The bulk ground-state energy of the planar Lennard–Jones system
For distinct points $x_1, \dots, x_N \in \mathbb{R}^2$ let $$E_N(x_1, \dots, x_N) = \sum_{1 \le i < j \le N} \left( |x_i - x_j|^{-12} - 2|x_i - x_j|^{-6} \right), \qquad e_\infty = \liminf_{N \to \infty} \frac{1}{N} \inf E_N,$$ the infimum being over all configurations of $N$ distinct points. Let $\Lambda_\triangle = \mathbb{Z}(1,0) + \mathbb{Z}(1/2, \sqrt{3}/2)$ be the triangular lattice and $$e_\triangle = \inf_{a > 0} \frac{1}{2} \sum_{p \in \Lambda_\triangle \setminus \{0\}} \left( |ap|^{-12} - 2|ap|^{-6} \right)$$ the least energy per point of its dilates. Heitmann and Radin proved crystallization on a triangular lattice for sticky disks (1980), and Theil proved that the ground-state energy per particle converges to that of a triangular lattice for a class of smooth potentials resembling the Lennard–Jones one, under restrictive hypotheses (2006). Among lattices alone, Bétermin proved with computer assistance that a triangular lattice minimizes the energy per point of this potential over all planar lattices (2023).
Problem. Prove or disprove $e_\infty = e_\triangle$. The finite configurations are unrestricted; they are not assumed to be lattices. Only the energy is asked for, not the convergence of minimizing configurations to a triangular lattice.
Source: The crystallization conjecture for the Lennard–Jones potential in the plane; see Blanc and Lewin (2015), §2. Stated by AIM — Open Applied Problems in this energy form.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MP — Mathematical Physics
- Secondary
- MG
- Keywords
- AIM Open Applied Problems, crystallization, lattice energies, Lennard–Jones potentials
References
@article{blanc2015,
author = {Blanc, Xavier and Lewin, Mathieu},
title = {The crystallization conjecture: a review},
journal = {EMS Surv. Math. Sci.},
volume = {2},
number = {2},
year = {2015},
pages = {255--306},
doi = {10.4171/EMSS/13},
eprint = {1504.01153},
archivePrefix = {arXiv},
}
@article{heitmann1980,
author = {Heitmann, R. C. and Radin, C.},
title = {The ground state for sticky disks},
journal = {J. Stat. Phys.},
volume = {22},
number = {3},
year = {1980},
pages = {281--287},
doi = {10.1007/BF01014644},
}
@article{theil2006,
author = {Theil, Florian},
title = {A proof of crystallization in two dimensions},
journal = {Comm. Math. Phys.},
volume = {262},
number = {1},
year = {2006},
pages = {209--236},
doi = {10.1007/s00220-005-1458-7},
}
@article{betermin2023,
author = {B{\'e}termin, Laurent},
title = {Optimality of the triangular lattice for {L}ennard--{J}ones type lattice energies: a computer-assisted method},
journal = {J. Phys. A},
volume = {56},
number = {14},
year = {2023},
pages = {145204},
doi = {10.1088/1751-8121/acc21d},
eprint = {2104.09795},
archivePrefix = {arXiv},
}
@misc{aim-lennard-jones-planar-crystallization,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The bulk ground-state energy of the planar Lennard–Jones system}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/083-lennard-jones-planar-crystallization.md},
note = {Entry 083 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00609, "The bulk ground-state energy of the planar Lennard–Jones system", clopen.solutions.