definitions of 'scale functions induced by units' and 'scale functions associated with physical parameters.' Without an independent benchmark, external validation against known solutions, or a demonstration that the classification can fail or be falsified, the scheme risks being tautological by construction rather than a derived result.

    Authors: We respectfully disagree that the classification is tautological. The scale functions induced by units follow strictly from the dimensional structure of each quantity (i.e., how its units transform under a change of measurement scale), which is fixed once the base dimensions are chosen. By contrast, the scale functions associated with physical parameters are determined by the invariance properties required by the governing equations, initial/boundary conditions, or conservation laws of the specific problem; these are independent of the unit system. Whether the two sets of functions coincide is therefore a non-trivial physical condition that holds precisely when dimensional analysis alone suffices. The further split of the second kind follows from the explicit functional form of the parameter-induced scaling (constant exponents versus exponents that depend on dimensionless groups), which corresponds to distinct solution behaviors documented in the literature. To address the request for external validation, we will add a dedicated section applying the classification to canonical examples (Sedov-Taylor blast wave for first kind; certain nonlinear diffusion problems for second kind with constant exponents; and problems with parameter-dependent exponents) and show consistency with known analytic results. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines scale functions from units and from physical parameters, then classifies self-similarity into three kinds according to whether those functions are equivalent and whether exponents depend on dimensionless numbers. This taxonomy is explicitly constructed from the introduced definitions and the decomposition of parameters into numerical values plus units; it does not reduce an independent prediction or theorem to a fit or self-citation. No load-bearing self-citations, ansatzes smuggled via prior work, or renaming of known results appear in the derivation chain. The framework is therefore self-contained as a conceptual reorganization rather than a circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on two domain assumptions about how physical quantities are decomposed and how scale transformations act on them; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Physical parameters consist of numerical values and units.
    Explicitly invoked as the starting point for applying scale invariance.
  • domain assumption Scale invariance can be formulated separately for units and for the functional dependence on physical parameters.
    Required to define 'partially shared' invariance and the first/second-kind distinction.

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discussion (0)

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Reference graph

Works this paper leans on

51 extracted references · 27 canonical work pages · 1 internal anchor

  1. [1]

    Galilei,Dialogues Concerning Two New Sciences(Elsevier, Leiden, 1638)

    G. Galilei,Dialogues Concerning Two New Sciences(Elsevier, Leiden, 1638)

  2. [2]

    Maxwell,A Treatise on Electricity and Magnetism(Clarendon Press, Oxford, 1873)

    J.C. Maxwell,A Treatise on Electricity and Magnetism(Clarendon Press, Oxford, 1873). P. 4

  3. [3]

    Rayleigh, The principle of similitude

    L. Rayleigh, The principle of similitude. Nature95(2368), 66–68 (1915). https: //doi.org/10.1038/095066c0. URL https://doi.org/10.1038/095066c0

    work page doi:10.1038/095066c0 1915

  4. [4]

    Mandelbrot,The Fractal Geometry of Nature(Macmillan, New York, 1983)

    B.B. Mandelbrot,The Fractal Geometry of Nature(Macmillan, New York, 1983)

    1983

  5. [5]

    Barenblatt,Scaling, Self-Similarity, and Intermediate Asymptotics

    G.I. Barenblatt,Scaling, Self-Similarity, and Intermediate Asymptotics. Cam- bridge Texts in Applied Mathematics (Cambridge University Press, 1996). https: //doi.org/10.1017/CBO9781107050242

    work page doi:10.1017/cbo9781107050242 1996

  6. [6]

    Barenblatt,Scaling

    G.I. Barenblatt,Scaling. Cambridge Texts in Applied Mathematics (Cambridge University Press, 2003). https://doi.org/10.1017/CBO9780511814921

    work page doi:10.1017/cbo9780511814921 2003

  7. [7]

    Lectures on Phase T ransitions and the Renormalization Group; Frontiers in Physics; Westview Press: Boca Raton, FL, USA, 1992; V olume 85

    N. Goldenfeld,Lectures on phase transitions and the renormalization group(CRC Press, 2018). https://doi.org/https://doi.org/10.1201/9780429493492

    work page doi:10.1201/9780429493492 2018

  8. [8]

    B¨ aumchen, M

    O. B¨ aumchen, M. Benzaquen, T. Salez, J.D. McGraw, M. Backholm, P. Fowler, E. Rapha¨ el, K. Dalnoki-Veress, Relaxation and intermediate asymptotics of a rectangular trench in a viscous film. Phys. Rev. E88, 035001 (2013). https: //doi.org/10.1103/PhysRevE.88.035001. URL https://link.aps.org/doi/10.1103/ PhysRevE.88.035001

    work page doi:10.1103/physreve.88.035001 2013

  9. [9]

    Benzaquen, T

    M. Benzaquen, T. Salez, E. Rapha¨ el, Intermediate asymptotics of the capillary- driven thin-film equation. The European Physical Journal E36(8), 82 (2013). https://doi.org/10.1140/epje/i2013-13082-3. URL https://doi.org/10. 1140/epje/i2013-13082-3 27

    work page doi:10.1140/epje/i2013-13082-3 2013

  10. [10]

    Boakye-Ansah, P

    Y.A. Boakye-Ansah, P. Grassia, Similarity solutions for early-time constant boundary flux imbibition in foams and soils. Eur. Phys. J. E

  11. [11]

    Z. Zhou, X. Wang, H. Li, J. Schulze, P. Hartmann, Z. Donk´ o, Y. Liu, Y. Fu, Similarity scaling for low-pressure capacitive radio-frequency cf 4 plasmas across operation modes. Phys. Rev. E113, 025202 (2026). https://doi.org/10.1103/ 7z84-ccpw. URL https://link.aps.org/doi/10.1103/7z84-ccpw

    work page doi:10.1103/7z84-ccpw 2026

  12. [12]

    Cabella, A.S

    B.C.T. Cabella, A.S. Martinez, F. Ribeiro, Data collapse, scaling functions, and analytical solutions of generalized growth models. Phys. Rev. E83, 061902 (2011). https://doi.org/10.1103/PhysRevE.83.061902. URL https://link.aps.org/doi/10. 1103/PhysRevE.83.061902

    work page doi:10.1103/physreve.83.061902 2011

  13. [13]

    Yokota, K

    M. Yokota, K. Okumura, Dimensional crossover in the coalescence dynam- ics of viscous drops confined in between two plates. Proceedings of the National Academy of Sciences108(16), 6395–6398 (2011). https://doi.org/ 10.1073/pnas.1017112108. URL https://www.pnas.org/doi/abs/10.1073/pnas. 1017112108. https://www.pnas.org/doi/pdf/10.1073/pnas.1017112108

    work page doi:10.1073/pnas.1017112108 2011

  14. [14]

    Murano, K

    M. Murano, K. Okumura, Rising bubble in a cell with a high aspect ratio cross- section filled with a viscous fluid and its connection to viscous fingering. Phys. Rev. Res.2, 013188 (2020). https://doi.org/10.1103/PhysRevResearch.2.013188. URL https://link.aps.org/doi/10.1103/PhysRevResearch.2.013188

    work page doi:10.1103/physrevresearch.2.013188 2020

  15. [15]

    Dresselhaus, B

    E.J. Dresselhaus, B. Sbierski, I.A. Gruzberg, Scaling collapse of longitudinal conductance near the integer quantum hall transition. Phys. Rev. Lett.129, 026801 (2022). https://doi.org/10.1103/PhysRevLett.129.026801. URL https: //link.aps.org/doi/10.1103/PhysRevLett.129.026801

    work page doi:10.1103/physrevlett.129.026801 2022

  16. [16]

    Watanabe, T

    R. Watanabe, T. Ishii, Y. Hirono, H. Maruoka, Physical Review E111, 024301 (2025)

    2025

  17. [17]

    Bertrand, Sur l’homog´ en´ eit´ e dans les formules de physique

    J. Bertrand, Sur l’homog´ en´ eit´ e dans les formules de physique. Comptes Rendus 86(15), 916–920 (1878)

  18. [18]

    Riabouchinsky, M´ ethode des variables de dimension z´ ero et son application en a´ erodynamique

    D. Riabouchinsky, M´ ethode des variables de dimension z´ ero et son application en a´ erodynamique. L’A´ erophile pp. 407–408 (1911)

    1911

  19. [19]

    Federman, On some general methods of integration of first-order partial differ- ential equations

    A. Federman, On some general methods of integration of first-order partial differ- ential equations. Proceedings of the Saint Petersburg Polytechnic Institute16(1), 97–155 (1911). Originally published in Russian

    1911

  20. [20]

    Buckingham, On physically similar systems; illustrations of the use of dimensional equations

    E. Buckingham, On physically similar systems; illustrations of the use of dimensional equations. Physical Review4(4), 345–376 (1914)

    1914

  21. [21]

    Gibbings,Dimensional Analysis, 1st edn

    J.C. Gibbings,Dimensional Analysis, 1st edn. (Springer London, London, 2011). https://doi.org/10.1007/978-1-84996-317-6. URL https://doi.org/10. 28 1007/978-1-84996-317-6

    work page doi:10.1007/978-1-84996-317-6 2011

  22. [22]

    Taylor, Proceedings of the Royal Society of London A201, 159–174, 175–186 (1950)

    G.I. Taylor, Proceedings of the Royal Society of London A201, 159–174, 175–186 (1950)

    1950

  23. [23]

    Diaz, European Journal of Physics42, 035803 (2021)

    S. Diaz, European Journal of Physics42, 035803 (2021)

    2021

  24. [24]

    Widom, Journal of Chemical Physics43, 3898 (1963)

    B. Widom, Journal of Chemical Physics43, 3898 (1963)

    1963

  25. [25]

    Holmes, L.G.V

    L.M. Holmes, L.G.V. Uitert, G.W. Hull, Magnetoelectric effect and critical behav- ior in the ising-like antiferromagnet fecl 2. Solid State Communications9(16), 1373–1376 (1971). https://doi.org/10.1016/0038-1098(71)90398-X

    work page doi:10.1016/0038-1098(71)90398-x 1971

  26. [26]

    Franke, J

    M. Ley-Koo, M.S. Green, Revised and extended scaling for coexisting densities of sf6. Phys. Rev. A16, 2483–2487 (1977). https://doi.org/10.1103/PhysRevA. 16.2483. URL https://link.aps.org/doi/10.1103/PhysRevA.16.2483

    work page doi:10.1103/physreva 1977

  27. [27]

    de Gennes, F

    P.G. de Gennes, F. Brochard-Wyart, D. Qu´ er´ e,Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves, 1st edn. (Springer, New York, 2004). https://doi.org/10.1007/978-0-387-21656-0. URL https://doi.org/10. 1007/978-0-387-21656-0

    work page doi:10.1007/978-0-387-21656-0 2004

  28. [28]

    Barenblatt,Flow, Deformation and Fracture

    G.I. Barenblatt,Flow, Deformation and Fracture. Cambridge Texts in Applied Mathematics (Cambridge University Press, 2014). https://doi.org/10.1017/ CBO9781139030014

    2014

  29. [29]

    Bandi, V.S

    M.M. Bandi, V.S. Akella, D.K. Singh, R.S. Singh, S. Mandre, Hydrodynamic sig- natures of stationary marangoni-driven surfactant transport. Phys. Rev. Lett. 119, 264501 (2017). https://doi.org/10.1103/PhysRevLett.119.264501. URL https://link.aps.org/doi/10.1103/PhysRevLett.119.264501

    work page doi:10.1103/physrevlett.119.264501 2017

  30. [30]

    Flory,Principles of Polymer Chemistry(Cornell University Press, Ithaca, NY, 1953)

    P.J. Flory,Principles of Polymer Chemistry(Cornell University Press, Ithaca, NY, 1953)

    1953

  31. [31]

    de Gennes,Scaling concepts in polymer physics(Cornell University Press, New York, 1979)

    P.G. de Gennes,Scaling concepts in polymer physics(Cornell University Press, New York, 1979). https://doi.org/10.1002/actp.1981.010320517

    work page doi:10.1002/actp.1981.010320517 1979

  32. [32]

    Maruoka, H

    H. Maruoka, H. Hayakawa, Early-stage impact dynamics in dense suspensions of millimeter-sized particles. Physics of Fluids37(11), 113335 (2025). https: //doi.org/10.1063/5.0278768. URL https://doi.org/10.1063/5.0278768

    work page doi:10.1063/5.0278768 2025

  33. [33]

    Maruoka, A framework for crossover of scaling law as a self-similar solution: dynamical impact of viscoelastic board

    H. Maruoka, A framework for crossover of scaling law as a self-similar solution: dynamical impact of viscoelastic board. The European Physical Journal E46(5), 35 (2023). https://doi.org/10.1140/epje/s10189-023-00292-9

    work page doi:10.1140/epje/s10189-023-00292-9 2023

  34. [34]

    Yokoyama, H

    Y. Yokoyama, H. Maruoka, K. Matsunuma, Y. Tagawa, Scaling crossover in droplet impact force on elastic substrates. Nature Communications17, 607 (2026). https://doi.org/10.1038/s41467-025-67790-6 29

    work page doi:10.1038/s41467-025-67790-6 2026

  35. [35]

    West, J.H

    G.B. West, J.H. Brown, B.J. Enquist, A general model for the origin of allometric scaling laws in biology. Science276(5309), 122–126 (1997). https://doi.org/10. 1126/science.276.5309.122

    1997

  36. [36]

    El-Nabulsi, W

    R.A. El-Nabulsi, W. Anukool, Foam drainage equation in fractal dimensions: breaking and instabilities. Eur. Phys. J. E46(11), 110 (2023). https:// doi.org/10.1140/epje/s10189-023-00368-6. URL https://doi.org/10.1140/epje/ s10189-023-00368-6

    work page doi:10.1140/epje/s10189-023-00368-6 2023

  37. [37]

    Lakhal, L

    S. Lakhal, L. Ponson, M. Benzaquen, J.P. Bouchaud, Wrapping and unwrapping multifractal fields: Application to fatigue and abrupt failure fracture surfaces. Phys. Rev. Res.7, L012003 (2025). https://doi.org/10.1103/PhysRevResearch. 7.L012003. URL https://link.aps.org/doi/10.1103/PhysRevResearch.7.L012003

    work page doi:10.1103/physrevresearch 2025

  38. [38]

    Lakhal, J

    S.E. Lakhal, J. Sardonia, M. Bandi, Collective and nonlinear structure of wind- power correlations. PRX Energy5, 023007 (2026). https://doi.org/10.1103/ vms3-ng8z. URL https://link.aps.org/doi/10.1103/vms3-ng8z

    work page doi:10.1103/vms3-ng8z 2026

  39. [39]

    Scaling Laws for Neural Language Models

    J. Kaplan, S. McCandlish, T. Henighan, T.B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu, D. Amodei, Scaling laws for neural language models. arXiv preprint arXiv:2001.08361 (2020). https://doi.org/10.48550/arXiv.2001.08361. arXiv:2001.08361 [cs.LG]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2001.08361 2001

  40. [40]

    Guderley, Starke kugelige und zylindrische verdichtungsst¨ oße in der n¨ ahe des kugelmittelpunktes bzw

    K.G. Guderley, Starke kugelige und zylindrische verdichtungsst¨ oße in der n¨ ahe des kugelmittelpunktes bzw. der zylinderachse. Luftfahrtforschung19, 302–312 (1942)

    1942

  41. [41]

    von Weizs¨ acker, Approximate representation of strong unsteady shock waves through homology solutions

    C.F. von Weizs¨ acker, Approximate representation of strong unsteady shock waves through homology solutions. Zeitschrift f¨ ur Naturforschung A9(4), 269–275 (1954)

    1954

  42. [42]

    Zel’dovich, The motion of a gas under the action of a short-term pressure shock

    Y.B. Zel’dovich, The motion of a gas under the action of a short-term pressure shock. Akusticheskii Zhurnal2(1), 28–38 (1956). In Russian

    1956

  43. [43]

    116.233901

    G.I. Barenblatt, Y.B. Zel’dovich, Self-similar solutions as intermediate asymp- totics. J. Sci. Comput.4, 285–312 (1972). https://doi.org/10.1103/PhysRevLett. 129.026801

    work page doi:10.1103/physrevlett 1972

  44. [44]

    Eggers, M.A

    J. Eggers, M.A. Fontelos,Singularities: Formation, Structure, and Propagation (Cambridge University Press, 2015)

    2015

  45. [45]

    Bridgman,Dimensional Analysis(Yale University Press, New Haven, 1922)

    P.W. Bridgman,Dimensional Analysis(Yale University Press, New Haven, 1922)

    1922

  46. [46]

    Ipsen,Units, Dimensions and Dimensionless Numbers(McGraw-Hill Book Company, New York, 1960) 30

    D.C. Ipsen,Units, Dimensions and Dimensionless Numbers(McGraw-Hill Book Company, New York, 1960) 30

    1960

  47. [47]

    Olver,Applications of Lie Groups to Differential Equations 2nd ed.1 (Springer, 1993)

    P.J. Olver,Applications of Lie Groups to Differential Equations 2nd ed.1 (Springer, 1993). https://doi.org/https://doi.org/10.1007/978-1-4684-0274-2

    work page doi:10.1007/978-1-4684-0274-2 1993

  48. [48]

    F.W. Hehl, C. L¨ ammerzahl, Physical dimensions/units and universal con- stants: Their invariance in special and general relativity. Annalen der Physik531(5), 1800407 (2019). https://doi.org/https://doi.org/10.1002/ andp.201800407. URL https://onlinelibrary.wiley.com/doi/abs/10.1002/andp. 201800407. https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.201800407

    work page doi:10.1002/andp 2019

  49. [49]

    Atkins, J

    P. Atkins, J. de Paula,Atkins’ Physical Chemistry, 8th edn. (W. H. Freeman and Company, New York, 2006)

    2006

  50. [50]

    L.Y. Chen, N. Goldenfeld, Y. Oono, Physical Review E54, 376 (1996)

    1996

  51. [51]

    Kunihiro, Y

    T. Kunihiro, Y. Kikuchi, K. Tsumura,Geometrical Formulation of Renormalization-Group Method as an Asymptotic Analysis(Springer, 2022) 31

    2022