  One of the outstanding problems in non-equilibrium physics is to precisely
understand when and how physically relevant observables in many-body systems
equilibrate under unitary time evolution. General equilibration results show
that equilibration is generic provided that the initial state has overlap with
sufficiently many energy levels. But results not referring to typicality which
show that natural initial states actually fulfill this condition are lacking.
In this work, we present stringent results for equilibration for systems in
which Renyi entanglement entropies in energy eigenstates with finite energy
density are extensive for at least some, not necessarily connected, sub-system.
Our results reverse the logic of common arguments, in that we derive
equilibration from a weak condition akin to the eigenstate thermalization
hypothesis, which is usually attributed to thermalization in systems that are
assumed to equilibrate in the first place. We put the findings into the context
of studies of many-body localization and many-body scars.
