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arXiv · 2509.12041

Leading order asymptotics for non-local energies and the Read-Shockley law

Peter J. Grabner·Florian Theil
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Abstract

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 $\mathbb{R} \times [-\pi, \pi]$, %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.

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Peter J. Grabner, Florian Theil. 2025-09-15. Leading order asymptotics for non-local energies and the Read-Shockley law. https://arxiv.org/abs/2509.12041

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