八分仪中的晶体生长
Crystal Growth Inside an Octant
摘要：
我们研究了立方晶格上无限八分圆内的晶体生长。生长通过将基本立方体沉积到内角中来进行。通过特征尺寸重新缩放后，界面在长时间限制中变得越来越确定。利用已知的结果内的二维角的晶体生长，我们提出了一个双曲型偏微分方程的极限形状的演变。该方程被解释为有助于找到解析解的Hamilton-Jacobi方程。生长过程的模拟与分析预测非常吻合。然后，我们研究的子领导校正的体积的晶体，渐近增长的方差的体积的晶体，和内外角的总数的演变。我们还展示了如何推广的结果，以任意的空间维度。
英文摘要：
We study crystal growth inside an infinite octant on a cubic lattice. The growth proceeds through the deposition of elementary cubes into inner corners. After re-scaling by the characteristic size, the interface becomes progressively more deterministic in the long-time limit. Utilizing known results for the crystal growth inside a two-dimensional corner, we propose a hyperbolic partial differential equation for the evolution of the limiting shape. This equation is interpreted as a Hamilton-Jacobi equation which helps in finding an analytical solution. Simulations of the growth process are in excellent agreement with analytical predictions. We then study the evolution of the sub-leading correction to the volume of the crystal, the asymptotic growth of the variance of the volume of the crystal, and the total number of inner and outer corners. We also show how to generalize the results to arbitrary spatial dimension.
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