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Optimal lattice configurations for interacting spatially extended particles

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Abstract

We investigate lattice energies for radially symmetric, spatially extended particles interacting via a radial potential and arranged on the sites of a two-dimensional Bravais lattice. We show the global minimality of the triangular lattice among Bravais lattices of fixed density in two cases: In the first case, the distribution of mass is sufficiently concentrated around the lattice points, and the mass concentration depends on the density we have fixed. In the second case, both interacting potential and density of the distribution of mass are described by completely monotone functions in which case the optimality holds at any fixed density.

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Acknowledgements

LB is grateful for the support of the Mathematics Center Heidelberg (MATCH) during his stay in Heidelberg. He also acknowledges support from ERC advanced Grant Mathematics of the Structure of Matter (Project No. 321029) and from VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). HK is grateful about support from DFG Grant 392124319. The authors also thank the referees for their interesting suggestions and comments.

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Authors and Affiliations

  1. QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100, Copenhagen Ø, Denmark

    Laurent Bétermin

  2. Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120, Heidelberg, Germany

    Hans Knüpfer

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  1. Laurent Bétermin
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  2. Hans Knüpfer
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Correspondence to Laurent Bétermin.

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Bétermin, L., Knüpfer, H. Optimal lattice configurations for interacting spatially extended particles. Lett Math Phys 108, 2213–2228 (2018). https://doi.org/10.1007/s11005-018-1077-9

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  • DOI: https://doi.org/10.1007/s11005-018-1077-9

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