Minimizing Lattice Energy and Hexagonal Crystallization
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915035524104192 |
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| author | Deng, Kaixin Luo, Senping |
| author_facet | Deng, Kaixin Luo, Senping |
| contents | Consider the energy per particle on the lattice given by $\min_{ Λ}\sum_{ \mathbb{P}\in Λ} \left|\mathbb{P}\right|^4 e^{-πα\left|\mathbb{P}\right|^2 }$, where $α>0$ and $Λ$ is a two dimensional lattice. We prove that for $α\geq\frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\fracπ{3}}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by Bétermin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_17199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimizing Lattice Energy and Hexagonal Crystallization Deng, Kaixin Luo, Senping Analysis of PDEs Consider the energy per particle on the lattice given by $\min_{ Λ}\sum_{ \mathbb{P}\in Λ} \left|\mathbb{P}\right|^4 e^{-πα\left|\mathbb{P}\right|^2 }$, where $α>0$ and $Λ$ is a two dimensional lattice. We prove that for $α\geq\frac{3}{2}$, among two dimensional lattices with unit density, such energy minimum is attained at $e^{i\fracπ{3}}$, corresponding to the hexagonal lattice. Our result partially answers some open questions proposed by Bétermin. |
| title | Minimizing Lattice Energy and Hexagonal Crystallization |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2411.17199 |