论文标题
关于具有电荷分布的晶格中岩石结构的最佳性
On the optimality of the rock-salt structure among lattices with charge distributions
论文作者
论文摘要
这项工作的目的是调查$ d $维岩石盐结构的最佳性，即Cuxic lattice $ v^{1/d} \ Mathbb {Z}^d $ of Volume $ v $的$ \ \ \ Mathbb {Z}^d $与电荷交替使用的费用$ \ pm 1 $在Lattice Points in Lattice Pointers of Chardic和Lattice结构的定期分配。我们假设这些电荷是根据两种类型的径向对称相互作用势来相互作用的。我们首先将研究限制为正交晶格的类别。我们证明，对于我们的能源模型，$ d $维岩石盐结构始终是固定密度周期性结构中的关键点。这具有巨大的潜力。然后，我们研究了正骨晶格之间的最小化问题，并以反功率定律和高斯相互作用电位的费用交替。高密度最小的结果和两种电势的低密度非密度结果均得出。从数字上讲，我们研究了几个特定案例，$ 2 $，$ 3 $和$ 8 $。数字支持以下猜想：岩盐结构是所有晶格和周期性电荷之间的全球最佳选择，从而满足了一些自然约束。对于$ d = 2 $，我们观察到随着密度降低的最小化器形状的“三角形 - 肉眼平方矩形”类型的相变。
The goal of this work is to investigate the optimality of the $d$-dimensional rock-salt structure, i.e., the cubic lattice $V^{1/d}\mathbb{Z}^d$ of volume $V$ with an alternation of charges $\pm 1$ at lattice points, among periodic distribution of charges and lattice structures. We assume that the charges are interacting through two types of radially symmetric interaction potentials, according to their signs. We first restrict our study to the class of orthorhombic lattices. We prove that, for our energy model, the $d$-dimensional rock-salt structure is always a critical point among periodic structures of fixed density. This holds for a large class of potentials. We then investigate the minimization problem among orthorhombic lattices with an alternation of charges for inverse power laws and Gaussian interaction potentials. High density minimality results and low-density non-optimality results are derived for both types of potentials. Numerically, we investigate several particular cases in dimensions $2$, $3$ and $8$. The numerics support the conjecture that the rock-salt structure is the global optimum among all lattices and periodic charges, satisfying some natural constraints. For $d=2$, we observe a phase transition of the type 'triangular-rhombic-square-rectangular' for the minimizer's shape as the density decreases.
