  Kinetically constrained models (KCMs) have been used to study and understand
the origin of glassy dynamics. Despite having trivial thermodynamic properties,
their dynamics slows down dramatically at low temperatures while displaying
dynamical heterogeneity as seen in glass forming supercooled liquids. This
dynamics has its origin in an ergodic-nonergodic first-order phase transition
between phases of distinct dynamical "activity". This is a "space-time"
transition as it corresponds to a singular change in ensembles of trajectories
of the dynamics rather than ensembles of configurations. Here we extend these
ideas to driven glassy systems by considering KCMs driven into non-equilibrium
steady states through non-conservative forces. By classifying trajectories
through their entropy production we prove that driven KCMs also display an
analogous first-order space-time transition between dynamical phases of finite
and vanishing entropy production. We also discuss how trajectories with rare
values of entropy production can be realized as typical trajectories of a
mapped system with modified forces.
