Crystal Growth Inside an Octant

TL;DR

This paper investigates the deterministic shape evolution of crystal growth inside an octant on a cubic lattice, proposing a PDE model, validating it with simulations, and exploring volume fluctuations and generalizations to higher dimensions.

Contribution

It introduces a hyperbolic PDE model for the limiting shape of crystal growth in an octant, extending known 2D results to 3D and higher dimensions.

Findings

Analytical solution for the limiting shape of the crystal.

Excellent agreement between simulations and the PDE model.

Insights into volume fluctuations and corner counts during growth.

Abstract

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.