A note on energy minimization in dimension 2

Markus Faulhuber; Irina Shafkulovska; Ilia Zlotnikov

doi:10.1090/bproc/247

journal article Open Access Dec 17, 2024

A note on energy minimization in dimension 2

Markus Faulhuber

Irina Shafkulovska Ilia Zlotnikov

Proceedings of the American Mathematical Society, Series B Vol. 11 No. 57 pp. 664-679 · American Mathematical Society (AMS)

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Abstract

Proving the
universal optimality of the hexagonal lattice
is one of the challenging endeavors of nowadays mathematics. In this note, we show that the hexagonal lattice outperforms certain “simple” classes of periodic configurations, including a natural nonlattice configuration—the honeycomb.

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References

25

[1]

Baernstein, Albert, II "A minimum problem for heat kernels of flat tori" (1997) 10.1090/conm/201/02604

[2]

Bernstein, Serge "Sur les fonctions absolument monotones" Acta Math. (1929) 10.1007/bf02547400

[3]

Blanc, Xavier "The crystallization conjecture: a review" EMS Surv. Math. Sci. (2015) 10.4171/emss/13

[4]

Borwein, J. M. "A cubic counterpart of Jacobi’s identity and the AGM" Trans. Amer. Math. Soc. (1991) 10.2307/2001551

[5]

Bétermin, Laurent "Maximal theta functions universal optimality of the hexagonal lattice for Madelung-like lattice energies" J. Anal. Math. (2023) 10.1007/s11854-022-0254-z

[6]

L. Bétermin, M. Faulhuber, and S. Steinerberger, A variational principle for Gaussian lattice sums, Preprint, arXiv:2110.06008, 2021.

[7]

Bétermin, Laurent "Dimension reduction techniques for the minimization of theta functions on lattices" J. Math. Phys. (2017) 10.1063/1.4995401

[8]

Borodachov, Sergiy V. 10.1007/978-0-387-84808-2

[9]

Universally optimal distribution of points on spheres

Henry Cohn, Abhinav Kumar

Journal of the American Mathematical Society 2007 10.1090/s0894-0347-06-00546-7

[10]

Cohn, Henry "Universal optimality of the 𝐸₈ and Leech lattices and interpolation formulas" Ann. of Math. (2) (2022) 10.4007/annals.2022.196.3.3

[11]

H. Cohn, A. Kumar, and A. Schuermann, Ground states and formal duality relations in the Gaussian core model, Phys. Rev. E, 80:061116, 2009. 10.1103/physreve.80.061116

[12]

Conway, J. H. (1999) 10.1007/978-1-4757-6568-7

[13]

Coulangeon, Renaud "Energy minimization, periodic sets and spherical designs" Int. Math. Res. Not. IMRN (2012) 10.1093/imrn/rnr048

[14]

Coulangeon, Renaud "Erratum: Energy minimization, periodic sets, and spherical designs" Int. Math. Res. Not. IMRN (2022) 10.1093/imrn/rnaa188

[15]

M. Faulhuber, A. Gumber, and I. Shafkulovska, The AGM of Gauss, Ramanujan’s corresponding theory, and spectral bounds of self-adjoint operators, Preprint, arXiv:2209.04202, 2022.

[16]

Faulhuber, Markus "Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions" J. Math. Anal. Appl. (2017) 10.1016/j.jmaa.2016.07.074

[17]

D. P. Hardin and N. J. Tenpas, Universally Optimal Periodic Configurations in the Plane, Preprint, arXiv:2307.15822, 2023.

[18]

Janssen, A. J. E. M. "Some Weyl-Heisenberg frame bound calculations" Indag. Math. (N.S.) (1996) 10.1016/0019-3577(96)85088-9

[19]

Montgomery, Hugh L. "Minimal theta functions" Glasgow Math. J. (1988) 10.1017/s0017089500007047

[20]

Stein, Elias M. (2003)

[21]

F. H. Stillinger, Phase transitions in the Gaussian core system, J. Chem. Phys., 65(10):3968–3974, 1976. 10.1063/1.432891

[22]

N. J. Tenpas, A Family of Universally Optimal Configurations on Rectangular Flat Tori, Preprint, arXiv:2311.05594, 2023.

[23]

Whittaker, E. T. (1996) 10.1017/cbo9780511608759

[24]

Widder, David Vernon (1941)

[25]

Wolfram Research, Inc., Champaign, IL. Mathematica, Version 12–13, 2019–2021.

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Details

Published: Dec 17, 2024
Vol/Issue: 11(57)
Pages: 664-679
License: View

Authors

Funding

Austrian Science Fund Award: 10.55776/P33217

Cite This Article

Markus Faulhuber, Irina Shafkulovska, Ilia Zlotnikov (2024). A note on energy minimization in dimension 2. Proceedings of the American Mathematical Society, Series B, 11(57), 664-679. https://doi.org/10.1090/bproc/247

A note on energy minimization in dimension 2

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