  A field theory approach for the nonequilibrium relaxation dynamics in open
systems at late times is developed. In the absence of conservation laws, all
excitations are subject to dissipation. Nevertheless, ordered stationary states
satisfy Goldstone's theorem. It implies a vanishing damping rate at small
momenta, which in turn allows for competition between environment-induced
dissipation and thermalization due to collisions. We derive the dynamic theory
in the symmetry-broken phase of an $O(N)$-symmetric field theory based on an
expansion of the two-particle irreducible (2PI) effective action to
next-to-leading order in $1/N$ and highlight the analogies and differences to
the corresponding theory for closed systems. A central result of this approach
is the systematic derivation of an \emph{open-system Boltzmann equation}, which
takes a very different form from its closed-system counterpart due to the
absence of well-defined quasiparticles. As a consequence of the general
structure of its derivation, it applies to open, gapless field theories that
satisfy certain testable conditions, which we identify here. Specifically for
the $O(N)$ model, we use scaling analysis and numerical simulations to show
that interactions are screened efficiently at small momenta and, therefore, the
late-time evolution is effectively collisionless. This implies that
fluctuations induced by a quench dissipate into the environment before they
thermalize. Goldstone's theorem also constrains the dynamics far from
equilibrium, which is used to show that the order parameter equilibrates more
quickly for quenches preserving the $O(N)$ symmetry than those breaking it
explicitly.
