  Relaxation dynamics of complex quantum systems with strong interactions
towards the steady state is a fundamental problem in statistical mechanics. The
steady state of subsystems weakly interacting with their environment is
described by the canonical ensemble which assumes the probability distribution
for energy to be of the Boltzmann form. The emergence of this probability
distribution is ensured by the detailed balance of the transitions induced by
the interaction with the environment. Here we consider relaxation of an open
correlated quantum system brought into contact with a reservoir in the vacuum
state. We refer to such a system as emissive since particles irreversibly
evaporate into the vacuum. The steady state of the system is a statistical
mixture of the stable eigenstates arising due to the binding energy. We found
that, despite the absence of the detailed balance, the stationary probability
distribution over these eigenstates is of the Boltzmann form in each
$N$-particle sector. A quantum statistical ensemble corresponding to the steady
state is characterized by different temperatures in the different sectors, in a
contrast to the Gibbs ensemble. We investigate the transition rates between the
eigenstates to understand the emergence of the Boltzmann distribution and find
their exponential dependence on the transition energy. We argue that this
property of transition rates is generic for a wide class of emissive quantum
many-body systems.
