Periodic Discrete Energy for Long-Range Potentials

Abstract

We consider periodic energy problems in Euclidean space with a special emphasis on long-range potentials that cannot be defined through the usual infinite sum. One of our main results builds on more recent developments of Ewald summation to define the periodic energy corresponding to a large class of long-range potentials. Two particularly interesting examples are the logarithmic potential and the Riesz potential when the Riesz parameter is smaller than the dimension of the space. For these examples, we use analytic continuation methods to provide concise formulas for the periodic kernel in terms of the Epstein Hurwitz Zeta function. We apply our energy definition to deduce several properties of the minimal energy including the asymptotic order of growth and the distribution of points in energy minimizing configurations as the number of points becomes large. We conclude with some detailed calculations in the case of one dimension, which shows the utility of this approach.

32 pages; Version 2: Minor changes and typos corrected, modified presentation of Theorem 5.4, one additional reference added, to appear in Journal of Mathematical Physics

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Cited by in corpus (9)

• Distributing many points on spheres: minimal energy and designs
• Coulomb and Riesz gases: The known and the unknown
• Dimension reduction techniques for the minimization of theta functions on lattices
• On Born's conjecture about optimal distribution of charges for an infinite ionic crystal
• Optimal lattice configurations for interacting spatially extended particles
• Expected Riesz energy of some determinantal processes on flat tori
• On energy ground states among crystal lattice structures with prescribed bonds
• 2D Theta Functions and Crystallization among Bravais Lattices
• Next order energy asymptotics for Riesz potentials on flat tori