On crystallization in the plane for pair potentials with an arbitrary norm
Laurent Bétermin1; Camille Furlanetto1
1Institut Camille Jordan and Université Claude Bernard Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
Mathematics Research Reports, Volume 7 (2026), pp. 27-44

We investigate two-dimensional crystallization phenomena, i.e., minimality of a lattice’s patch for interaction energies, with pair potentials of type $(x,y)\mapsto V(\Vert x-y\Vert )$ where $\Vert \cdot \Vert $ is an arbitrary norm on $\mathbb{R}^2$ and $V:\mathbb{R}_+^*\rightarrow \mathbb{R}$ is a function. For the Heitmann–Radin sticky disk potential $V=V_{\textnormal{HR}}$, we prove, using Brass’ key result from Computational Geometry 1996, that crystallization occurs for any fixed norm, with a classification of minimizers and minimal energies according to the kissing number associated to $\Vert \cdot \Vert $. The minimizer is proved to be, up to affine transform, a patch of the triangular or the square lattice, which shows how to easily get anisotropy in a crystallization phenomenon. We apply this result to the $p$-norms $\Vert \cdot \Vert _p$, $p\ge 1$, which allows us to construct an explicit family of norms for which crystallization holds on any given lattice. We also solve part of a crystallization problem studied in [Arch. Ration. Mech. Anal., 240:987–1053] where points are constrained to be on $\mathbb{Z}^2$. Moreover, we numerically investigate the minimization problem for the energy per point among lattices for the Lennard–Jones potential $V=V_{\textnormal{LJ}}:r\mapsto r^{-12}-2r^{-6}$ as well as the Epstein zeta function associated to a $p$-norm $\Vert \cdot \Vert _p$, i.e., when $V=V_s:r\mapsto r^{-s}$, $s>2$. Our simulations show a new and unexpected phase transition for the minimizers with respect to $p$.

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DOI: 10.5802/mrr.25
Classification: 74G65, 82B20, 52C15
Keywords: Crystallization, Norm, Lennard–Jones potential, Sticky disk potential, Lattices, Minimal energy, Epstein zeta functions, Anisotropy

Laurent Bétermin  1 ; Camille Furlanetto  1

1 Institut Camille Jordan and Université Claude Bernard Lyon 1, 43 boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
Laurent Bétermin; Camille Furlanetto. On crystallization in the plane for pair potentials with an arbitrary norm. Mathematics Research Reports, Volume 7 (2026), pp. 27-44. doi: 10.5802/mrr.25
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[1] Laurent Bétermin Two-dimensional theta functions and crystallization among Bravais Lattices, SIAM Journal on Mathematical Analysis, Volume 48 (2016) no. 5, pp. 3236-3269 | DOI | Zbl | MR

[2] Laurent Bétermin Local variational study of 2D lattice energies and application to Lennard–Jones type interactions, Nonlinearity, Volume 31 (2018) no. 9, pp. 3973-4005 | DOI | Zbl | MR

[3] Laurent Bétermin Effect of periodic arrays of defects on lattice energy minimizers, Annales Henri Poincaré, Volume 22 (2021), pp. 2995-3023 | DOI | Zbl | MR

[4] Laurent Bétermin On energy ground states among crystal lattice structures with prescribed bonds, Journal of Physics A, Volume 54 (2021) no. 24, 245202 | Zbl | MR

[5] Laurent Bétermin Optimality of the triangular lattice for Lennard–Jones type lattice energies: a computer-assisted method, Journal of Physics A: Mathematical and General, Volume 56 (2023), 145204 | Zbl | MR

[6] Laurent Bétermin; Lorenzo De Luca; Mircea Petrache Crystallization to the square lattice for a two-body potential, Archive for Rational Mechanics and Analysis, Volume 240 (2021), pp. 987-1053 | DOI | Zbl | MR

[7] Laurent Bétermin; Michael Faulhuber Maximal theta functions – universal optimality of the hexagonal lattice for Madelung-like lattice energies, Journal d’Analyse Mathématique, Volume 149 (2023), pp. 307-341 | DOI | Zbl | MR

[8] Laurent Bétermin; Mircea Petrache Optimal and non-optimal lattices for non-completely monotone interaction potentials, Analysis and Mathematical Physics, Volume 9 (2019) no. 4, pp. 2033-2073 | DOI | Zbl | MR

[9] Laurent Bétermin; Ping Zhang Minimization of energy per particle among Bravais lattices in ℝ 2 : Lennard–Jones and Thomas–Fermi cases, Communications in Contemporary Mathematics, Volume 17 (2015) no. 6, 1450049 | MR | Zbl

[10] Xavier Blanc; Claude Le Bris Periodicity of the infinite-volume ground state of a one-dimensional quantum model, Nonlinear Analysis: Theory, Methods & Applications, Volume 48 (2002) no. 6, pp. 791-803 | DOI | Zbl | MR

[11] Xavier Blanc; Mathieu Lewin The Crystallization Conjecture: A Review, EMS Surveys in Mathematical Sciences, Volume 2 (2015), pp. 255-306 | DOI | MR

[12] Peter Brass Erdös distance problems in normed spaces, Computational Geometry, Volume 6 (1996), pp. 195-214 | DOI | Zbl | MR

[13] José A. Cañizo; Alejandro Ramos-Lora Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review, Active Particles, Volume 4 (J. A. Carrillo; E. Tadmor, eds.) (Modeling and Simulation in Science, Engineering and Technology), Birkhäuser, 2024, pp. 55-78 | DOI

[14] J. W. S. Cassels On a problem of Rankin about the Epstein zeta function, Proceedings of the Glasgow Mathematical Association, Volume 4 (1959), pp. 73-80 | DOI | Zbl | MR

[15] Marco Cicalese; Gian Paolo Leonardi Maximal fluctuations on periodic lattices: an approach via quantitative Wulff inequalities, Communications in Mathematical Physics, Volume 375 (2020), pp. 1931-1944 | Zbl | DOI | MR

[16] O. Ciftja Results for an anisotropic Coulomb interaction potential, Results in Physics, Volume 43 (2022), 106052 | DOI

[17] Edoardo Davoli; Paolo Piovano; Ulisse Stefanelli Sharp N 3/4 law for the minimizers of the edge-isoperimetric problem on the triangular lattice, Journal of Nonlinear Science, Volume 27 (2017) no. 2, pp. 627-660 | DOI | Zbl | MR

[18] Lorenzo De Luca; Gero Friesecke Classification of particle numbers with unique Heitmann–Radin minimizer, Journal of Statistical Physics, Volume 167 (2017), pp. 1586-1592 | DOI | Zbl | MR

[19] Lorenzo De Luca; Gero Friesecke Crystallization in two dimensions and a discrete Gauss–Bonnet Theorem, Journal of Nonlinear Science, Volume 28 (2018) no. 1, pp. 69-90 | DOI | MR | Zbl

[20] Giovanni Del Nin; Lorenzo De Luca A crystallization result in two dimensions for a soft disc affine potential, Anisotropic Isoperimetric Problems and Related Topics (Springer INdAM Series), Volume 62, Springer (2022), pp. 201-212 | DOI | MR

[21] Giovanni Del Nin; Lorenzo De Luca The square sticky disk: crystallization and Gamma-convergence to the octagonal anisotropic perimeter (2025) | arXiv | Zbl

[22] Giovanni Del Nin; Mircea Petrache Continuum limits of discrete isoperimetric problems and Wulff shapes in lattices and quasicrystal tilings, Calculus of Variations and Partial Differential Equations, Volume 61 (2022), 226, 44 pages | MR | Zbl

[23] P. H. Diananda notes on two lemmas concerning the Epstein zeta-function, Proceedings of the Glasgow Mathematical Association, Volume 6 (1964), pp. 202-204 | DOI | MR | Zbl

[24] Paul Engel geometric crystallography. an axiomatic introduction to crystallography, Reidel, 1942

[25] Veikko Ennola A lemma about the Epstein zeta-function, Proceedings of the Glasgow Mathematical Association, Volume 6 (1964), pp. 198-201 | DOI | MR | Zbl

[26] Matthias Friedrich; Lukas Kreutz; Bernd Schmidt emergence of rigid polycrystals from atomistic systems with Heitmann–Radin sticky disk energy, Archive for Rational Mechanics and Analysis, Volume 240 (2021), pp. 627-698 | DOI | MR | Zbl

[27] Gero Friesecke; Bernd Schmidt; Yuen A. Yeung Minimizing atomic configurations of short range pair potentials in two dimensions: crystallization in the Wulff shape, Calculus of Variations and Partial Differential Equations, Volume 44 (2012) no. 1–2, pp. 81-100 | MR | Zbl

[28] Clément Furlanetto Réseaux et cristallisation dans le plan pour un potentiel à sphères dures, Rapport de TIPE, Université Claude Bernard Lyon 1, 2024 https://licence-math.univ-lyon1.fr/...

[29] C. S. Gardner; Charles Radin The infinite-volume ground state of the Lennard–Jones potential, Journal of Statistical Physics, Volume 20 (1979), pp. 719-724 | DOI | MR

[30] Branko Grünbaum On a conjecture of H. Hadwiger, Pacific Journal of Mathematics, Volume 11 (1961), pp. 215-219 | DOI | MR | Zbl

[31] Heiko Harborth Lösung zu problem 664a, Elemente der Mathematik, Volume 29 (1974), pp. 14-15

[32] R. C. Heitmann; Charles Radin The ground state for sticky disks, Journal of Statistical Physics, Volume 22 (1980), pp. 281-287 | DOI | MR

[33] Xavier Lamy Uniaxial symmetry in nematic liquid crystals, Annales de l’Institut Henri Poincaré, Volume 32 (2015), pp. 1125-1144 | Numdam | Zbl

[34] Senping Luo; Juncheng Wei On minima of difference of Epstein zeta functions and exact solutions to Lennard–Jones lattice energy, Journal of the European Mathematical Society (2025) (Online First) | DOI

[35] Elisabetta Mainini; Paolo Piovano; Bernd Schmidt; Ulisse Stefanelli N 3/4 law in the cubic lattice, Journal of Statistical Physics, Volume 176 (2019) no. 6, pp. 1480-1499 | DOI | MR | Zbl

[36] Émile Marcotte; Frank H. Stillinger; Salvatore Torquato Optimized monotonic convex pair potentials stabilize low-coordinated crystals, Soft Matter, Volume 7 (2011), pp. 2332-2335 | DOI

[37] Hugh L. Montgomery Minimal theta functions, Glasgow Mathematical Journal, Volume 30 (1988) no. 1, pp. 75-85 | DOI | MR | Zbl

[38] Charles Radin The ground state for soft disks, Journal of Statistical Physics, Volume 26 (1981) no. 2, pp. 365-373 | DOI | MR

[39] Charles Radin Low temperature and the origin of crystalline symmetry, International Journal of Modern Physics B, Volume 1 (1987) no. 5–6, pp. 1157-1191 | DOI | MR

[40] R. A. Rankin A minimum problem for the Epstein zeta-function, Proceedings of the Glasgow Mathematical Association, Volume 1 (1953), pp. 149-158 | DOI | MR | Zbl

[41] Etienne Sandier; Sylvia Serfaty From the Ginzburg–Landau model to vortex lattice problems, Communications in Mathematical Physics, Volume 313 (2012) no. 3, pp. 635-743 | DOI | MR | Zbl

[42] Bernd Schmidt Ground states of the 2D sticky disc model: fine properties and N 3/4 law for the deviation from the asymptotic Wulff shape, Journal of Statistical Physics, Volume 153 (2013), pp. 727-738 | DOI | MR | Zbl

[43] Tor Stensola; Hanne Stensola; May-Britt Moser; Edvard I. Moser Shearing-induced asymmetry in entorhinal grid cells, Nature, Volume 518 (2015), pp. 207-212 | DOI

[44] Jun Sun; Jun Wei; Wenming Zou On lattice energy minimization problem for non-completely monotone functions and applications, Analysis and Mathematical Physics, Volume 9 (2019), pp. 2033-2073

[45] Konrad J. Swanepoel Combinatorial distance geometry in normed spaces, Handbook of Discrete and Computational Geometry, Springer, 2018, pp. 407-458 | Zbl

[46] Florian Theil A proof of crystallization in two dimensions, Communications in Mathematical Physics, Volume 262 (2006) no. 1, pp. 209-236 | DOI | MR | Zbl

[47] Salvatore Torquato Inverse optimization techniques for targeted self-assembly, Soft Matter, Volume 6 (2009), pp. 1157-1173 | DOI

[48] Ivan Travěnec; Ladislav Šamaj Two-dimensional Wigner crystals of classical Lennard–Jones particles, Journal of Physics A: Mathematical and Theoretical, Volume 52 (2019) no. 20, 205002 | MR | Zbl

[49] W. J. Ventevogel On the configuration of a one-dimensional system of interacting particles with minimum potential energy per particle, Physica A, Volume 92 (1978) no. 3-4, pp. 343-361 | DOI

[50] W. J. Ventevogel; B. R. A. Nijboer On the configuration of systems of interacting particle with minimum potential energy per particle, Physica A, Volume 98 (1979), pp. 274-288 | DOI

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