Optimal transportation of grain boundaries: A forward model for predicting migration mechanisms

It has been hypothesized that the most likely atomic rearrangement mechanism during grain boundary (GB) migration is the one that minimizes the lengths of atomic displacements in the dichromatic pattern. In this work, we recast the problem of atomic displacement minimization during GB migration as an optimal transport (OT) problem. Under the assumption of a small potential energy barrier for atomic rearrangement, the principle of stationary action applied to GB migration is reduced to the determination of the Wasserstein metric for two point sets. In order to test the minimum distance hypothesis, optimal displacement patterns predicted on the basis of a regularized OT based forward model are compared to molecular dynamics (MD) GB migration data for a variety of GB types and temperatures. Limits of applicability of the minimum distance hypothesis and interesting consequences of the OT formulation are discussed in the context of MD data analysis for twist GBs, general $\Sigma 3$ twin boundaries and a tilt GB that exhibits shear coupling. The forward model may be used to predict atomic displacement patterns for arbitrary disconnection modes and a variety of metastable states, facilitating the analysis of multimodal GB migration data.