  We show how coupling an ensemble of bistable systems to a common cavity field
affects the collective stochastic behavior of this ensemble. In particular, the
cavity provides an effective interaction between the systems, and
parametrically modifies the transition rates between the metastable states. We
predict that the cavity induces a collective phase transition at a critical
temperature which depends linearly on the number of systems. It shows up as a
spontaneous symmetry breaking where the stationary states of the bistable
system bifurcate. We observe that the transition rates slow down independently
of the phase transition, but the rate modification vanishes for alternating
signs of the system-cavity couplings, corresponding to a disordered ensemble of
dipoles. Our results are of particular relevance in polaritonic chemistry where
the presence of a cavity has been suggested to affect chemical reactions.
