Leading order asymptotics for non-local energies and the Read-Shockley law
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| Format: | Preprint |
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2025
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| author | Grabner, Peter J. Theil, Florian |
| author_facet | Grabner, Peter J. Theil, Florian |
| contents | We study an energy minimization problem $\sum_{i \neq j} W(z_i - z_j)$ for $N$ points $\left\{z_1, \dots, z_N\right\}$ with applications in dislocation theory. The $N$ points lie in the two-dimensional domain $\mathbb{R} \times [-π, π]$, %who are trying to minimize their interaction energy where
where the kernel $W$ is derived from the Volterra potential $V(x,y) = \frac{x^2}{x^2+y^2}-\frac12\log(x^2+y^2)$. We prove that the minimum energy is given by $- N \log{N} +\mathcal{O}(N)$. This lower bound recovers the leading order term of the Read-Shockley law characterizing the energy of small angle grain boundaries in polycrystals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12041 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Leading order asymptotics for non-local energies and the Read-Shockley law Grabner, Peter J. Theil, Florian Classical Analysis and ODEs We study an energy minimization problem $\sum_{i \neq j} W(z_i - z_j)$ for $N$ points $\left\{z_1, \dots, z_N\right\}$ with applications in dislocation theory. The $N$ points lie in the two-dimensional domain $\mathbb{R} \times [-π, π]$, %who are trying to minimize their interaction energy where where the kernel $W$ is derived from the Volterra potential $V(x,y) = \frac{x^2}{x^2+y^2}-\frac12\log(x^2+y^2)$. We prove that the minimum energy is given by $- N \log{N} +\mathcal{O}(N)$. This lower bound recovers the leading order term of the Read-Shockley law characterizing the energy of small angle grain boundaries in polycrystals. |
| title | Leading order asymptotics for non-local energies and the Read-Shockley law |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2509.12041 |