  A cornerstone of the theory of phase transitions is the observation that
many-body systems exhibiting a spontaneous symmetry breaking in the
thermodynamic limit generally show extensive fluctuations of an order parameter
in large but finite systems. In this work, we introduce the dynamical analogue
of such a theory. Specifically, we consider local dissipative dynamics
preparing a steady-state of quantum spins on a lattice exhibiting a discrete or
continuous symmetry but with extensive fluctuations in a local order parameter.
We show that for all such processes satisfying detailed balance, there exist
metastable symmetry-breaking states, i.e., states that become stationary in the
thermodynamic limit and give a finite value to the order parameter. We give
results both for discrete and continuous symmetries and explicitly show how to
construct the symmetry-breaking states. Our results show in a simple way that,
in large systems, local dissipative dynamics satisfying detailed balance cannot
uniquely and efficiently prepare states with extensive fluctuations with
respect to local operators. We discuss the implications of our results for
quantum simulators and dissipative state preparation.
