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

Field theory for the dynamics of the open $O(N)$ model