| Summary: | Weakly pumped systems with approximate conservation laws can be efficiently
described by a generalized Gibbs ensemble if the steady state of the system is
unique. However, such a description can fail if there are multiple steady state
solutions, for example, a bistability. In this case domains and domain walls
may form. In one-dimensional (1D) systems any type of noise (thermal or
non-thermal) will in general lead to a proliferation of such domains. We study
this physics in a 1D spin chain with two approximate conservation laws, energy
and the $z$-component of the total magnetization. A bistability in the
magnetization is induced by the coupling to suitably chosen Lindblad operators.
We analyze the theory for a weak coupling strength $\epsilon$ to the
non-equilibrium bath. In this limit, we argue that one can use hydrodynamic
approximations which describe the system locally in terms of space- and
time-dependent Lagrange parameters. Here noise terms enforce the creation of
domains, where the typical width of a domain wall goes as $\sim
1/\sqrt{\epsilon}$ while the density of domain walls is exponentially small in
$1/\sqrt{\epsilon}$. This is shown by numerical simulations of a simplified
hydrodynamic equation in the presence of noise.
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