Note on Crystallization for Alternating Particle Chains
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Note on Crystallization for Alternating Particle Chains Laurent Bétermin1
· Hans Knüpfer2 · Florian Nolte2
Received: 23 February 2020 / Accepted: 25 June 2020 © The Author(s) 2020
Abstract We investigate one-dimensional periodic chains of alternate type of particles interacting through mirror symmetric potentials. The optimality of the equidistant configuration at fixed density—also called crystallization—is shown in various settings. In particular, we prove the crystallization at any scale for neutral and non-neutral systems with inverse power laws interactions, including the three-dimensional Coulomb potential. We also show the minimality of the equidistant configuration at high density for systems involving inverse power laws and repulsion at the origin. Furthermore, we derive a necessary condition for crystallization at high density based on the positivity of the Fourier transform of the interaction potentials sum. Keywords Crystallization · Ionic crystals · Convexity · Energy minimization Mathematics Subject Classification 82B05 · 26A51 · 74E15
1 Introduction A fundamental question in the theory of crystallization is to understand why many large systems of interacting particles exhibit the spontaneous formation of periodic structures and how they can be explained by energy minimization [10]. Such periodic structures are observed in systems consisting of identical particles but also appear in models composed of different types of particles. For example, ionic compounds exhibit periodic structures, even though different attractive and repulsive interaction potentials between the ions are present
Communicated by Alessandro Giuliani.
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Laurent Bétermin [email protected] Hans Knüpfer [email protected] Florian Nolte [email protected]
1
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria
2
Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
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L. Bétermin et al.
[35]. In this paper, we consider prototypical alternating chains of particles where particles of different (resp. same) kind repel (resp. attract) each other at short distance and investigate necessary and sufficient conditions for the optimality of periodic (equidistant) configurations. This consideration is motivated for instance by alternating chains of magnetic domain walls (see e.g. [24]). We note that while one-dimensional model systems do not occur commonly in nature, they can be created by confinement (see e.g. [32]). In this paper, once the charges are fixed, as well as the interaction between species, we show the optimality of the equidistant configuration at fixed density, among one-dimensional periodic configurations of alternating species in different settings. The novelty of the paper consists in the systematic analysis for the ground state energy of alternating two-particle systems. We assume repulsive interaction at short distances between different species in order to avoid a collapsing of t
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