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

2025/09/15 by Peter J. Grabner, Grabner, Peter J., Florian Theil +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2509.12041

openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

We study an energy minimization problem ∑_{i ≠ j} W(z_i - z_j) for N points \z₁, …, z_N\ with applications in dislocation theory. The N points lie in the two-dimensional domain ℝ × [-π, π], %who are trying to minimize their interaction energy where where the kernel W is derived from the Volterra potential V(x,y) = (x²)/(x²+y²)-\frac12log(x²+y²). We prove that the minimum energy is given by - N logN +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.

Citations

• Why Grain Growth is Not Curvature Flow
• The ellipse law: Kirchhoff meets dislocations
• The equilibrium measure for a nonlocal dislocation energy
• An Energy Estimate for Dislocation Configurations and the Emergence of Cosserat-Type Structures in Metal Plasticity
• Convergence of interaction-driven evolutions of dislocations with Wasserstein dissipation and slip-plane confinement
• 1D Log Gases and the Renormalized Energy: Crystallization at Vanishing Temperature
• Gradient theory for plasticity via homogenization of discrete dislocations

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• The Gamma-limit of the two-dimensional Ohta-Kawasaki energy. II. Droplet arrangement at the sharp interface level via the renormalized energy