Bétermin Laurent, Knüpfer Hans, Nolte Florian
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.
Institute of Applied Mathematics and IWR, University of Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany.
J Stat Phys. 2020;181(3):803-815. doi: 10.1007/s10955-020-02603-2. Epub 2020 Jul 13.
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.
我们研究了通过镜像对称势相互作用的交替类型粒子的一维周期链。在各种情况下，都证明了固定密度下等距构型的最优性——也称为结晶。特别是，我们证明了具有逆幂律相互作用的中性和非中性系统在任何尺度下的结晶，包括三维库仑势。我们还表明，对于涉及逆幂律和原点处排斥的系统，在高密度下等距构型是最小的。此外，我们基于相互作用势之和的傅里叶变换的正性，推导出了高密度下结晶的必要条件。
