  We present the theory of spontaneous symmetry breaking (SSB) of discrete time
translations as recently realized in the space-time crystals of an atomic
Bose-Einstein condensate. The non-equilibrium physics related to such a
driven-dissipative system is discussed in both the Langevin as well as the
Fokker-Planck formulation. We consider a semi-classical and a fully quantum
approach, depending on the dissipation being either frequency independent or
linearly dependent on frequency, respectively. For both cases, the Langevin
equation and Fokker-Planck equation are derived, and the resulting equilibrium
distribution is studied. We also study the time evolution of the space-time
crystal and focus in particular on its formation and the associated dynamics of
the spontaneous breaking of a Z2 symmetry out of the symmetry unbroken phase,
i.e., the equilibrium Bose-Einstein condensate before the periodic drive is
turned on. Finally, we compare our results with experiments and conclude that
our theory provides a solid foundation for the observations.
