Crystallization in the Winterbottom shape and sharp fluctuation laws
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908522401234944 |
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| author | Friedrich, Manuel Kreutz, Leonard Stefanelli, Ulisse |
| author_facet | Friedrich, Manuel Kreutz, Leonard Stefanelli, Ulisse |
| contents | We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength $β>0$ couples the particles and the substrate. Our first main result proves crystallization for all $β>0$, corresponding to
the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations.
Our second main result concerns fluctuations estimates for $β\in (0,1)$. We obtain bounds on the distance between distinct minimizers with the same number $N$ of particles, showing a sharp scaling law $N^{3/4}$ when $β$ is rational, and $N^{1/3}$ when $β$ is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_05642 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Crystallization in the Winterbottom shape and sharp fluctuation laws Friedrich, Manuel Kreutz, Leonard Stefanelli, Ulisse Mesoscale and Nanoscale Physics Mathematical Physics Analysis of PDEs We address finite crystallization in two dimensions in the presence of a flat crystalline substrate. Particles interact through short-range two- and three-body potentials favoring local square-lattice arrangements. An additional interaction term of relative strength $β>0$ couples the particles and the substrate. Our first main result proves crystallization for all $β>0$, corresponding to the onset of discrete Winterbottom configurations. The proof relies on a stratification technique from [31], characterizing the topology of the bond graph of minimizing configurations. Our second main result concerns fluctuations estimates for $β\in (0,1)$. We obtain bounds on the distance between distinct minimizers with the same number $N$ of particles, showing a sharp scaling law $N^{3/4}$ when $β$ is rational, and $N^{1/3}$ when $β$ is irrational and algebraic. This reveals a genuine substrate-driven effect on fluctuation laws. As a corollary, we derive a discrete-to-continuum convergence of minimizers towards the Winterbottom equilibrium shape in the large-particle limit. |
| title | Crystallization in the Winterbottom shape and sharp fluctuation laws |
| topic | Mesoscale and Nanoscale Physics Mathematical Physics Analysis of PDEs |
| url | https://arxiv.org/abs/2509.05642 |