  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.
