  We introduce a dislocation density tensor and derive its kinematic evolution
law from a phase field description of crystal deformations in three dimensions.
The phase field crystal (PFC) model is used to define the lattice distortion,
including topological singularities, and the associated configurational
stresses. We derive an exact expression for the velocity of dislocation line
determined by the phase field evolution, and show that dislocation motion in
the PFC is driven by a Peach-Koehler force. As is well known from earlier PFC
model studies, the configurational stress is not divergence free for a general
field configuration. Therefore, we also present a method (PFCMEq) to constrain
the diffusive dynamics to mechanical equilibrium by adding an independent and
integrable distortion so that the total resulting stress is divergence free. In
the PFCMEq model, the far-field stress agrees very well with the predictions
from continuum elasticity, while the near-field stress around the dislocation
core is regularized by the smooth nature of the phase-field. We apply this
framework to study the rate of shrinkage of an dislocation loop seeded in its
glide plane.
