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REVIEW 3 major objections 4 minor 54 references

Exploring the energy landscape of the Thomson problem: local minima and stationary states

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims the number of distinct local minima in the Thomson problem grows exponentially with N faster than previous estimates, and that stationary states grow even faster, roughly as e^{0.52 N}.

desk verdict Bigger enumerations, same old completeness problem: the new counts are valuable, but the exponential growth rates and extrapolations rest on incomplete searches and a too-coarse energy filter. read the letter →

arxiv 2506.08398 v1 pith:JU7BSNUE submitted 2025-06-10 cond-mat.soft physics.comp-ph

Paolo Amore , Victor Figueroa , Enrique Diaz , Jorge A. López , Trevor Vincent This is my paper
classification cond-mat.softphysics.comp-ph
keywords ThomsonproblemlocalminimastationarystatesenergylandscapegapsMorseindexCoulombpotentialexponentialgrowth
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

Placing N equal point charges on a sphere and minimizing their Coulomb energy — the Thomson problem — yields many distinct local minima, not just one global optimum. This paper reports a much more extensive numerical census for N ≤ 150 (with a partial run at N = 180) and claims that the number of distinct local minima grows exponentially with N faster than earlier studies suggested, roughly as $N_{\mathrm{conf}}(N)=0.00017\,e^{0.12061 N}$ for $100\le N\le 150$. It also finds that the smallest and average energy gaps between neighboring minima shrink exponentially with N, so the landscape becomes a dense stack of nearly degenerate configurations that are still genuinely different. For stationary states (saddles as well as minima), it introduces a squared-force potential whose zero minima are exactly the stationary configurations, and reports that their number also grows exponentially, roughly as $0.06898\,e^{0.52295 N}$, much faster than the minima. If these counts hold, exhaustive enumeration of the Thomson landscape is out of reach for moderate N, and any claim of having found the global minimum must contend with exponentially many nearly degenerate alternatives.

What carries the argument

The key objects are two. For local minima, the deterministic upgrade and downgrade algorithms: starting from a known minimum with $N-1$ charges, the upgrade step inserts a new charge at a vertex of the Voronoi diagram and minimizes; starting from a minimum with $N+1$ charges, the downgrade step removes one charge and minimizes. These generate ansatzes that stay close to equilibrium and are much cheaper to relax than random initial conditions. For stationary states, the paper defines the squared total tangential-force functional $V^{(s)}=\sum_i |\vec F_i^{(s)}|^2$, whose global minima (at value zero) are exactly the stationary configurations of the Coulomb energy; this converts the search for saddles into a minimization problem. The numerical pipeline combines Newton-conjugate-gradient minimization, refinement of each candidate to very small gradients, and a duplicate filter that compares per-particle energies and distance tables as well as total energies.

What would settle it

Perform a certified exhaustive enumeration of all stationary states for a small fixed N, such as N = 24, using an independent method with rigorous isolation (for example interval-arithmetic Newton or homotopy continuation with certified roots) and compare the total with the fit from Eq. (17); a certified total that differs from the fit by an order of magnitude would show the exponential rate is biased. Alternatively, find, for any N in the fitted range 100 ≤ N ≤ 150, a distinct local minimum whose energy differs from an already listed configuration by less than the $10^{{-8}}$ duplicate-filter tolerance and whose distance-based comparison rejects the merge; that would show the published counts omit real configurations.

Watch

Extended reading notes

Core claim

The central discovery is that the Thomson problem has far more stationary solutions than previously counted. The paper's enumerations give an exponential fit $N_{\mathrm{conf}}(N)=0.00017\,e^{0.12061 N}$ for $100\le N\le 150$, implying about $4.55\times 10^5$ local minima at $N=180$ and about $1.5\times 10^{17}$ at $N=400$; for $N=147$ it finds 8,356 minima, more than the 6,644 previously reported. It further observes that the smallest gap $\Delta_{\min}$ and the average gap $\Delta_{\mathrm{avg}}$ between sorted configurations decay exponentially with $N$, while the total energy span grows only linearly, so the density of distinct states in energy explodes. For stationary states, the number found for $15\le N\le 24$ fits $N_{\mathrm{conf}}\approx 0.06898\,e^{0.52295 N}$, with the counts over Morse indices following a Gaussian-shaped distribution peaking just below $N/2$. The paper presents these results as evidence that the Thomson energy landscape is exponentially more complex than previous fits suggested.

Load-bearing premise

The paper assumes that the number of distinct configurations its heuristic searches happened to find is a reliable proxy for the true number, even though it never completes or certifies the enumeration and explicitly states that the searches at N = 180 and N = 30 were partial.

Editorial extensions

If this is right

  • At $N=180$ the fitted count predicts about $4.55\times10^5$ distinct local minima, far more than the hundreds or thousands previously observed or estimated.
  • At $N=400$ the same fit gives about $1.5\times10^{17}$ minima, which would make exhaustive or near-exhaustive landscape exploration impossible.
  • The smallest and average energy gaps shrink exponentially with $N$ while the energy span grows linearly, so distinct configurations become unresolvable in double precision for large $N$.
  • Stationary states are far more numerous than local minima, growing roughly as $0.06898\,e^{0.52295 N}$, with a Gaussian-shaped index distribution peaking below $N/2$.
  • Configurations with nearly identical energies can still be geometrically very different, so energy differences alone systematically under-resolve the landscape.

Reading between the lines

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

  • The paper explicitly states that the searches at $N=180$ (local minima) and $N=30$ (stationary states) were partial, so the exponential fits are more safely read as estimated lower bounds on the true counts rather than certified totals.
  • Because the duplicate filter uses a total-energy tolerance of $10^{-8}$ and the fitted smallest gaps fall below that tolerance for large $N$, some genuinely distinct configurations may have been merged; applying the paper's distance-based comparison to every near-degenerate pair would quantify the bias.
  • Equation (17) evaluated at $N=30$ gives about $4.5\times10^5$ stationary states, whereas the text reports a lower bound of $4.4\times10^4$; the extrapolated count should be checked before the fit is used beyond its fitted range.
  • The same squared-force potential could be applied to other repulsive interactions, such as logarithmic or general power-law potentials, to test whether the exponential growth of stationary states is a universal feature of spherical repulsive landscapes.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper reports an extensive numerical exploration of the Thomson problem, focusing on two quantities: the number of distinct local minima for N ≤ 150 (with a partial search at N = 180) and the number of stationary configurations for N ≤ 24 (with a partial search at N = 30). The central quantitative claims are that the number of local minima grows as Nconf(N) = 0.00017 e^{0.12061 N} for 100 ≤ N ≤ 150, that the average and smallest energy gaps between minima decay exponentially with N, and that the number of stationary states grows as Nconf ≈ 0.06898 e^{0.52295 N} for 15 ≤ N ≤ 24. The paper also introduces a squared-force functional V(s) whose global minima coincide with stationary points of the Thomson energy, and uses it to enumerate stationary states. Additional results include Burr XII and Weibull fits to energy and gap distributions, Morse-index statistics, and PCA-based visualizations of the landscape.

Significance. If the quantitative claims are fully supported, the paper significantly revises current understanding of the Thomson problem: it would imply a much faster exponential growth of local minima than previous estimates, exponentially shrinking energy gaps, and a new computational tool for enumerating stationary points. The squared-force reformulation is mathematically sound and practically useful, and the observed counts already exceed earlier reported counts, so the qualitative direction of the result is credible. The main value of the paper lies in the new data set and the stationary-point methodology. However, the quantitative growth rates and extrapolations rest on incomplete heuristic enumeration and on a duplicate filter that can merge distinct states, which directly affects the central claims.

major comments (3)
  1. [§3.1, Eq. (11)] The identification of the fitted Nconf(N) with the true number of distinct local minima is not established. The text states that the search for N = 180 is partial (§1) and that for N = 150 the 10^6 random trials produced only about half of the independent configurations identified (§3.1, discussion of Fig. 7). No completeness estimator of the kind used by Calef et al. [37] is applied. Consequently the fitted exponent 0.12061 in Eq. (11) is at best a lower bound for the growth rate of the found set, not a statistically supported estimate of the true growth rate, and the extrapolations Nconf(180) ≈ 4.55 × 10^5 and Nconf(400) ≈ 1.5 × 10^17 carry no error bar. The paper should either present the counts as lower bounds, apply a completeness correction, or explicitly restrict the claim to the observed configurations.
  2. [§2.1 and Fig. 3] The duplicate filter rejects configurations whose energies differ by less than the tolerance, usually 10^-8, because the text says their energies cannot be reliably distinguished. However, the fitted smallest-gap curve in Fig. 3 falls below 10^-8 in the upper part of the fitted range (near N ≈ 145 and beyond), so near-degenerate but geometrically distinct minima may be merged in exactly the regime used for the fit. The additional geometric comparison tests of §2.3 are applied only to selected smallest-gap pairs in Fig. 5, not globally to all near-degenerate candidates. This can systematically undercount Nconf and censor the smallest gaps, biasing both Eq. (11) and Eq. (12). The authors should quantify how many distinct pairs have energy separations below the tolerance and verify them with the geometric comparison, or use a stricter energy tolerance with error-controlled arithmetic.
  3. [§3.2, Eq. (17)] The stationary-state count fit suffers from the same completeness problem as Eq. (11), and the text itself states that the N = 30 exploration was partial. Moreover, the sentence following Eq. (17) says 'Nconf(30) > 4.4 × 10^4, using the fit', but Eq. (17) predicts Nconf(30) ≈ 4.5 × 10^5 and the paper reports having already identified 145,404 configurations at N = 30. The stated inequality is therefore internally inconsistent and the fitted growth rate is not supported without a completeness estimate or a demonstration that the found counts have converged as a function of computational effort.
minor comments (4)
  1. [§3.1, Fig. 7 discussion] The sentence 'the 10^6 configurations shown in Fig. 7 represent only about 50% of the total number of distinct configurations we identified for N = 150' is ambiguous, because Fig. 7 plots all trials without filtering repeated configurations; please clarify what exactly is being compared.
  2. [§2.1] There is a typographical error: 'applyied' should be 'applied' in the description of the Kahan summation algorithm.
  3. [§3.1, Fig. 10] The text refers to 'partial component analysis'; the standard term is 'principal component analysis' (PCA).
  4. [§3.1, data availability] The paper states that the results are available for download at Zenodo but provides no link or identifier; please add the DOI or URL for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the exponential counts are empirical fits and the squared-force functional is an identity; the sole self-citation (ref. [48]) is illustrative and not load-bearing.

full rationale

The paper's central quantitative claims are empirical summaries of its own heuristic searches, not predictions derived from fitted inputs. Equation (11) is introduced as 'the exponential fit obtained using the results for 100 ≤ N ≤ 150', and Eqs. (12) and (17) are likewise fits to measured gaps and found stationary-state counts; the extrapolations to N=180, N=400 and N=30 are clearly projections of those fits, not independent outputs of the fits. The squared-force functional in Eq. (10), V(s) = Σ_i |F_i(s)|^2, is an algebraic identity: Eq. (9) requires each tangent force to vanish, so the nonnegative V(s) is exactly zero at stationary configurations. This is a reformulation presented as a 'specifically designed potential', not a derivation of one quantity from another. The only self-citation, ref. [48], is used to say that a similar PCA analysis 'was carried out by one of us in ref. [48]' and does not support the exponential-growth or stationary-state claims. The manuscript also flags its own limits: 'we have performed a systematic search for N ≤ 150 and partial searches for N = 180' (Section 1), 'The exploration for N = 30 has been partial' (Section 3.2), and the duplicate filter uses an energy tolerance 'usually we have set this tolerance at 10^{-8}' (Section 2.1). These are completeness/precision risks for the extrapolated rates, but they are not circularity: the counts and fits are not constructed to equal a fitted parameter by definition. The honest finding is therefore no significant circularity.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The quantitative claims rest on fitted scaling laws, hand-set search hyperparameters, and unverified completeness of the enumeration. The squared-force identity itself is exact, but using it to count stationary states assumes basin coverage and no spurious local minima.

free parameters (8)
  • Local-minima count fit prefactor A = 0.00017
    Fitted prefactor in Eq. (11); depends on observed incomplete counts for 100<=N<=150.
  • Local-minima count exponent b = 0.12061
    Fitted exponent in Eq. (11); the claim of faster-than-previous growth rests on this rate.
  • Smallest-gap fit amplitude and exponent = 3410, -0.183498
    Eq. (12) fit to observed smallest gaps; gaps near or below the 1e-8 filter tolerance are censored.
  • Average-gap fit amplitude and exponent = 16.615, -0.0782718
    Eq. (12) fit to average gaps of found configurations; a lower bound on true gap density.
  • Stationary-count fit amplitude and exponent = 0.06898, 0.52295
    Eq. (17) fit to observed stationary states for 15<=N<=24; search is not shown complete.
  • Burr XII parameters c, k, lambda = 6.36, 4.39, 0.777
    Eq. (15) fit to N=150 energy distribution; descriptive only.
  • Weibull parameters k, lambda = 0.947, 3.530e-5
    Eq. (16) fit to N=150 energy-gap distribution; descriptive only.
  • Energy filter tolerance = 1e-8
    Hand-set in Section 2.1; determines which configurations are discarded as duplicates, so it shapes the gap statistics and counts.
assumptions (4)
  • domain assumption All stationary points of the Thomson energy are nondegenerate, so Morse index classification by Hessian eigenvalues is valid.
    The paper classifies configurations by Hessian eigenvalues and fits Gaussian index distributions; degeneracies would alter counts and indices. No proof of genericity is given.
  • ad hoc to paper Minimizing the squared-force functional V(s) from random starts reaches every stationary configuration without being trapped by spurious local minima at V>0.
    Section 2.2 constructs V(s) whose zeros are exactly the stationary points, but the completeness of the enumeration depends on the basin structure of V, which is not analyzed.
  • ad hoc to paper The observed counts are a faithful enough sample of the true number of configurations to support exponential fits.
    Eqs. (11) and (17) fit exponential laws to observed counts; the paper does not provide a completeness estimator as in Calef et al. [37].
  • ad hoc to paper The duplicate filter (energy tolerance plus distance tests) never merges two genuinely distinct configurations.
    The tolerance is 1e-8, while Fig. 3 shows smallest gaps can be smaller than this for large N; the paper's additional tests may not catch all cases.

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Cite this review

Pith. "Pith review of Exploring the energy landscape of the Thomson problem: local minima and stationary states." pith.science (2026). https://pith.science/paper/JU7BSNUE

@misc{pith2026250608398,
  author       = {Pith},
  title        = {Pith review of: Exploring the energy landscape of the Thomson problem: local minima and stationary states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JU7BSNUE}},
  note         = {Machine review of arXiv:2506.08398}
}
abstract

We conducted a comprehensive numerical investigation of the energy landscape of the Thomson problem for systems up to $N=150$. Our results show the number of distinct configurations grows exponentially with $N$, but significantly faster than previously reported. Furthermore, we find that the average energy gap between independent configurations at a given $N$ decays exponentially with $N$, dramatically increasing the computational complexity for larger systems. Finally, we developed a novel approach that reformulates the search for stationary points in the Thomson problem (or similar systems) as an equivalent minimization problem using a specifically designed potential. Leveraging this method, we performed a detailed exploration of the solution landscape for $N\leq24$ and estimated the growth of the number of stationary states to be exponential in $N$.

Figures

Figures reproduced from arXiv: 2506.08398 by the authors.

Figure 1
Figure 1. Number of local minima for the Thomson problem. [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Number of local minima for the Thomson problem (detail). [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Smallest energy gap for configurations with 80 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Energy span defined as Emax(N) − Emin(N) for configurations with 80 ≤ N ≤ 150. 80 100 120 140 N 10−3 10−2 10−1 100 101 ∆, dE ∆ dE [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Smallest energy gap for configurations with 80 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]
Figure 6
Figure 6. Figure 6: Energy comparison for almost degenerate configuration [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Histogram for the energies for N = 150 (106 random trials). appear very often, certain energy regions remain unpopulated—even though [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Histogram for the energies of independent configuration [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]
Figure 9
Figure 9. Figure 9: Histogram for the energy gaps (right) for [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: PCA for local minima with 140 charges. In [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Number of stationary configurations for the Thomson p [PITH_FULL_IMAGE:figures/full_fig_p021_11.png]
Figure 12
Figure 12. Figure 12: Number of stationary configurations for the Thomson p [PITH_FULL_IMAGE:figures/full_fig_p021_12.png]
Figure 13
Figure 13. Figure 13: PCA for stationary states with 18 charges. [PITH_FULL_IMAGE:figures/full_fig_p022_13.png]

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Works this paper leans on

54 extracted references · 54 canonical work pages

  1. [37]

    ”Esti- mating the number of stable configurations for the generalized Thomson problem.” Journal of Statistical Physics 160.1 (2015): 239-253

    Calef, Matthew, Whitney Griffiths, and Alexia Schulz. ”Esti- mating the number of stable configurations for the generalized Thomson problem.” Journal of Statistical Physics 160.1 (2015): 239-253

    work page 2015

  2. [1]

    Thomson, J.J., ”On the structure of the atom: an investigation of the stability of the periods of oscillation of a number of cor- puscles arranged at equal intervals around the circumference of a circle with application of the results to the theory of atomic structure”, Philosophical Magazine, Series 6, vol. 7, no. 39, pp. 237–265 (1904)

    work page 1904

  3. [2]

    ”Stabile Anordnungen von Elektronen im Atom.” Journal f¨ ur die reine und angewandte Mathematik 141 (1912): 251-302

    F¨ oppl, L. ”Stabile Anordnungen von Elektronen im Atom.” Journal f¨ ur die reine und angewandte Mathematik 141 (1912): 251-302. 23

    work page 1912

  4. [3]

    ”Stability configurations of electrons on a spher e.” Mathematical Tables and Other Aids to Computation 10.55 (1956): 117-120

    Cohn, Harvey. ”Stability configurations of electrons on a spher e.” Mathematical Tables and Other Aids to Computation 10.55 (1956): 117-120

    work page 1956

  5. [4]

    ”Stability configurations of electrons on a sphere.” Mathematics of Computation 23.108 (1969): 785-786

    Goldberg, Michael. ”Stability configurations of electrons on a sphere.” Mathematics of Computation 23.108 (1969): 785-786

    work page 1969

  6. [5]

    ”On the origin of number and arrangement of the places of exit on the surface of pollen- grains.” Recueil des travaux botaniques n´ eerlandais 27.1 (1930): 1-84

    Tammes, Pieter Merkus Lambertus. ”On the origin of number and arrangement of the places of exit on the surface of pollen- grains.” Recueil des travaux botaniques n´ eerlandais 27.1 (1930): 1-84

    work page 1930

  7. [6]

    ”On the sum of distances determined by a pointset.” Acta Mathematica Hungarica 7.3-4 (1956): 397-401

    T´ oth, L. ”On the sum of distances determined by a pointset.” Acta Mathematica Hungarica 7.3-4 (1956): 397-401

    work page 1956

  8. [7]

    Erber, T., and G. M. Hockney. ”Equilibrium configurations of N equal charges on a sphere.” Journal of Physics A: Mathematical and General 24.23 (1991): L1369

    work page 1991

Show all 54 references
  1. [8]

    method of constrained global optimization

    Erber, T., and G. M. Hockney. ”Comment on “method of constrained global optimization”.” Physical review letters 74.8 (1995): 1482

    1995

  2. [9]

    Erber, T., and G. M. Hockney. ”Complex systems: equilibrium configurations of n equal charges on a sphere (2 < n < 112).” Advances in chemical physics 98 (1997): 495-594

    1997

  3. [10]

    Glasser, L., and A. G. Every. ”Energies and spacings of point charges on a sphere.” Journal of Physics A: Mathematical and General 25.9 (1992): 2473

    1992

  4. [11]

    Rakhmanov, Evguenii A., E. B. Saff, and Y. M. Zhou. ”Electrons on the sphere.” Computational methods and function theory,

  5. [12]

    Boal, and P

    Bergersen, B., D. Boal, and P. Palffy-Muhoray. ”Equilibrium configurations of particles on a sphere: the case of logarithmic interactions.” Journal of Physics A: Mathematical and General 27.7 (1994): 2579

    1994

  6. [13]

    ”The five-electron case of Thomson ’s problem.” Experimental Mathematics 22.2 (2013): 157-186

    Schwartz, Richard Evan. ”The five-electron case of Thomson ’s problem.” Experimental Mathematics 22.2 (2013): 157-186

    2013

  7. [14]

    ”Five point energy minimization: a synopsis.” Constructive Approximation 51.3 (2020): 537-564

    Schwartz, Richard Evan. ”Five point energy minimization: a synopsis.” Constructive Approximation 51.3 (2020): 537-564

    2020

  8. [15]

    Rakhmanov, Evguenii A., E. B. Saff, and Y. M. Zhou. ”Minimal discrete energy on the sphere.” Mathematical Research Letters 1.6 (1994): 647-662. 24

    1994

  9. [16]

    ”Distributing many points on a sphere.” The mathematical intelligencer 19.1 (1997): 5-11

    Saff, Edward B., and Amo BJ Kuijlaars. ”Distributing many points on a sphere.” The mathematical intelligencer 19.1 (1997): 5-11

    1997

  10. [17]

    Hardin, and Edward B

    Borodachov, Sergiy V., Douglas P. Hardin, and Edward B. Saff. ”Discrete energy on rectifiable sets”. Springer New York, 2019

    2019

  11. [18]

    ”Mathematical problems for the next century.” The mathematical intelligencer 20 (1998): 7-15

    Smale, Steve. ”Mathematical problems for the next century.” The mathematical intelligencer 20 (1998): 7-15

    1998

  12. [19]

    ”The state of the art in Smale’s 7th prob- lem.” Foundations of computational mathematics, Budapest 403 (2011): 1-15

    Beltr´ an, Carlos. ”The state of the art in Smale’s 7th prob- lem.” Foundations of computational mathematics, Budapest 403 (2011): 1-15

    2011

  13. [20]

    ”Method of constrained global op- timization.” Physical review letters 72.17 (1994): 2671

    Altschuler, Eric Lewin, et al. ”Method of constrained global op- timization.” Physical review letters 72.17 (1994): 2671

    1994

  14. [21]

    Stillinger, Frank H., and Thomas A. Weber. ”Packing structures and transitions in liquids and solids.” Science 225.4666 (1984): 983-989

    1984

  15. [22]

    Morris, J. R., D. M. Deaven, and K. M. Ho. ”Genetic-algorithm energy minimization for point charges on a sphere.” Physical Review B 53.4 (1996): R1740

    1996

  16. [23]

    ”Possible global minimum lattice configurations for Thomson’s problem of charges on a sphere.” Physical Review Letters 78.14 (1997): 2681

    Altschuler, Eric Lewin, et al. ”Possible global minimum lattice configurations for Thomson’s problem of charges on a sphere.” Physical Review Letters 78.14 (1997): 2681

    1997

  17. [24]

    ”Generalized simulated annealing algorithm and its application to the Thomson model.” Physics Letters A 233.3 (1997): 216-220

    Xiang, Yang, et al. ”Generalized simulated annealing algorithm and its application to the Thomson model.” Physics Letters A 233.3 (1997): 216-220

    1997

  18. [25]

    P´ erez-Garrido, Antonio, M. J. W. Dodgson, and M. A. Moore. ”Influence of dislocations in thomson’s problem.” Physical Re- view B 56.7 (1997): 3640

    1997

  19. [26]

    P´ erez-Garrido, A., and M. A. Moore. ”Symmetric patterns of dislocations in Thomson’s problem.” Physical Review B 60.23 (1999): 15628

    1999

  20. [27]

    Structure and dynamics of spher- ical crystals characterized for the Thomson problem

    Wales, D. J., and Ulker, S. , “Structure and dynamics of spher- ical crystals characterized for the Thomson problem”, Physical Review B 74.21 (2006): 212101

    2006

  21. [28]

    Altschuler

    Wales, David J., Hayley McKay, and Eric L. Altschuler. ”Defect motifs for spherical topologies.” Physical Review B 79.22 (2009): 224115. 25

    2009

  22. [29]

    Scheraga

    Li, Zhenqin, and Harold A. Scheraga. ”Monte Carlo- minimization approach to the multiple-minima problem in pro- tein folding.” Proceedings of the National Academy of Sciences 84.19 (1987): 6611-6615

    1987

  23. [30]

    Wales, David J., and Jonathan PK Doye. ”Global optimization by basin-hopping and the lowest energy structures of Lennard- Jones clusters containing up to 110 atoms.” The Journal of Phys- ical Chemistry A 101.28 (1997): 5111-5116

    1997

  24. [31]

    Scheraga

    Wales, David J., and Harold A. Scheraga. ”Global optimization of clusters, crystals, and biomolecules.” Science 285.5432 (1999): 1368-1372

    1999

  25. [32]

    Energy landscapes: Applications to clusters, biomolecules and glasses

    Wales, David. Energy landscapes: Applications to clusters, biomolecules and glasses. Cambridge University Press, 2003

    2003

  26. [33]

    D. J. Wales, GMIN: A Program for Basin-Hopping Global Optimization, Basin-Sampling, and Parallel Tempering, http://www-wales.ch.cam.ac.uk/software.html

  27. [34]

    David Wales, https://www-wales.ch.cam.ac.uk/ wales/C- CD/Thomson/table.html

  28. [35]

    David Wales, https://www-wales.ch.cam.ac.uk/ wales/C- CD/Thomson2/table.html

  29. [36]

    Lai, Xiangjing, et al. ”Global optimization and structural anal- ysis of Coulomb and logarithmic potentials on the unit sphere using a population-based heuristic approach.” Expert Systems with Applications 238 (2024): 121704

    2024

  30. [38]

    ”Kinetic transition networks for the Thomson problem and Smale’s seventh problem.” Physical Re- view Letters 117.2 (2016): 028301

    Mehta, Dhagash, et al. ”Kinetic transition networks for the Thomson problem and Smale’s seventh problem.” Physical Re- view Letters 117.2 (2016): 028301

    2016

  31. [39]

    ”Network topology of a potential energy landscape: A static scale-free network.” Physical review letters 88.23 (2002): 238701

    Doye, Jonathan PK. ”Network topology of a potential energy landscape: A static scale-free network.” Physical review letters 88.23 (2002): 238701

    2002

  32. [40]

    ”Stationary points and dynamics in high-dimensional systems.” The Journal of chemical physics 119.23 (2003): 12409-12416

    Wales, David J., and Jonathan PK Doye. ”Stationary points and dynamics in high-dimensional systems.” The Journal of chemical physics 119.23 (2003): 12409-12416. 26

    2003

  33. [41]

    ”Exploring the potential energy land- scape of the Thomson problem via Newton homotopies.” The Journal of chemical physics 142.19 (2015)

    Mehta, Dhagash, et al. ”Exploring the potential energy land- scape of the Thomson problem via Newton homotopies.” The Journal of chemical physics 142.19 (2015)

    2015

  34. [42]

    ”Constrained high- index saddle dynamics for the solution landscape with equality constraints.” Journal of Scientific Computing 91.2 (2022): 62

    Yin, Jianyuan, Zhen Huang, and Lei Zhang. ”Constrained high- index saddle dynamics for the solution landscape with equality constraints.” Journal of Scientific Computing 91.2 (2022): 62

    2022

  35. [43]

    Cheviakov

    Ridgway, Wesley JM, and Alexei F. Cheviakov. ”An iterative procedure for finding locally and globally optimal arrangements of particles on the unit sphere.” Computer Physics Communica- tions 233 (2018): 84-109

    2018

  36. [44]

    Van Rossum, Guido, and Fred L. Drake. Python reference man - ual. Vol. 111. Amsterdam: Centrum voor Wiskunde en Infor- matica, 1995

    1995

  37. [45]

    ”SciPy 1.0: fundamental algorithms for scientific computing in Python.” Nature methods 17.3 (2020): 261-272

    Virtanen, Pauli, et al. ”SciPy 1.0: fundamental algorithms for scientific computing in Python.” Nature methods 17.3 (2020): 261-272

    2020

  38. [46]

    ”Numba: A llvm-based python jit compiler.” Proceedings of the Second Workshop on the LL VM Compiler Infrastructure in HPC

    Lam, Siu Kwan, Antoine Pitrou, and Stanley Seibert. ”Numba: A llvm-based python jit compiler.” Proceedings of the Second Workshop on the LL VM Compiler Infrastructure in HPC. 2015

    2015

  39. [47]

    ”mpmath: a Python library for arbitrary-precision floating-point arithmetic.” Zenodo (2013)

    Johansson, Fredrik, et al. ”mpmath: a Python library for arbitrary-precision floating-point arithmetic.” Zenodo (2013)

    2013

  40. [48]

    ”Circle packing on spherical caps.” Physics of Fluids 36.9 (2024)

    Amore, Paolo. ”Circle packing on spherical caps.” Physics of Fluids 36.9 (2024)

    2024

  41. [49]

    ”Some physical features of the Burr-type-X II distribution.” Physical Review E 99.2 (2019): 022123

    Sanchez, Ewin. ”Some physical features of the Burr-type-X II distribution.” Physical Review E 99.2 (2019): 022123

    2019

  42. [50]

    G., and T

    Tenchov, B. G., and T. K. Yanev. ”Weibull distribution of par- ticle sizes obtained by uniform random fragmentation.” Journal of colloid and interface science 111.1 (1986): 1-7

    1986

  43. [51]

    Patterson, and Malcolm E

    Fang, Zhigang, Burton R. Patterson, and Malcolm E. Turner Jr . ”Modeling particle size distributions by the Weibull distribution function.” Materials characterization 31.3 (1993): 177-182

    1993

  44. [52]

    ”F rac- ture statistics of brittle materials: Weibull or normal distribu- tion.” Physical Review E 65.6 (2002): 067102

    Lu, Chunsheng, Robert Danzer, and Franz Dieter Fischer. ”F rac- ture statistics of brittle materials: Weibull or normal distribu- tion.” Physical Review E 65.6 (2002): 067102

    2002

  45. [53]

    Brouwers, H. J. H. ”Packing fraction of particles with a Weibull size distribution.” Physical Review E 94.1 (2016): 012905. 27

    2016

  46. [54]

    ”Weibull function to describe the cumula- tive size distribution of clumps formed by two-dimensional grains randomly arranged on a plane.” Physical Review E 109.4 (2024): 044131

    Fanfoni, M., et al. ”Weibull function to describe the cumula- tive size distribution of clumps formed by two-dimensional grains randomly arranged on a plane.” Physical Review E 109.4 (2024): 044131. 28

    2024

Pith tools

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